Research ArticleA Simple Empirical Likelihood Ratio Test for Normality Basedon the Moment Constraints of a Half-Normal Distribution

C S Marange and Y Qin

Department of Statistics Faculty of Science and Agriculture Fort Hare University East London Campus 5200 South Africa

Correspondence should be addressed to C S Marange cmarangeufhacza

Received 12 May 2018 Revised 5 July 2018 Accepted 26 July 2018 Published 12 September 2018

Academic Editor Elio Chiodo

Copyright copy 2018 C S Marange and Y Qin This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

A simple and efficient empirical likelihood ratio (ELR) test for normality based on moment constraints of the half-normaldistribution was developedThe proposed test can also be easily modified to test for departures from half-normality and is relativelysimple to implement in various statistical packages with no ordering of observations required Using Monte Carlo simulationsour test proved to be superior to other well-known existing goodness-of-fit (GoF) tests considered under symmetric alternativedistributions for small to moderate sample sizes A real data example revealed the robustness and applicability of the proposed testas well as its superiority in power over other common existing tests studied

1 Introduction

Testing for distributional assumptions for normality is ofparamount importance in applied statistical modelling Sev-eral well-known numerical tests for normality are widelyused by investigators to supplement the graphical techniquesin assessing departures from normality Amongst othersthese tests include the Kolmogorov-Smirnov (KS) test [1] theLilliefors (LL) test [2] the Anderson-Darling (AD) test [3 4]the Shapiro-Wilks (SW) test [5] the Jarque-Bera (JB) test [6]and the DAgostino and Pearson (DP) test [7] These testsdiffer on certain characteristics of the normal distributionon which they focus That is some focus on the empiricaldistribution function (EDF) some are moment based andsome are based on regression as well as correlation Of thesetests some use normalized sample data whilst some useobserved values However though these tests are commonlyused in practice they do have major drawbacks For examplesome of these tests require complete specification of the nulldistribution some require computation of critical values tobe done for each specified null distribution and some requireordering of the sample data when computing the test statisticGenerally most of these tests are not supported when certaincombinations of parameters of a specified distribution areestimated

Of these the most well-known goodness-of-fit (GoF)test is the SW test but it was originally restricted to smallsample sizes (ie 119899 le 50) Several modifications have beenproposed by several researchers These include Royston [8]who suggested a normalized transformation for the test inorder to resolve the limitations on the sample size Shapiroand Francia [9] who also modified the test so that it canbe ideal for large sample sizes Chen and Shapiro [10] whoproposed normalized spacings for an alternative test of theSW test and Rahman and Govindarajulu [11] who definednew weights for the SW test statistic However the majordrawback of the SW test is computation time in dealing withlarge samples when computing the covariance matrix thatcorresponds to order statistics of the vector of weights andthe standard normal distribution

However we also have GoF tests that are based onmoment constraints such as the skewness and kurtosiscoefficients and these are well known to be efficient toolsfor evaluating normality These moment based tests includethe skewness test the kurtosis test the DP test and theJB test These tests combine moment constraints to checkfor deviations from normality They are often referred to asomnibus tests because of their ability to detect departuresfrom normality whilst not depending upon the parameters ofthe normal distribution The adoption of the use of moment

HindawiJournal of Probability and StatisticsVolume 2018 Article ID 8094146 10 pageshttpsdoiorg10115520188094146

2 Journal of Probability and Statistics

based tests coupled with the empirical likelihood method-ology has recently attracted the attention of researchersin developing GoF tests for normality [12 13] Dong andGiles [12] proposed an empirical likelihood ratio (ELR) testutilizing the empirical likelihood (EL) methodology of Owen[14] They monitored the first four moment conditions ofthe normal distribution and their test outperformed alternatecommon existing tests studied against several alternativedistributions Our study followed from the works of Shanet al [13] who proposed a simple ELR test for normalitybased on moment constraints using a standardized normalvariable Their test proved to be more powerful than otherwell-known GoF tests on small to moderate sample sizes forseveral alternative distributions In this study we adoptedtheir approach and focused on the construction of a simpleELR test for normality using the moment constraints ofthe half-normal distribution The next section will outlinethe development of our proposed test followed by MonteCarlo simulations A real data example will be presentedDiscussions and conclusion of the findings aswell as potentialareas of future research will be highlighted

2 ELR Test Development

Let us assume we have independent and identically dis-tributed (119894119894119889) nonordered random variables1198831 1198832 119883119899The intention being to assess whether the observed data isnormally distributed Thus we intend testing the followingnull hypothesis

1198670 119883 sim 119873(120583 1205902) (1)

where 120583 and 1205902 are considered to be unknown parametersWe proposed using the standardized random variables of thenormal distribution by using the following transformations

119885⋆119894 = 119883119894 minus 120583119878119863 119894 = 1 2 119899 (2)

where 120583 = 119883 = (1119899)sum119899119894=1119883119894 and 119878119863 is the standarddeviation to be estimated by an unbiased quantity 1199042 =119878(119899minus1) One can also decide to use themaximum likelihoodestimate (MLE) 2 = 119878119899 where 119878 = sum119899119894=1(119883119894 minus 119883)2 and119883 = (1119899)sum119899119894=1119883119894 Both quantities 1199042 and 2 are known toconverge to1205902 as 119899 approachesinfinWe also used an alternativetransformation following Lin and Mudholkarrsquos [17] workwhich also eliminates the dependency that exists between 120583and 120590 on the data distributionThus we also transformed ourobservations using

119885lowast119894 = radic119899 (119899 minus 1) (119883119894 minus 119883)119878119863minus119894 119894 = 1 2 119899 (3)

where 119883 = (1119899)sum119899119895=1119883119895 1198781198632minus119894 = (1(119899 minus 2))sum119899119895=1119895 =119894(119883119895 minus119883minus119894)2 and 119883minus119894 = (1(119899 minus 1))sum119899119895=1119895 =119894119883119895 As 119899 gets large thestandardized data points 1198851 1198852 119885119899 become asymptoti-cally independent If 119883 sim 119873(0 1205902) then the absolute value|119883| sim 119867119873(120583 1205902) It also follows that if119883 sim 119873(120583 1205902) then the

modulus of the standardized normal random variables 119885⋆and 119885lowast follows a standardized half-normal random variablewith mean = radic2120587 and variance = 1 The standardized formof the half-normal distribution is also known as the 1205942-distribution with ] = 1 The standardized half-normalrandom variable has a PDF that is given by

119891119885 (119911) = 2radic2120587119890minus(12)119911

2

for 119911 ge 00 for 119911 lt 0 (4)

and we denote it as 119885 sim 119867119873(120583 1205902) Following Prudnikovet al [18] the 119896th moment of the standardized half-normalvariable for some integer 119896 gt 0 is as outlined in theproposition below

Proposition 1 Let 119885 sim 119867119873(radic2120587 1) for k = 1 2 n andthen the 119896119905ℎ moments are given by

119864 (119885119896) = 120583119896 = 1radic12058721198962Γ(119896 + 12 ) (5)

where Γ(sdot) denotes the gamma function

We then derived the first four moments using the func-tion given in (5) These moments are easily obtained asfollows

Corollary 2 Let 119885 sim 119867119873(120583 1205902) The first two moments of 119885that is 120583 and 120590 are given by

119864 (119885) = 120583 = radic 2120587Γ (1) = radic 2120587 asymp 07979 (6)

var (119885) = 2radic120587Γ(32) = 1 (7)

Corollary 3 Let 119885 sim 119867119873(120583 1205902) The skewness and kurtosiscoefficients of 119885 are given by

120574 (1198851) = 119864 (1198853) = 1205833 = radic23120587 Γ (2) = 2radic 2120587 asymp 15958 (8)

120574 (1198852) = 119864 (1198854) = 1205834 = 4radic120587Γ(52) = 3 (9)

In this study we used the first four moment constraints ofthe standardized half-normal distribution

21 The ELR Based Test Statistic We used an empiricallikelihood ratio test (ELR) to construct our test statisticOur aim was to compare the GoF test under 1198670 against thealternative (119867119886) In order to achieve this we constructed ourtest statistic as follows Let us consider 119899 nonordered observa-tions1198831 1198832 119883119899 that are independent and identically dis-tributed and assumed to have unknown 120583 and 120590 The inten-tion is to perform aGoF test for the distributional assumptionthat1198831 1198832 119883119899 are consistent with a normal distributionNow consider that the random variables 1198851 1198852 119885119899 areabsolute standardized normal variables from the random

Journal of Probability and Statistics 3

variables1198831 1198832 119883119899 Thus the transformedstandardizedobservations have a moment function given in Proposition 1above Following the EL methodology we assigned 119901119894 whichis a probability parameter to each transformed observation119885119894 and then formulated the EL function that is given by

119871 (119865) = 119899prod119894=1

119901119894 (10)

where 119901119894rsquos satisfy the fundamental properties of probabilitythat is 0 le 119901119894 le 1 and sum119899119894=1 119901119894 = 1 Probabilityparameters 119901119894rsquos will then be chosen subject to unbiasedmoment conditions and the EL method will utilize these119901119894rsquos in order to maximize the EL function Following thisEL technique 119864(119885119896) has sample moments sum119899119894=1 119901119894119885119896119894 and theprobability parameters (119901119894rsquos) are elements of the EL functionUnder 1198670 the four unbiased empirical moment equationshave the form

119899sum119894=1

119901119894119885119896119894 minus 120583119896 = 0 119896 = 1 2 119899 (11)

The composite hypotheses for the ELR test are given by

1198670 1199111015840119894 119904 sim 119867119873(120583 1205902)vs 119867119886 1199111015840119894 119904 ≁ 119867119873(120583 1205902) (12)

Alternatively considering the above unbiased empiricalmoment equations the hypotheses for the ELR test can bewritten as

1198670 119864 (119885119896) = 120583119896vs 119867119886 119864 (119885119896) = 120583119896

(13)

The nonparametric empirical likelihood function corre-sponding to the given hypotheses has the form

119871 (119865) = 119871 (1198851 1198852 119885119899 | 120583119896) = 119899prod119894=1

119901119894 (14)

where the unknown probability parameters and 119901119894rsquos areattained under 1198670 and 119867119886 Under 1198670 the EL function ismaximized with respect to the 119901119894rsquos subject to two constraints

119899sum119894=1

119901119894 = 1119899sum119894=1

119901119894119885119896119894 = 120583119896(15)

Following this the weights of 119901119894rsquos are identified as

1199011 1199012 119901119899 = sup119899prod119894=1

119886119894 | 119899sum119894=1

119886119894 = 1119899sum119894=1

119886119894119885119896119894 = 120583119896(16)

where 0 le 119886119895 le 1 for 119895 = 1 2 119899 If we then use theLagrangian multipliers technique it can be shown that themaximum EL function under 1198670 can be expressed by thegiven form

119871 (1198651198670) = 119871 (1198851 1198852 119885119899 | 120583119896)= 119899prod119894=1

1119899 (1 + 120582119896 (119885119896119894 minus 120583119896))

(17)

where 120582119896 is a root of119899sum119894=1

(119885119896119894 minus 120583119896)1 + 120582119896 (119885119896119894 minus 120583119896) = 0 (18)

Under the alternative hypothesis sum119899119894=1 119901119894119885119896119894 = 120583119896 is notrequired to identify the weights 119901119894 in order to maximizethe EL function but only sum119899119894=1 119901119894 = 1 Thus under 119867119886 thenonparametric EL function is given by

119871 (119865119867119886) = 119871 (1198851 1198852 119885119899) = 119899prod119894=1

(1119899) = (1119899)119899 (19)

Now let us consider (minus2119871119871119877)119896 to be -2 log likelihood teststatistic for the hypotheses 1198670 119864(119885119896) = 120583119896 vs 119867119886 119864(119885119896) = 120583119896 It should be noted that under 1198670 minustwo times the log ELR has an asmymptotic 1205942 limitingdistribution [19] Thus considering the null and alternativehypotheses the above test statistic will simply be transformedto

(minus2119871119871119877)119896 = minus2 log 119871 (1198651198670)119871 (119865119867119886)= minus2 log 119871 (1198851 1198852 119885119899 | 120583119896)119871 (1198851 1198852 119885119899)

(20)

With simple substitution the above can be simplified to

(minus2119871119871119877)119896 = minus2 log prod119899119894=1 (1119899 (1 + 120582119896 (119885119896119894 minus 120583119896)))prod119899119894=1 (1119899)= 2 119899sum119894=1

log [1 + 120582119896 (119885119896119894 minus 120583119896)] (21)

We used the likelihood ratio to compare to size adjustedcritical values in order to decide whether or not to reject1198670We then proposed to reject the null hypothesis if

ELR119885 = max119896isin119866

(minus2119871119871119877)119896 gt 119862120572 (22)

where119862120572 is the test threshold and is 100(1minus120572)percentile ofthe1205942(1) distributionwhilst119866 are integer values representingthe set ofmoment constraints thatmaximizes the test statisticAs recommended by Dong and Giles [12] we used the firstfour moment constraints that is we set 119866 = 1 2 3 4 Inthis study we used the abbreviation ELR1198851 to refer to the

4 Journal of Probability and Statistics

first test where we transformed data using (2) and we usedthe abbreviation ELR1198852 to refer to the second alternativetest where we transformed data using (3) Our test statisticELR119885 = max119896isin119866(minus2119871119871119877)119896 is a CUSUM-type statistic asclassified by Vexler and Wu [20] In their article Vexler andWu [20] stated that based on the change point literatureanother commonalternative is to utilize the Shiryaev-Roberts(SR) statistic in replacement of the CUSUM-type statistic(see for example [21 22]) In our case the classical SRstatistic was of the form sum119896isin119866 exp(minus2119871119871119877)119896 Vexler Liu andPollak [23] showed that the classical SR statistic and thesimple CUSUM-type statistic have almost equivalent optimalstatistical properties due to their common null-martingalebasis Moreover the classical SR statistic is adapted from theCUSUM-type statistic

Shan et al [13] usedMonte Carlo experiments to comparethe CUSUM-type statistic for their ELR test for normalitywith an equivalent classical SR statistic and based on therelative simplicity of the CUSUM-type statistic as well asits power properties the authors opted to use the CUSUM-type statistic for their study We conducted a numericalexperiment to compare power for the CUSUM-type andSR statistic for our proposed test statistics with increasedmoment constraints and based on the same reasons given byShan et al [13] we decided to use the CUSUM-type statisticfor our Monte Carlo comparisons Also from the resultsELR1198852 outperformed ELR1198851 hence ELR1198852 was our preferredtest For all further comparisons ELR1198851 was excluded inthis study Findings for this Monte Carlo experiment arepresented in Table 4 However it should be noted from thesefindings that ELR1198851 has the potential to be superior to ELR1198852under certain alternatives Further investigations to uncoverthe alternatives in which ELR1198851 is superior to ELR1198852 are apotential area of future research which will not be furtheraddressed in this study The next section will outline theMonte Carlo simulation procedures using the R statisticalpackage

3 Monte Carlo Simulation Study

We used the R statistical package to implement our MonteCarlo simulation procedures in power comparisons as well asassessment of our preferred proposed test (ELR1198852) It shouldbe noted that other standard statistical packages can easilybe used to implement our proposed tests In order for us toconduct any assessments and evaluations of the proposed testfirstly we had to determine the size adjusted critical values

31 Size Adjusted Critical Values Since the proposed ELR testis an asymptotic test we therefore computed the unknownactual sizes for finite samples using Monte Carlo simulationswith 50000 replications Motivated by practical applicationswe considered critical values for relatively small sample sizesie 10 le 119899 le 200 because most applied statistical sciencesdatasets fall within this range The actual rejection rate fora given sample size (119899) is considered to be the total numberof the rejections divided by the total number of replicationsData was simulated from a standard normal distributionThestored ordered test statistics were then used to determine

the percentiles of the empirical distribution This makes itpossible to obtain the 30 25 20 1 size adjustedcritical values

32 ELR Test Assessment The power of the proposed test(ELR1198852) was compared to that of common existing GoF teststhat include the Anderson-Darling (AD) test [3 4] test themodified Kolmogorov-Smirnov (KS) test [2] the Cramer-von Mises (CVM) test [24ndash26] the Jarque-Bera (JB) test [6]the Shapiro-Wilk (SW) test [5] the density based empiricallikelihood ratio based (DB) test [16] and the simple and exactempirical likelihood test based onmoment relations (SEELR)[13] at the 5 significance level Power simulations were doneusing 5000 replications for all tests with varying samplesizes (119899 = 20 30 50 and 80) against different alternativedistributions We adopted alternative distributions used byShan et al [13] which covers a wide range of both symmetricand asymmetric applied distributions To assess robustnessand applicability of our proposed test (ELR1198852) we conducteda bootstrap study using some real data

4 Results of the Monte Carlo Simulations

This section presents the findings of the power comparisonsfor the different categories of the alternative distributionsconsidered The results of the power comparisons are pre-sented in Tables 5ndash8 Under symmetric cases defined on(minusinfininfin) our new test ELR1198852 outperformed all other studiedtests against the considered alternative distributions butslightly inferior to the JB test For symmetric distributionsdefined on (0 1) our proposed test (ELR1198852) was comparableto the DB test and significantly outperformed other alternatetests studied However when the alternative is Beta (05 05)the ELR1198852 test is comparable to the SW and SEELR testswhilst only outperforming the KS test the CVM test and theJB test

As for asymmetric distributions defined on (0infin) theSW and SEELR are the most powerful tests and should bethe preferred tests under these cases The AD and DB testsare comparable and they performed better than the proposedtest as well as the KS and CVM tests Lastly in the category ofasymmetric alternative distributions defined on (minusinfininfin) theELR1198852 test was comparable to the SEELR test at low samplesizes (ie 119899 = 20 30) for the non-central 119905-distributionsThe SW test outperformed all the tests considered in thisstudy under these asymmetric alternative distributions Forthe ELR based tests only the SEELR test was comparable tothe common existing tests studied that is the AD test theKS119872 test the CVM test and the JB test

Overall when considering all the normality tests withrespect to all of the alternative distributions considered it canbe seen that the JB the ELR1198852 and the SW tests are generallythe most powerful tests given symmetric alternatives definedon (minusinfininfin) whilst the DB and the ELR1198852 tests are the mostpowerful tests for symmetric alternatives defined on (0 1)On the other hand the SEELR and the SW tests are the mostpowerful tests for asymmetric alternatives defined on (0infin)whereas the JB and SW tests are the most powerful tests forasymmetric alternatives defined on (minusinfininfin)

Journal of Probability and Statistics 5

Table 1 Comparisons of computational times (in seconds) for the studied tests

Test Replications Elapsed Relative Userself SysselfAD 5000 114 2000 114 000CVM 5000 082 1439 081 000DB 5000 1700 29825 1694 004ELR1198852 5000 4483 78649 4478 001JB 5000 25264 443228 25260 000KS119872 5000 089 1561 090 000SEELR 5000 4542 79684 4541 000SW 5000 057 1000 058 000

Table 2 The baby boom data

Times between births (in min)59 14 37 62 68 2 15 9 157 27 37 2 55 86 144 40 36 47 9 61 1 26 13 28 77 26 45 25 1829 15 38 2 2 19 27 14 13 19 54 70 28Note Data appeared in the newspaper the Sunday Mail on December 21 1997 [15]

It was of paramount importance for us to determine thecomputational cost of the new algorithms by focusing on thecomputation time of the proposed test as compared to thatof the considered existing tests To assess this we used the Rbenchmark tool on a notebook installed with 64 BitWindows10 Home addition Equipped with a 4th generation Intel Corei5-4210U processor which has a speed of 17 GHz cache andmemory (RAM) of 4GB PC3 DDR3L SDRAM we set oursimulations to 5000 for each test with sample size set at 119899 =80 The results (see Table 1) show only a clear advantage ofour proposed approach to that of the widely known JB testAlso from the results our proposed methods are comparableto the SEELR test but inferior to the DB test The SW CVMKS and AD tests are computationally more efficient in termsof time than the rest of the studied tests

5 A Real Data Example

In this example we used baby boom data from an observa-tional study with records of forty-four (44) babies born ata 24-hour hospital in Brisbane Australia We opted for thisdataset because it can be used to demonstrate applicabilityof various statistical procedures to some common applieddistributions which include the normal (by modelling thebirth weights) the binomial (inferences in the number ofboysgirls born) the geometric (by considering the numberof births until a boygirl is born) the Poisson (births perhour for each hour) and the exponential (inference ontimes between births) Recently Miecznikowski et al [27]used the baby boom dataset in a resampling study on theapplication of their ELR based goodness-of-fit test For moreinformation regarding this dataset one can refer toDunn [28]For our application we opted to make use of the exponentialdistribution thuswewere interested in inference on the timesbetween births Table 2 shows the times between births whichwere computed by taking the differences between successivetimes of birth after midnight of birth times

Histogram of BabyFrequency

05

1015

50 100 1500Baby

Figure 1 Histogram for times between births for baby boom data

The goal of this examplewas to carry out a bootstrap studyin assessing the robustness and applicability of our proposedtest (ELR1198852) on uniformly distributed data However thetimes between births are known to be consistent with theexponential distribution (see Figure 1) By assessing thehistogram one can easily see that the data resembles theexponential distribution revealing that the times betweenbirths are exponentially consistent We used the inverseexponential distribution to transform the times betweenbirths so that they can be uniformly distributed We thenused the density based empirical likelihood ratio based test(dbEmpLikeGOF) to check if the transformed baby boomdata are uniformly distributed The dbEmpLikeGOF testreturned a 119901 value of 06950 suggesting that the transformeddata are consistent with the uniform distribution

6 Journal of Probability and Statistics

Table 3 Bootstrapping using the inverse exponential transformed baby boom data

Bootstrap power comparisons1198670 data is normally distributedObservations removed AD KS119872 CVM JB SW DB SEELR ELR11988523 00000 00000 00000 00000 01012 06040 00000 061328 00164 00006 00012 00000 01182 04486 00146 0405413 00281 00073 00066 00011 01023 03488 00568 02714

Table 4 A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests (ELR1198851and ELR1198852) with increased moment constraints at 120572 = 005119899 ELR1198851 ELR1198852119896 = 1 2 3 4 119896 = 1 2 3 4 5 119896 = 1 2 3 4 119896 = 1 2 3 4 5

S-R C-t S-R C-t S-R C-t S-R C-t119905(2)

30 00416 00330 00020 00010 06980 06998 06166 0591250 05142 04112 01666 01356 08766 08774 08320 0803080 08732 08336 07476 07184 09718 09684 09544 09488

Cauchy(01)30 03262 03438 00000 00000 09560 09556 09248 0919250 09538 09344 07246 06754 09970 09974 09928 0990080 09996 09996 09964 09940 10000 10000 09998 09996

Uniform(01)30 07230 01958 07208 07206 05772 05986 06996 0700450 09458 05222 09532 09474 09032 09122 09434 0939880 09966 08462 09978 09980 09940 09956 09986 09976

Exp(1)30 00094 00304 00070 00068 04638 04818 03874 0377250 00836 08096 00022 00042 06274 06306 05628 0538080 03764 09972 02504 02346 07942 08070 07558 07506

119905(120575 = 1 ] = 2)30 00476 00136 00012 00028 07168 07230 06476 0628050 05204 04172 01932 01676 08904 08908 08450 0829480 08752 08610 07736 07700 09714 09766 09636 09558

SN(015)30 00514 00520 00486 00442 01394 01242 01048 0094450 00404 00362 00350 00352 01408 01432 01114 0090480 00358 00338 00272 00204 01592 01646 01158 01226Note Our proposed tests are maximized on 119896 isin 119866 where 119866 can take any integer to represent the moment constraints used to maximise the test statistics forspecified sample sizes at 5 level of significance using 5000 simulations 119899 is the sample size Bold represents the powerful test statistic for the given simulationscenarios

For the resampling study we performed a power simu-lation study by randomly removing 3 8 and 13 observationsfrom the transformed baby boomdata at 5 significance levelusing 20000 replications for each simulation For compari-sonrsquos sakewe considered theAD test themodifiedKS test theCVM test the JB test the SW test theDB test the SEELR testand our proposed test (ELR1198852) The Monte Carlo bootstrapsimulation results are presented in Table 3 It is undeniablyclear that our test outperformed all the common existingtests and therefore suggests its robustness and applicabilityon real data It should be noted that we opted for uniformlydistributed data for our application since our proposedtest (ELR1198852) proved to be more powerful for symmetricalternative distributions which are defined on (0 1)

6 Conclusion

An empirical likelihood ratio test for normality based onmoment constraints of the half-normal distribution has beendeveloped Overall the proposed ELR test has good powerproperties and significantly outperformed the consideredwell-known common existing tests against the studied alter-native symmetric distributions In our case the attractivepower properties of the proposed ELR test resulted fromthe EL method being able to integrate most of the availableinformation by utilizing the first four moment constraintsand also through the utilization of the EL function whichleads to additional power benefits We advocate for ourproposed test (ELR1198852) to be the preferred choice when one

Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

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Journal of Probability and Statistics 3

variables1198831 1198832 119883119899 Thus the transformedstandardizedobservations have a moment function given in Proposition 1above Following the EL methodology we assigned 119901119894 whichis a probability parameter to each transformed observation119885119894 and then formulated the EL function that is given by

119871 (119865) = 119899prod119894=1

119901119894 (10)

where 119901119894rsquos satisfy the fundamental properties of probabilitythat is 0 le 119901119894 le 1 and sum119899119894=1 119901119894 = 1 Probabilityparameters 119901119894rsquos will then be chosen subject to unbiasedmoment conditions and the EL method will utilize these119901119894rsquos in order to maximize the EL function Following thisEL technique 119864(119885119896) has sample moments sum119899119894=1 119901119894119885119896119894 and theprobability parameters (119901119894rsquos) are elements of the EL functionUnder 1198670 the four unbiased empirical moment equationshave the form

119899sum119894=1

119901119894119885119896119894 minus 120583119896 = 0 119896 = 1 2 119899 (11)

The composite hypotheses for the ELR test are given by

1198670 1199111015840119894 119904 sim 119867119873(120583 1205902)vs 119867119886 1199111015840119894 119904 ≁ 119867119873(120583 1205902) (12)

Alternatively considering the above unbiased empiricalmoment equations the hypotheses for the ELR test can bewritten as

1198670 119864 (119885119896) = 120583119896vs 119867119886 119864 (119885119896) = 120583119896

(13)

The nonparametric empirical likelihood function corre-sponding to the given hypotheses has the form

119871 (119865) = 119871 (1198851 1198852 119885119899 | 120583119896) = 119899prod119894=1

119901119894 (14)

where the unknown probability parameters and 119901119894rsquos areattained under 1198670 and 119867119886 Under 1198670 the EL function ismaximized with respect to the 119901119894rsquos subject to two constraints

119899sum119894=1

119901119894 = 1119899sum119894=1

119901119894119885119896119894 = 120583119896(15)

Following this the weights of 119901119894rsquos are identified as

1199011 1199012 119901119899 = sup119899prod119894=1

119886119894 | 119899sum119894=1

119886119894 = 1119899sum119894=1

119886119894119885119896119894 = 120583119896(16)

where 0 le 119886119895 le 1 for 119895 = 1 2 119899 If we then use theLagrangian multipliers technique it can be shown that themaximum EL function under 1198670 can be expressed by thegiven form

119871 (1198651198670) = 119871 (1198851 1198852 119885119899 | 120583119896)= 119899prod119894=1

1119899 (1 + 120582119896 (119885119896119894 minus 120583119896))

(17)

where 120582119896 is a root of119899sum119894=1

(119885119896119894 minus 120583119896)1 + 120582119896 (119885119896119894 minus 120583119896) = 0 (18)

Under the alternative hypothesis sum119899119894=1 119901119894119885119896119894 = 120583119896 is notrequired to identify the weights 119901119894 in order to maximizethe EL function but only sum119899119894=1 119901119894 = 1 Thus under 119867119886 thenonparametric EL function is given by

119871 (119865119867119886) = 119871 (1198851 1198852 119885119899) = 119899prod119894=1

(1119899) = (1119899)119899 (19)

Now let us consider (minus2119871119871119877)119896 to be -2 log likelihood teststatistic for the hypotheses 1198670 119864(119885119896) = 120583119896 vs 119867119886 119864(119885119896) = 120583119896 It should be noted that under 1198670 minustwo times the log ELR has an asmymptotic 1205942 limitingdistribution [19] Thus considering the null and alternativehypotheses the above test statistic will simply be transformedto

(minus2119871119871119877)119896 = minus2 log 119871 (1198651198670)119871 (119865119867119886)= minus2 log 119871 (1198851 1198852 119885119899 | 120583119896)119871 (1198851 1198852 119885119899)

(20)

With simple substitution the above can be simplified to

(minus2119871119871119877)119896 = minus2 log prod119899119894=1 (1119899 (1 + 120582119896 (119885119896119894 minus 120583119896)))prod119899119894=1 (1119899)= 2 119899sum119894=1

log [1 + 120582119896 (119885119896119894 minus 120583119896)] (21)

We used the likelihood ratio to compare to size adjustedcritical values in order to decide whether or not to reject1198670We then proposed to reject the null hypothesis if

ELR119885 = max119896isin119866

(minus2119871119871119877)119896 gt 119862120572 (22)

where119862120572 is the test threshold and is 100(1minus120572)percentile ofthe1205942(1) distributionwhilst119866 are integer values representingthe set ofmoment constraints thatmaximizes the test statisticAs recommended by Dong and Giles [12] we used the firstfour moment constraints that is we set 119866 = 1 2 3 4 Inthis study we used the abbreviation ELR1198851 to refer to the

4 Journal of Probability and Statistics

first test where we transformed data using (2) and we usedthe abbreviation ELR1198852 to refer to the second alternativetest where we transformed data using (3) Our test statisticELR119885 = max119896isin119866(minus2119871119871119877)119896 is a CUSUM-type statistic asclassified by Vexler and Wu [20] In their article Vexler andWu [20] stated that based on the change point literatureanother commonalternative is to utilize the Shiryaev-Roberts(SR) statistic in replacement of the CUSUM-type statistic(see for example [21 22]) In our case the classical SRstatistic was of the form sum119896isin119866 exp(minus2119871119871119877)119896 Vexler Liu andPollak [23] showed that the classical SR statistic and thesimple CUSUM-type statistic have almost equivalent optimalstatistical properties due to their common null-martingalebasis Moreover the classical SR statistic is adapted from theCUSUM-type statistic

Shan et al [13] usedMonte Carlo experiments to comparethe CUSUM-type statistic for their ELR test for normalitywith an equivalent classical SR statistic and based on therelative simplicity of the CUSUM-type statistic as well asits power properties the authors opted to use the CUSUM-type statistic for their study We conducted a numericalexperiment to compare power for the CUSUM-type andSR statistic for our proposed test statistics with increasedmoment constraints and based on the same reasons given byShan et al [13] we decided to use the CUSUM-type statisticfor our Monte Carlo comparisons Also from the resultsELR1198852 outperformed ELR1198851 hence ELR1198852 was our preferredtest For all further comparisons ELR1198851 was excluded inthis study Findings for this Monte Carlo experiment arepresented in Table 4 However it should be noted from thesefindings that ELR1198851 has the potential to be superior to ELR1198852under certain alternatives Further investigations to uncoverthe alternatives in which ELR1198851 is superior to ELR1198852 are apotential area of future research which will not be furtheraddressed in this study The next section will outline theMonte Carlo simulation procedures using the R statisticalpackage

3 Monte Carlo Simulation Study

We used the R statistical package to implement our MonteCarlo simulation procedures in power comparisons as well asassessment of our preferred proposed test (ELR1198852) It shouldbe noted that other standard statistical packages can easilybe used to implement our proposed tests In order for us toconduct any assessments and evaluations of the proposed testfirstly we had to determine the size adjusted critical values

31 Size Adjusted Critical Values Since the proposed ELR testis an asymptotic test we therefore computed the unknownactual sizes for finite samples using Monte Carlo simulationswith 50000 replications Motivated by practical applicationswe considered critical values for relatively small sample sizesie 10 le 119899 le 200 because most applied statistical sciencesdatasets fall within this range The actual rejection rate fora given sample size (119899) is considered to be the total numberof the rejections divided by the total number of replicationsData was simulated from a standard normal distributionThestored ordered test statistics were then used to determine

the percentiles of the empirical distribution This makes itpossible to obtain the 30 25 20 1 size adjustedcritical values

32 ELR Test Assessment The power of the proposed test(ELR1198852) was compared to that of common existing GoF teststhat include the Anderson-Darling (AD) test [3 4] test themodified Kolmogorov-Smirnov (KS) test [2] the Cramer-von Mises (CVM) test [24ndash26] the Jarque-Bera (JB) test [6]the Shapiro-Wilk (SW) test [5] the density based empiricallikelihood ratio based (DB) test [16] and the simple and exactempirical likelihood test based onmoment relations (SEELR)[13] at the 5 significance level Power simulations were doneusing 5000 replications for all tests with varying samplesizes (119899 = 20 30 50 and 80) against different alternativedistributions We adopted alternative distributions used byShan et al [13] which covers a wide range of both symmetricand asymmetric applied distributions To assess robustnessand applicability of our proposed test (ELR1198852) we conducteda bootstrap study using some real data

4 Results of the Monte Carlo Simulations

This section presents the findings of the power comparisonsfor the different categories of the alternative distributionsconsidered The results of the power comparisons are pre-sented in Tables 5ndash8 Under symmetric cases defined on(minusinfininfin) our new test ELR1198852 outperformed all other studiedtests against the considered alternative distributions butslightly inferior to the JB test For symmetric distributionsdefined on (0 1) our proposed test (ELR1198852) was comparableto the DB test and significantly outperformed other alternatetests studied However when the alternative is Beta (05 05)the ELR1198852 test is comparable to the SW and SEELR testswhilst only outperforming the KS test the CVM test and theJB test

As for asymmetric distributions defined on (0infin) theSW and SEELR are the most powerful tests and should bethe preferred tests under these cases The AD and DB testsare comparable and they performed better than the proposedtest as well as the KS and CVM tests Lastly in the category ofasymmetric alternative distributions defined on (minusinfininfin) theELR1198852 test was comparable to the SEELR test at low samplesizes (ie 119899 = 20 30) for the non-central 119905-distributionsThe SW test outperformed all the tests considered in thisstudy under these asymmetric alternative distributions Forthe ELR based tests only the SEELR test was comparable tothe common existing tests studied that is the AD test theKS119872 test the CVM test and the JB test

Overall when considering all the normality tests withrespect to all of the alternative distributions considered it canbe seen that the JB the ELR1198852 and the SW tests are generallythe most powerful tests given symmetric alternatives definedon (minusinfininfin) whilst the DB and the ELR1198852 tests are the mostpowerful tests for symmetric alternatives defined on (0 1)On the other hand the SEELR and the SW tests are the mostpowerful tests for asymmetric alternatives defined on (0infin)whereas the JB and SW tests are the most powerful tests forasymmetric alternatives defined on (minusinfininfin)

Journal of Probability and Statistics 5

Table 1 Comparisons of computational times (in seconds) for the studied tests

Test Replications Elapsed Relative Userself SysselfAD 5000 114 2000 114 000CVM 5000 082 1439 081 000DB 5000 1700 29825 1694 004ELR1198852 5000 4483 78649 4478 001JB 5000 25264 443228 25260 000KS119872 5000 089 1561 090 000SEELR 5000 4542 79684 4541 000SW 5000 057 1000 058 000

Table 2 The baby boom data

Times between births (in min)59 14 37 62 68 2 15 9 157 27 37 2 55 86 144 40 36 47 9 61 1 26 13 28 77 26 45 25 1829 15 38 2 2 19 27 14 13 19 54 70 28Note Data appeared in the newspaper the Sunday Mail on December 21 1997 [15]

It was of paramount importance for us to determine thecomputational cost of the new algorithms by focusing on thecomputation time of the proposed test as compared to thatof the considered existing tests To assess this we used the Rbenchmark tool on a notebook installed with 64 BitWindows10 Home addition Equipped with a 4th generation Intel Corei5-4210U processor which has a speed of 17 GHz cache andmemory (RAM) of 4GB PC3 DDR3L SDRAM we set oursimulations to 5000 for each test with sample size set at 119899 =80 The results (see Table 1) show only a clear advantage ofour proposed approach to that of the widely known JB testAlso from the results our proposed methods are comparableto the SEELR test but inferior to the DB test The SW CVMKS and AD tests are computationally more efficient in termsof time than the rest of the studied tests

5 A Real Data Example

In this example we used baby boom data from an observa-tional study with records of forty-four (44) babies born ata 24-hour hospital in Brisbane Australia We opted for thisdataset because it can be used to demonstrate applicabilityof various statistical procedures to some common applieddistributions which include the normal (by modelling thebirth weights) the binomial (inferences in the number ofboysgirls born) the geometric (by considering the numberof births until a boygirl is born) the Poisson (births perhour for each hour) and the exponential (inference ontimes between births) Recently Miecznikowski et al [27]used the baby boom dataset in a resampling study on theapplication of their ELR based goodness-of-fit test For moreinformation regarding this dataset one can refer toDunn [28]For our application we opted to make use of the exponentialdistribution thuswewere interested in inference on the timesbetween births Table 2 shows the times between births whichwere computed by taking the differences between successivetimes of birth after midnight of birth times

Histogram of BabyFrequency

05

1015

50 100 1500Baby

Figure 1 Histogram for times between births for baby boom data

The goal of this examplewas to carry out a bootstrap studyin assessing the robustness and applicability of our proposedtest (ELR1198852) on uniformly distributed data However thetimes between births are known to be consistent with theexponential distribution (see Figure 1) By assessing thehistogram one can easily see that the data resembles theexponential distribution revealing that the times betweenbirths are exponentially consistent We used the inverseexponential distribution to transform the times betweenbirths so that they can be uniformly distributed We thenused the density based empirical likelihood ratio based test(dbEmpLikeGOF) to check if the transformed baby boomdata are uniformly distributed The dbEmpLikeGOF testreturned a 119901 value of 06950 suggesting that the transformeddata are consistent with the uniform distribution

6 Journal of Probability and Statistics

Table 3 Bootstrapping using the inverse exponential transformed baby boom data

Bootstrap power comparisons1198670 data is normally distributedObservations removed AD KS119872 CVM JB SW DB SEELR ELR11988523 00000 00000 00000 00000 01012 06040 00000 061328 00164 00006 00012 00000 01182 04486 00146 0405413 00281 00073 00066 00011 01023 03488 00568 02714

Table 4 A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests (ELR1198851and ELR1198852) with increased moment constraints at 120572 = 005119899 ELR1198851 ELR1198852119896 = 1 2 3 4 119896 = 1 2 3 4 5 119896 = 1 2 3 4 119896 = 1 2 3 4 5

S-R C-t S-R C-t S-R C-t S-R C-t119905(2)

30 00416 00330 00020 00010 06980 06998 06166 0591250 05142 04112 01666 01356 08766 08774 08320 0803080 08732 08336 07476 07184 09718 09684 09544 09488

Cauchy(01)30 03262 03438 00000 00000 09560 09556 09248 0919250 09538 09344 07246 06754 09970 09974 09928 0990080 09996 09996 09964 09940 10000 10000 09998 09996

Uniform(01)30 07230 01958 07208 07206 05772 05986 06996 0700450 09458 05222 09532 09474 09032 09122 09434 0939880 09966 08462 09978 09980 09940 09956 09986 09976

Exp(1)30 00094 00304 00070 00068 04638 04818 03874 0377250 00836 08096 00022 00042 06274 06306 05628 0538080 03764 09972 02504 02346 07942 08070 07558 07506

119905(120575 = 1 ] = 2)30 00476 00136 00012 00028 07168 07230 06476 0628050 05204 04172 01932 01676 08904 08908 08450 0829480 08752 08610 07736 07700 09714 09766 09636 09558

SN(015)30 00514 00520 00486 00442 01394 01242 01048 0094450 00404 00362 00350 00352 01408 01432 01114 0090480 00358 00338 00272 00204 01592 01646 01158 01226Note Our proposed tests are maximized on 119896 isin 119866 where 119866 can take any integer to represent the moment constraints used to maximise the test statistics forspecified sample sizes at 5 level of significance using 5000 simulations 119899 is the sample size Bold represents the powerful test statistic for the given simulationscenarios

For the resampling study we performed a power simu-lation study by randomly removing 3 8 and 13 observationsfrom the transformed baby boomdata at 5 significance levelusing 20000 replications for each simulation For compari-sonrsquos sakewe considered theAD test themodifiedKS test theCVM test the JB test the SW test theDB test the SEELR testand our proposed test (ELR1198852) The Monte Carlo bootstrapsimulation results are presented in Table 3 It is undeniablyclear that our test outperformed all the common existingtests and therefore suggests its robustness and applicabilityon real data It should be noted that we opted for uniformlydistributed data for our application since our proposedtest (ELR1198852) proved to be more powerful for symmetricalternative distributions which are defined on (0 1)

6 Conclusion

An empirical likelihood ratio test for normality based onmoment constraints of the half-normal distribution has beendeveloped Overall the proposed ELR test has good powerproperties and significantly outperformed the consideredwell-known common existing tests against the studied alter-native symmetric distributions In our case the attractivepower properties of the proposed ELR test resulted fromthe EL method being able to integrate most of the availableinformation by utilizing the first four moment constraintsand also through the utilization of the EL function whichleads to additional power benefits We advocate for ourproposed test (ELR1198852) to be the preferred choice when one

Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

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4 Journal of Probability and Statistics

first test where we transformed data using (2) and we usedthe abbreviation ELR1198852 to refer to the second alternativetest where we transformed data using (3) Our test statisticELR119885 = max119896isin119866(minus2119871119871119877)119896 is a CUSUM-type statistic asclassified by Vexler and Wu [20] In their article Vexler andWu [20] stated that based on the change point literatureanother commonalternative is to utilize the Shiryaev-Roberts(SR) statistic in replacement of the CUSUM-type statistic(see for example [21 22]) In our case the classical SRstatistic was of the form sum119896isin119866 exp(minus2119871119871119877)119896 Vexler Liu andPollak [23] showed that the classical SR statistic and thesimple CUSUM-type statistic have almost equivalent optimalstatistical properties due to their common null-martingalebasis Moreover the classical SR statistic is adapted from theCUSUM-type statistic

Shan et al [13] usedMonte Carlo experiments to comparethe CUSUM-type statistic for their ELR test for normalitywith an equivalent classical SR statistic and based on therelative simplicity of the CUSUM-type statistic as well asits power properties the authors opted to use the CUSUM-type statistic for their study We conducted a numericalexperiment to compare power for the CUSUM-type andSR statistic for our proposed test statistics with increasedmoment constraints and based on the same reasons given byShan et al [13] we decided to use the CUSUM-type statisticfor our Monte Carlo comparisons Also from the resultsELR1198852 outperformed ELR1198851 hence ELR1198852 was our preferredtest For all further comparisons ELR1198851 was excluded inthis study Findings for this Monte Carlo experiment arepresented in Table 4 However it should be noted from thesefindings that ELR1198851 has the potential to be superior to ELR1198852under certain alternatives Further investigations to uncoverthe alternatives in which ELR1198851 is superior to ELR1198852 are apotential area of future research which will not be furtheraddressed in this study The next section will outline theMonte Carlo simulation procedures using the R statisticalpackage

3 Monte Carlo Simulation Study

We used the R statistical package to implement our MonteCarlo simulation procedures in power comparisons as well asassessment of our preferred proposed test (ELR1198852) It shouldbe noted that other standard statistical packages can easilybe used to implement our proposed tests In order for us toconduct any assessments and evaluations of the proposed testfirstly we had to determine the size adjusted critical values

31 Size Adjusted Critical Values Since the proposed ELR testis an asymptotic test we therefore computed the unknownactual sizes for finite samples using Monte Carlo simulationswith 50000 replications Motivated by practical applicationswe considered critical values for relatively small sample sizesie 10 le 119899 le 200 because most applied statistical sciencesdatasets fall within this range The actual rejection rate fora given sample size (119899) is considered to be the total numberof the rejections divided by the total number of replicationsData was simulated from a standard normal distributionThestored ordered test statistics were then used to determine

the percentiles of the empirical distribution This makes itpossible to obtain the 30 25 20 1 size adjustedcritical values

32 ELR Test Assessment The power of the proposed test(ELR1198852) was compared to that of common existing GoF teststhat include the Anderson-Darling (AD) test [3 4] test themodified Kolmogorov-Smirnov (KS) test [2] the Cramer-von Mises (CVM) test [24ndash26] the Jarque-Bera (JB) test [6]the Shapiro-Wilk (SW) test [5] the density based empiricallikelihood ratio based (DB) test [16] and the simple and exactempirical likelihood test based onmoment relations (SEELR)[13] at the 5 significance level Power simulations were doneusing 5000 replications for all tests with varying samplesizes (119899 = 20 30 50 and 80) against different alternativedistributions We adopted alternative distributions used byShan et al [13] which covers a wide range of both symmetricand asymmetric applied distributions To assess robustnessand applicability of our proposed test (ELR1198852) we conducteda bootstrap study using some real data

4 Results of the Monte Carlo Simulations

This section presents the findings of the power comparisonsfor the different categories of the alternative distributionsconsidered The results of the power comparisons are pre-sented in Tables 5ndash8 Under symmetric cases defined on(minusinfininfin) our new test ELR1198852 outperformed all other studiedtests against the considered alternative distributions butslightly inferior to the JB test For symmetric distributionsdefined on (0 1) our proposed test (ELR1198852) was comparableto the DB test and significantly outperformed other alternatetests studied However when the alternative is Beta (05 05)the ELR1198852 test is comparable to the SW and SEELR testswhilst only outperforming the KS test the CVM test and theJB test

As for asymmetric distributions defined on (0infin) theSW and SEELR are the most powerful tests and should bethe preferred tests under these cases The AD and DB testsare comparable and they performed better than the proposedtest as well as the KS and CVM tests Lastly in the category ofasymmetric alternative distributions defined on (minusinfininfin) theELR1198852 test was comparable to the SEELR test at low samplesizes (ie 119899 = 20 30) for the non-central 119905-distributionsThe SW test outperformed all the tests considered in thisstudy under these asymmetric alternative distributions Forthe ELR based tests only the SEELR test was comparable tothe common existing tests studied that is the AD test theKS119872 test the CVM test and the JB test

Overall when considering all the normality tests withrespect to all of the alternative distributions considered it canbe seen that the JB the ELR1198852 and the SW tests are generallythe most powerful tests given symmetric alternatives definedon (minusinfininfin) whilst the DB and the ELR1198852 tests are the mostpowerful tests for symmetric alternatives defined on (0 1)On the other hand the SEELR and the SW tests are the mostpowerful tests for asymmetric alternatives defined on (0infin)whereas the JB and SW tests are the most powerful tests forasymmetric alternatives defined on (minusinfininfin)

Journal of Probability and Statistics 5

Table 1 Comparisons of computational times (in seconds) for the studied tests

Test Replications Elapsed Relative Userself SysselfAD 5000 114 2000 114 000CVM 5000 082 1439 081 000DB 5000 1700 29825 1694 004ELR1198852 5000 4483 78649 4478 001JB 5000 25264 443228 25260 000KS119872 5000 089 1561 090 000SEELR 5000 4542 79684 4541 000SW 5000 057 1000 058 000

Table 2 The baby boom data

Times between births (in min)59 14 37 62 68 2 15 9 157 27 37 2 55 86 144 40 36 47 9 61 1 26 13 28 77 26 45 25 1829 15 38 2 2 19 27 14 13 19 54 70 28Note Data appeared in the newspaper the Sunday Mail on December 21 1997 [15]

It was of paramount importance for us to determine thecomputational cost of the new algorithms by focusing on thecomputation time of the proposed test as compared to thatof the considered existing tests To assess this we used the Rbenchmark tool on a notebook installed with 64 BitWindows10 Home addition Equipped with a 4th generation Intel Corei5-4210U processor which has a speed of 17 GHz cache andmemory (RAM) of 4GB PC3 DDR3L SDRAM we set oursimulations to 5000 for each test with sample size set at 119899 =80 The results (see Table 1) show only a clear advantage ofour proposed approach to that of the widely known JB testAlso from the results our proposed methods are comparableto the SEELR test but inferior to the DB test The SW CVMKS and AD tests are computationally more efficient in termsof time than the rest of the studied tests

5 A Real Data Example

In this example we used baby boom data from an observa-tional study with records of forty-four (44) babies born ata 24-hour hospital in Brisbane Australia We opted for thisdataset because it can be used to demonstrate applicabilityof various statistical procedures to some common applieddistributions which include the normal (by modelling thebirth weights) the binomial (inferences in the number ofboysgirls born) the geometric (by considering the numberof births until a boygirl is born) the Poisson (births perhour for each hour) and the exponential (inference ontimes between births) Recently Miecznikowski et al [27]used the baby boom dataset in a resampling study on theapplication of their ELR based goodness-of-fit test For moreinformation regarding this dataset one can refer toDunn [28]For our application we opted to make use of the exponentialdistribution thuswewere interested in inference on the timesbetween births Table 2 shows the times between births whichwere computed by taking the differences between successivetimes of birth after midnight of birth times

Histogram of BabyFrequency

05

1015

50 100 1500Baby

Figure 1 Histogram for times between births for baby boom data

The goal of this examplewas to carry out a bootstrap studyin assessing the robustness and applicability of our proposedtest (ELR1198852) on uniformly distributed data However thetimes between births are known to be consistent with theexponential distribution (see Figure 1) By assessing thehistogram one can easily see that the data resembles theexponential distribution revealing that the times betweenbirths are exponentially consistent We used the inverseexponential distribution to transform the times betweenbirths so that they can be uniformly distributed We thenused the density based empirical likelihood ratio based test(dbEmpLikeGOF) to check if the transformed baby boomdata are uniformly distributed The dbEmpLikeGOF testreturned a 119901 value of 06950 suggesting that the transformeddata are consistent with the uniform distribution

6 Journal of Probability and Statistics

Table 3 Bootstrapping using the inverse exponential transformed baby boom data

Bootstrap power comparisons1198670 data is normally distributedObservations removed AD KS119872 CVM JB SW DB SEELR ELR11988523 00000 00000 00000 00000 01012 06040 00000 061328 00164 00006 00012 00000 01182 04486 00146 0405413 00281 00073 00066 00011 01023 03488 00568 02714

Table 4 A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests (ELR1198851and ELR1198852) with increased moment constraints at 120572 = 005119899 ELR1198851 ELR1198852119896 = 1 2 3 4 119896 = 1 2 3 4 5 119896 = 1 2 3 4 119896 = 1 2 3 4 5

S-R C-t S-R C-t S-R C-t S-R C-t119905(2)

30 00416 00330 00020 00010 06980 06998 06166 0591250 05142 04112 01666 01356 08766 08774 08320 0803080 08732 08336 07476 07184 09718 09684 09544 09488

Cauchy(01)30 03262 03438 00000 00000 09560 09556 09248 0919250 09538 09344 07246 06754 09970 09974 09928 0990080 09996 09996 09964 09940 10000 10000 09998 09996

Uniform(01)30 07230 01958 07208 07206 05772 05986 06996 0700450 09458 05222 09532 09474 09032 09122 09434 0939880 09966 08462 09978 09980 09940 09956 09986 09976

Exp(1)30 00094 00304 00070 00068 04638 04818 03874 0377250 00836 08096 00022 00042 06274 06306 05628 0538080 03764 09972 02504 02346 07942 08070 07558 07506

119905(120575 = 1 ] = 2)30 00476 00136 00012 00028 07168 07230 06476 0628050 05204 04172 01932 01676 08904 08908 08450 0829480 08752 08610 07736 07700 09714 09766 09636 09558

SN(015)30 00514 00520 00486 00442 01394 01242 01048 0094450 00404 00362 00350 00352 01408 01432 01114 0090480 00358 00338 00272 00204 01592 01646 01158 01226Note Our proposed tests are maximized on 119896 isin 119866 where 119866 can take any integer to represent the moment constraints used to maximise the test statistics forspecified sample sizes at 5 level of significance using 5000 simulations 119899 is the sample size Bold represents the powerful test statistic for the given simulationscenarios

For the resampling study we performed a power simu-lation study by randomly removing 3 8 and 13 observationsfrom the transformed baby boomdata at 5 significance levelusing 20000 replications for each simulation For compari-sonrsquos sakewe considered theAD test themodifiedKS test theCVM test the JB test the SW test theDB test the SEELR testand our proposed test (ELR1198852) The Monte Carlo bootstrapsimulation results are presented in Table 3 It is undeniablyclear that our test outperformed all the common existingtests and therefore suggests its robustness and applicabilityon real data It should be noted that we opted for uniformlydistributed data for our application since our proposedtest (ELR1198852) proved to be more powerful for symmetricalternative distributions which are defined on (0 1)

6 Conclusion

An empirical likelihood ratio test for normality based onmoment constraints of the half-normal distribution has beendeveloped Overall the proposed ELR test has good powerproperties and significantly outperformed the consideredwell-known common existing tests against the studied alter-native symmetric distributions In our case the attractivepower properties of the proposed ELR test resulted fromthe EL method being able to integrate most of the availableinformation by utilizing the first four moment constraintsand also through the utilization of the EL function whichleads to additional power benefits We advocate for ourproposed test (ELR1198852) to be the preferred choice when one

Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

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Journal of Probability and Statistics 5

Table 1 Comparisons of computational times (in seconds) for the studied tests

Test Replications Elapsed Relative Userself SysselfAD 5000 114 2000 114 000CVM 5000 082 1439 081 000DB 5000 1700 29825 1694 004ELR1198852 5000 4483 78649 4478 001JB 5000 25264 443228 25260 000KS119872 5000 089 1561 090 000SEELR 5000 4542 79684 4541 000SW 5000 057 1000 058 000

Table 2 The baby boom data

Times between births (in min)59 14 37 62 68 2 15 9 157 27 37 2 55 86 144 40 36 47 9 61 1 26 13 28 77 26 45 25 1829 15 38 2 2 19 27 14 13 19 54 70 28Note Data appeared in the newspaper the Sunday Mail on December 21 1997 [15]

It was of paramount importance for us to determine thecomputational cost of the new algorithms by focusing on thecomputation time of the proposed test as compared to thatof the considered existing tests To assess this we used the Rbenchmark tool on a notebook installed with 64 BitWindows10 Home addition Equipped with a 4th generation Intel Corei5-4210U processor which has a speed of 17 GHz cache andmemory (RAM) of 4GB PC3 DDR3L SDRAM we set oursimulations to 5000 for each test with sample size set at 119899 =80 The results (see Table 1) show only a clear advantage ofour proposed approach to that of the widely known JB testAlso from the results our proposed methods are comparableto the SEELR test but inferior to the DB test The SW CVMKS and AD tests are computationally more efficient in termsof time than the rest of the studied tests

5 A Real Data Example

In this example we used baby boom data from an observa-tional study with records of forty-four (44) babies born ata 24-hour hospital in Brisbane Australia We opted for thisdataset because it can be used to demonstrate applicabilityof various statistical procedures to some common applieddistributions which include the normal (by modelling thebirth weights) the binomial (inferences in the number ofboysgirls born) the geometric (by considering the numberof births until a boygirl is born) the Poisson (births perhour for each hour) and the exponential (inference ontimes between births) Recently Miecznikowski et al [27]used the baby boom dataset in a resampling study on theapplication of their ELR based goodness-of-fit test For moreinformation regarding this dataset one can refer toDunn [28]For our application we opted to make use of the exponentialdistribution thuswewere interested in inference on the timesbetween births Table 2 shows the times between births whichwere computed by taking the differences between successivetimes of birth after midnight of birth times

Histogram of BabyFrequency

05

1015

50 100 1500Baby

Figure 1 Histogram for times between births for baby boom data

The goal of this examplewas to carry out a bootstrap studyin assessing the robustness and applicability of our proposedtest (ELR1198852) on uniformly distributed data However thetimes between births are known to be consistent with theexponential distribution (see Figure 1) By assessing thehistogram one can easily see that the data resembles theexponential distribution revealing that the times betweenbirths are exponentially consistent We used the inverseexponential distribution to transform the times betweenbirths so that they can be uniformly distributed We thenused the density based empirical likelihood ratio based test(dbEmpLikeGOF) to check if the transformed baby boomdata are uniformly distributed The dbEmpLikeGOF testreturned a 119901 value of 06950 suggesting that the transformeddata are consistent with the uniform distribution

6 Journal of Probability and Statistics

Table 3 Bootstrapping using the inverse exponential transformed baby boom data

Bootstrap power comparisons1198670 data is normally distributedObservations removed AD KS119872 CVM JB SW DB SEELR ELR11988523 00000 00000 00000 00000 01012 06040 00000 061328 00164 00006 00012 00000 01182 04486 00146 0405413 00281 00073 00066 00011 01023 03488 00568 02714

Table 4 A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests (ELR1198851and ELR1198852) with increased moment constraints at 120572 = 005119899 ELR1198851 ELR1198852119896 = 1 2 3 4 119896 = 1 2 3 4 5 119896 = 1 2 3 4 119896 = 1 2 3 4 5

S-R C-t S-R C-t S-R C-t S-R C-t119905(2)

30 00416 00330 00020 00010 06980 06998 06166 0591250 05142 04112 01666 01356 08766 08774 08320 0803080 08732 08336 07476 07184 09718 09684 09544 09488

Cauchy(01)30 03262 03438 00000 00000 09560 09556 09248 0919250 09538 09344 07246 06754 09970 09974 09928 0990080 09996 09996 09964 09940 10000 10000 09998 09996

Uniform(01)30 07230 01958 07208 07206 05772 05986 06996 0700450 09458 05222 09532 09474 09032 09122 09434 0939880 09966 08462 09978 09980 09940 09956 09986 09976

Exp(1)30 00094 00304 00070 00068 04638 04818 03874 0377250 00836 08096 00022 00042 06274 06306 05628 0538080 03764 09972 02504 02346 07942 08070 07558 07506

119905(120575 = 1 ] = 2)30 00476 00136 00012 00028 07168 07230 06476 0628050 05204 04172 01932 01676 08904 08908 08450 0829480 08752 08610 07736 07700 09714 09766 09636 09558

SN(015)30 00514 00520 00486 00442 01394 01242 01048 0094450 00404 00362 00350 00352 01408 01432 01114 0090480 00358 00338 00272 00204 01592 01646 01158 01226Note Our proposed tests are maximized on 119896 isin 119866 where 119866 can take any integer to represent the moment constraints used to maximise the test statistics forspecified sample sizes at 5 level of significance using 5000 simulations 119899 is the sample size Bold represents the powerful test statistic for the given simulationscenarios

For the resampling study we performed a power simu-lation study by randomly removing 3 8 and 13 observationsfrom the transformed baby boomdata at 5 significance levelusing 20000 replications for each simulation For compari-sonrsquos sakewe considered theAD test themodifiedKS test theCVM test the JB test the SW test theDB test the SEELR testand our proposed test (ELR1198852) The Monte Carlo bootstrapsimulation results are presented in Table 3 It is undeniablyclear that our test outperformed all the common existingtests and therefore suggests its robustness and applicabilityon real data It should be noted that we opted for uniformlydistributed data for our application since our proposedtest (ELR1198852) proved to be more powerful for symmetricalternative distributions which are defined on (0 1)

6 Conclusion

An empirical likelihood ratio test for normality based onmoment constraints of the half-normal distribution has beendeveloped Overall the proposed ELR test has good powerproperties and significantly outperformed the consideredwell-known common existing tests against the studied alter-native symmetric distributions In our case the attractivepower properties of the proposed ELR test resulted fromthe EL method being able to integrate most of the availableinformation by utilizing the first four moment constraintsand also through the utilization of the EL function whichleads to additional power benefits We advocate for ourproposed test (ELR1198852) to be the preferred choice when one

Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

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6 Journal of Probability and Statistics

Table 3 Bootstrapping using the inverse exponential transformed baby boom data

Bootstrap power comparisons1198670 data is normally distributedObservations removed AD KS119872 CVM JB SW DB SEELR ELR11988523 00000 00000 00000 00000 01012 06040 00000 061328 00164 00006 00012 00000 01182 04486 00146 0405413 00281 00073 00066 00011 01023 03488 00568 02714

Table 4 A numerical assessment on power using the Shiryaev-Roberts (S-R) and CUSUM-type (C-t) statistics for the proposed tests (ELR1198851and ELR1198852) with increased moment constraints at 120572 = 005119899 ELR1198851 ELR1198852119896 = 1 2 3 4 119896 = 1 2 3 4 5 119896 = 1 2 3 4 119896 = 1 2 3 4 5

S-R C-t S-R C-t S-R C-t S-R C-t119905(2)

30 00416 00330 00020 00010 06980 06998 06166 0591250 05142 04112 01666 01356 08766 08774 08320 0803080 08732 08336 07476 07184 09718 09684 09544 09488

Cauchy(01)30 03262 03438 00000 00000 09560 09556 09248 0919250 09538 09344 07246 06754 09970 09974 09928 0990080 09996 09996 09964 09940 10000 10000 09998 09996

Uniform(01)30 07230 01958 07208 07206 05772 05986 06996 0700450 09458 05222 09532 09474 09032 09122 09434 0939880 09966 08462 09978 09980 09940 09956 09986 09976

Exp(1)30 00094 00304 00070 00068 04638 04818 03874 0377250 00836 08096 00022 00042 06274 06306 05628 0538080 03764 09972 02504 02346 07942 08070 07558 07506

119905(120575 = 1 ] = 2)30 00476 00136 00012 00028 07168 07230 06476 0628050 05204 04172 01932 01676 08904 08908 08450 0829480 08752 08610 07736 07700 09714 09766 09636 09558

SN(015)30 00514 00520 00486 00442 01394 01242 01048 0094450 00404 00362 00350 00352 01408 01432 01114 0090480 00358 00338 00272 00204 01592 01646 01158 01226Note Our proposed tests are maximized on 119896 isin 119866 where 119866 can take any integer to represent the moment constraints used to maximise the test statistics forspecified sample sizes at 5 level of significance using 5000 simulations 119899 is the sample size Bold represents the powerful test statistic for the given simulationscenarios

For the resampling study we performed a power simu-lation study by randomly removing 3 8 and 13 observationsfrom the transformed baby boomdata at 5 significance levelusing 20000 replications for each simulation For compari-sonrsquos sakewe considered theAD test themodifiedKS test theCVM test the JB test the SW test theDB test the SEELR testand our proposed test (ELR1198852) The Monte Carlo bootstrapsimulation results are presented in Table 3 It is undeniablyclear that our test outperformed all the common existingtests and therefore suggests its robustness and applicabilityon real data It should be noted that we opted for uniformlydistributed data for our application since our proposedtest (ELR1198852) proved to be more powerful for symmetricalternative distributions which are defined on (0 1)

6 Conclusion

An empirical likelihood ratio test for normality based onmoment constraints of the half-normal distribution has beendeveloped Overall the proposed ELR test has good powerproperties and significantly outperformed the consideredwell-known common existing tests against the studied alter-native symmetric distributions In our case the attractivepower properties of the proposed ELR test resulted fromthe EL method being able to integrate most of the availableinformation by utilizing the first four moment constraintsand also through the utilization of the EL function whichleads to additional power benefits We advocate for ourproposed test (ELR1198852) to be the preferred choice when one

Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

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Journal of Probability and Statistics 7

Table 5 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Symmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852t(2) 20 05068 04482 05138 05632 05282 02806 03774 05268

30 06834 05832 06552 07016 06908 03946 04228 0700450 08538 07782 08370 08812 08572 05640 04800 0872680 09602 09200 09554 09646 09566 08010 05420 09658

t(4) 20 02270 01768 02114 02898 02410 00922 01698 0245030 03002 02182 02764 03788 03338 01084 02164 0339850 04150 03176 03794 05400 04520 01388 02468 0478480 05558 03994 05210 07064 06282 02094 02784 06760

t(7) 20 01162 00952 01006 01670 01398 00492 01066 0134630 01404 01008 01306 02222 01806 00552 01188 0166450 01806 01272 01578 02954 02362 00502 01422 0227680 02380 01618 02086 04010 03122 00650 01590 03324

Cauchy(01) 20 08780 08386 08898 08622 08674 07012 06368 0845030 09672 09410 09622 09574 09610 08606 06910 0954250 09976 09950 09964 09954 09958 09712 07424 0997680 10000 10000 09998 09998 09998 09992 08882 10000

Cauchy(05) 20 08778 08374 08796 08650 08704 06902 06454 0855030 09628 09414 09648 09512 09590 08664 06950 0954250 09968 09948 09976 09968 09966 09730 07468 0996280 10000 10000 10000 09998 10000 09996 08872 10000

Logistic 20 01090 00872 00982 01460 01138 00436 00944 0115830 01176 00908 01220 01982 01474 00452 01044 0148250 01562 01184 01456 02620 01986 00414 01216 0190080 02098 01406 01870 03474 02662 00468 01266 02908

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 6 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from symmetric alternative distributions definedon (0 1) at 120572 = 005

Symmetric alternative distributions defined on (0 1) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Beta(22) 20 00564 00544 00594 00052 00516 01310 00696 00970

30 00786 00520 00812 00012 00768 02004 00550 0196250 01222 00852 01172 00010 01528 03468 00628 0425280 02340 01256 01834 00128 03170 05978 01128 07204

Beta(33) 20 00404 00474 00408 00076 00372 00780 00518 0062030 00786 00520 00812 00046 00768 01112 00392 0103050 00736 00524 00650 00014 00682 01654 00326 0190680 01076 00762 00826 00022 01128 02772 00298 03458

Beta(0505) 20 06160 03098 05058 00066 07190 09094 07092 0701530 08576 04998 07332 00052 09392 09914 08830 0896050 09902 07976 09568 03822 09992 10000 09916 0995680 10000 09724 09990 09872 10000 10000 10000 10000

Uniform(01) 20 01640 01014 01396 00040 01886 04064 02598 0333230 03004 01422 02262 00020 03894 06622 03202 0600250 05780 02532 04282 00118 07546 09358 05624 0912080 08636 04578 07092 03706 09688 09990 08730 09944

Logit-norm(01) 20 00648 00442 00562 00056 00578 01294 00700 0101030 00858 00574 00748 00024 00796 01974 00658 0199050 01394 00812 01220 00010 01612 03420 00676 0415680 02630 01368 02114 00126 03408 05830 01094 07108

Logit-norm(02) 20 03758 01844 02934 00046 04366 07034 04806 0534830 06092 02884 04822 00030 07342 09150 06512 0825850 09016 05412 07814 01174 09742 09976 09006 0981880 09942 08170 09644 08594 10000 10000 09958 09996

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

8 Journal of Probability and Statistics

Table 7 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (0infin) at 120572 = 005

Asymmetric alternative distributions defined on (0infin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852Exp(1) 20 07850 05722 07222 06230 08334 08384 08522 03642

30 09296 07780 08922 08286 09646 09754 09996 0475250 09972 09594 09878 09756 09998 09992 10000 0640080 10000 09990 09998 09998 10000 10000 10000 08114

Gamma(21) 20 04590 03066 04136 04080 05380 04420 05684 0226430 06662 04776 06072 05852 07502 06876 08094 0284450 08960 06926 08436 08242 09500 09180 09668 0382280 09840 08962 09682 09782 09976 09914 09984 05210

Lognorm(01) 20 09080 07760 08846 08172 09350 09210 09418 0603630 09838 09304 09730 09466 09888 09906 10000 0741850 10000 09942 09998 09976 10000 10000 10000 0906880 10000 10000 10000 10000 10000 10000 10000 09838

Lognorm(02) 20 09986 09904 09970 09840 09990 09998 09999 0889430 09998 09998 10000 09994 10000 10000 10000 0968450 10000 10000 10000 10000 10000 10000 10000 0998880 10000 10000 10000 10000 10000 10000 10000 10000

Weibull(21) 20 01348 00980 01142 01258 01582 01264 01626 0093230 01828 01306 01654 01704 02274 01958 02718 0089250 03050 02000 02530 02738 04086 03446 05202 0112080 04954 03186 04200 04346 06644 05634 07812 01080

Weibull(051) 20 09962 09810 09954 09562 09990 09996 09986 0801430 10000 09990 10000 09972 10000 10000 10000 0916850 10000 10000 10000 10000 10000 10000 10000 0986680 10000 10000 10000 10000 10000 10000 10000 10000

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Table 8 Results of the Monte Carlo power comparisons based on samples with sizes (119899) from asymmetric alternative distributions definedon (minusinfininfin) at 120572 = 005

Asymmetric alternative distributions defined on (minusinfininfin) at 120572 = 005Distribution 119899 AD KS119872 CVM JB SW DB SEELR ELR1198852119905(120575 = 1 ] = 2) 20 06446 05692 06440 06556 06498 04612 05688 05542

30 08060 07178 07934 08080 08072 06210 06678 0724250 09492 08900 09394 09414 09410 07820 07782 0887280 09928 09762 09892 09924 09924 09294 08410 09726119905(120575 = 1 ] = 4) 20 03180 02368 02848 03606 03142 01638 02790 0274430 04086 03246 03884 04810 04518 02262 03584 0371850 05912 04618 05538 06592 06360 03202 04830 0534480 07642 06370 07296 08290 08108 04626 05826 07084119905(120575 = 1 ] = 7) 20 01490 01138 01404 01934 01692 00766 01420 0149230 01958 01424 01736 02722 02318 00940 01876 0193650 02846 01968 02522 03834 03372 01194 02556 0273880 03848 02756 03542 05102 04556 01620 03300 03798

SN(012) 20 00896 00756 00882 01054 01068 00636 00978 0071030 01194 00912 01062 01214 01422 00784 01336 0079250 01666 01260 01488 01810 01968 01164 02080 0080080 02434 01918 02258 02712 02940 01550 03074 00890

SN(015) 20 02406 01744 02212 02060 02660 02092 02810 0114430 03586 02604 03152 03056 04230 03346 04742 0127850 05796 04098 05430 04768 06672 05250 07110 0137280 08080 06084 07554 07378 08888 07394 09020 01722

SC(025) 20 09660 09360 09658 09410 09736 09436 09462 0848230 09978 09884 09954 09910 09970 09882 09774 0952450 10000 09998 10000 09998 09998 09992 09844 0994080 10000 10000 10000 10000 10000 10000 09976 09998

Anderson-Darling (AD) test Modified Kolmogorov-Smirnov (KSM) test [2] Cramer-von Mises test (CVM) test Jarque-Bera (JB) test Shapiro-Wilk (SW)test density based empirical likelihood ratio based (DB) test [16] simple and exact empirical likelihood ratio based (SEELR) test [13] and the proposed testELR1198852

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Journal of Probability and Statistics 9

is testing for departures from normality against symmetricalternative distributions for small to moderate sample sizesHowever our test has low power in the considered asymmet-ric alternatives and further modifications in improving thepower of the test under these alternatives would be muchappreciated

In this study we used the moment constraints of thestandardized variables of the half-normal distribution It willbe of interest for one to use the raw moments (nonstandard-ized data points) of the half-normal distribution Howeveraccording to Dong and Giles [12] the power of the ELR testusing standardized observations is within the same rangeas it is when using nonstandardized data points Also ofinterest are the findings by Mittelhammer et al [29] wherethey suggested that the power of ELR based tests increasesas the moment constraints increase From our numericalexperiment we did not extensively explore this conjectureand this is a potential area of future research and it might beinteresting to carry out a more detailed investigation for theproposed tests We focused on tests for normality which is acommon distribution to test in applied statistical modellingand we believe that our proposed test will assist investigatorsto use empirical likelihood approaches using moment con-straints for goodness-of-fit tests of other applied distributionsin practice By simply ignoring the absolute values of thetransformed observations and utilizing standardized half-normal data points our proposed test will simply transformto a GoF test for assessing departures from half-normality

Data Availability

The data appeared in an article entitled ldquoBabies by the Dozenfor Christmas 24-Hour Baby Boomrdquo in the newspaper theSunday Mail on December 21 1997 [15] One can get the datain the package lsquodbEmpLikeGOFrsquo in R

Conflicts of Interest

The authors declare that they have no conflicts of interest

Acknowledgments

The authors would like to extend their gratitude to ProfessorAlbert Vexler for his patience and assistance in attendingto our questions and queries on research gate They wouldalso like to thank the National Research Foundation of SouthAfrica and the Govan Mbeki Research Unit of the hostingUniversity for sponsoring this study

References

[1] A N Kolmogorov ldquoSulla determinazione empirica di una leggedi distribuzionerdquo Giornalle dellrsquoInstituto Italiano degli Attuarivol 4 pp 83ndash91 1933

[2] HW Lilliefors ldquoOn the Kolmogorov-Smirnov test for normal-ity with mean and variance unknownrdquo Journal of the AmericanStatistical Association vol 62 no 318 pp 399ndash402 1967

[3] T W Anderson and D A Darling ldquoAsymptotic theory ofcertain goodness of fit criteria based on stochastic processesrdquoAnnals of Mathematical Statistics vol 23 pp 193ndash212 1952

[4] T W Anderson and D A Darling ldquoA test of goodness of fitrdquoJournal of the American Statistical Association vol 49 pp 765ndash769 1954

[5] S S Shapiro and M B Wilk ldquoAn analysis of variance test fornormality Complete samplesrdquo Biometrika vol 52 pp 591ndash6111965

[6] C M Jarque and A K Bera ldquoA test for normality of observa-tions and regression residualsrdquo International Statistical Reviewvol 55 no 2 pp 163ndash172 1987

[7] R DAgostino and E S Pearson ldquoTests for departure fromnormality Empirical results for the distributions of b2 and b1rdquoBiometrika vol 60 no 3 pp 613ndash622 1973

[8] P Royston ldquoApproximating the Shapiro-Wilk W-test for non-normalityrdquo Statistics and Computing vol 2 no 3 pp 117ndash1191992

[9] S S Shapiro and R S Francia ldquoAn approximate analysis ofvariance test for normalityrdquo Journal of the American StatisticalAssociation vol 67 no 337 pp 215-216 1972

[10] L Chen and S S Shapiro ldquoAn alernative test for normality basedon normalized spacingsrdquo Journal of Statistical Computation andSimulation vol 53 no 3-4 pp 269ndash287 1995

[11] M M Rahman and Z Govindarajulu ldquoA modification of thetest of Shapiro and Wilk for normalityrdquo Journal of AppliedStatistics vol 24 no 2 pp 219ndash235 1997

[12] L B Dong and D E Giles ldquoAn empirical likelihood ratio testfor normalityrdquo Communications in StatisticsmdashSimulation andComputation vol 36 no 1ndash3 pp 197ndash215 2007

[13] G Shan A Vexler G E Wilding and A D Hutson ldquoSimpleand exact empirical likelihood ratio tests for normality based onmoment relationsrdquo Communications in StatisticsmdashSimulationand Computation vol 40 no 1 pp 129ndash146 2010

[14] A B Owen Empirical Likelihood Chapman and Hall NewYork NY USA 2001

[15] S Steele Babies by the Dozen for Christmas 24-Hour BabyBoom The Sunday Mail (Brisbane) 1997

[16] A Vexler and G Gurevich ldquoEmpirical likelihood ratios appliedto goodness-of-fit tests based on sample entropyrdquo Computa-tional Statistics ampData Analysis vol 54 no 2 pp 531ndash545 2010

[17] C C Lin and G S Mudholkar ldquoA simple test for normalityagainst asymmetric alternativesrdquo Biometrika vol 67 no 2 pp455ndash461 1980

[18] A P Prudnikov Y A Brychkov and O I Marichev Integralsand Series vol 1 Gordon and Breach Science Publishers 1986

[19] A B Owen ldquoEmpirical likelihood ratio confidence intervals fora single functionalrdquo Biometrika vol 75 no 2 pp 237ndash249 1988

[20] A Vexler and C Wu ldquoAn optimal retrospective change pointdetection policyrdquo Scandinavian Journal of Statistics vol 36 no3 pp 542ndash558 2009

[21] G Lorden and M Pollak ldquoNonanticipating estimation appliedto sequential analysis and changepoint detectionrdquoTheAnnals ofStatistics vol 33 no 3 pp 1422ndash1454 2005

[22] A Vexler ldquoGuaranteed testing for epidemic changes of a linearregression modelrdquo Journal of Statistical Planning and Inferencevol 136 no 9 pp 3101ndash3120 2006

[23] A Vexler A Liu and M Pollak ldquoTransformation of change-point detection methods into a Shiryayev-Roberts formrdquo TechRep Department of Biostatistics The New York State Univer-sity at Buffalo 2006

[24] H Cramer ldquoOn the composition of elementary errors firstpaper mathematical deductionsrdquo Scandinavian Actuarial Jour-nal vol 11 pp 13ndash74 1928

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

10 Journal of Probability and Statistics

[25] R VonMises ldquoWahrscheinlichkeitsrechnung und Ihre Anwen-dung in der Statistik und Theoretischen Physikrdquo F DeutickeLeipzig Vol 61 1931

[26] N V Smirnov ldquoSui la distribution de w2 (Criterium de MRvMises)rdquo Comptes Rendus Mathematique Academie des SciencesParis vol 202 pp 449ndash452 1936

[27] J C Miecznikowski A Vexler and L Shepherd ldquoDbEmp-LikeGOF An R package for nonparametric likelihood ratiotests for goodness-of-fit and two-sample comparisons based onsample entropyrdquo Journal of Statistical Software vol 54 no 3 pp1ndash19 2013

[28] P K Dunn ldquoA simple data set for demonstrating commondistributionsrdquo Journal of Statistics Education vol 7 no 3 1999

[29] R C Mittelhammer G G Judge and D Miller EconometricFoundations Cambridge University Press 2000

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom

Hindawiwwwhindawicom Volume 2018

MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Mathematical Problems in Engineering

Applied MathematicsJournal of

Hindawiwwwhindawicom Volume 2018

Probability and StatisticsHindawiwwwhindawicom Volume 2018

Journal of

Hindawiwwwhindawicom Volume 2018

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawiwwwhindawicom Volume 2018

OptimizationJournal of

Hindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom Volume 2018

Engineering Mathematics

International Journal of

Hindawiwwwhindawicom Volume 2018

Operations ResearchAdvances in

Journal of

Hindawiwwwhindawicom Volume 2018

Function SpacesAbstract and Applied AnalysisHindawiwwwhindawicom Volume 2018

International Journal of Mathematics and Mathematical Sciences

Hindawiwwwhindawicom Volume 2018

Hindawi Publishing Corporation httpwwwhindawicom Volume 2013Hindawiwwwhindawicom

The Scientific World Journal

Volume 2018

Hindawiwwwhindawicom Volume 2018Volume 2018

Numerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisNumerical AnalysisAdvances inAdvances in Discrete Dynamics in

Nature and SocietyHindawiwwwhindawicom Volume 2018

Hindawiwwwhindawicom

Dierential EquationsInternational Journal of

Volume 2018

Hindawiwwwhindawicom Volume 2018

Decision SciencesAdvances in

Hindawiwwwhindawicom Volume 2018

AnalysisInternational Journal of

Hindawiwwwhindawicom Volume 2018

Stochastic AnalysisInternational Journal of

Submit your manuscripts atwwwhindawicom