Post on 02-Oct-2020
Research ArticleAn Improved Class of Chain Ratio-Product Type Estimators inTwo-Phase Sampling Using Two Auxiliary Variables
Gajendra K Vishwakarma and Manish Kumar
Department of Applied Mathematics Indian School of Mines Dhanbad Jharkhand 826004 India
Correspondence should be addressed to Manish Kumar manishstats88gmailcom
Received 13 September 2013 Accepted 23 January 2014 Published 6 March 2014
Academic Editor Zhidong Bai
Copyright copy 2014 G K Vishwakarma and M Kumar This is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in any medium provided the original work isproperly cited
This paper presents a technique for estimating finite populationmean of the study variable in the presence of two auxiliary variablesusing two-phase sampling scheme when the regression line does not pass through the neighborhood of the origin The propertiesof the proposed class of estimators are studied under large sample approximation In addition bias and efficiency comparisons arecarried out to study the performances of the proposed class of estimators over the existing estimators It has also been shown that theproposed technique has greater applicability in survey research An empirical study is carried out to demonstrate the performanceof the proposed estimators
1 Introduction
The use of auxiliary information for estimating populationmean of the study variable has greater applicability in surveyresearch It is utilized at the estimation stage and designstage to obtain an improved estimator compared to thosenot utilizing auxiliary information The use of ratio andproduct strategies in survey sampling solely depends uponthe knowledge of populationmean119883 of the auxiliary variable119883
The ratio estimator was developed by Cochran [1] toestimate the population mean 119884 of the study variable 119884
by using information on auxiliary variable 119883 positivelycorrelated with 119884 The ratio estimator is most effective whenthe relationship between 119884 and 119883 is linear through theorigin and the variance of 119884 is proportional to 119883 Robson [2]defined a product estimator that was revisited by Murthy [3]The product estimator is used when the auxiliary variable 119883is negatively correlated with the study variable 119884
When the population mean 119883 of the auxiliary variable119883 is not known before the start of a survey then a first-phase sample of size 119899
1015840 is selected from the population ofsize 119873 on which only the auxiliary variable 119883 is measuredin order to furnish a good estimate of 119883 And then a
second-phase sample of size 119899 is selected from the first-phasesample of size 119899
1015840 on which both the study variable 119884 andthe auxiliary variable 119883 are measured This procedure ofselecting the samples from the given population is knownas two-phase sampling (or double sampling) The conceptof double sampling was first introduced by Neyman [4]Some contribution to two-phase sampling has been madeby Sukhatme [5] Hidiroglou and Sarndal [6] Fuller [7]Hidiroglou [8] Singh and Vishwakarma [9] and Sahoo et al[10]
We can use either one or two (ormore than two) auxiliaryvariables while estimating population mean of the studyvariable keeping this fact Chand [11] introduced chain ratioestimators This led various authors including Kiregyera [12]Singh and Upadhyaya [13] Prasad et al [14] Singh et al [15]Singh and Choudhury [16] and Vishwakarma and Gangele[17] to modify the chain type estimators and discuss theirproperties
When the population mean 119885 of another auxiliary vari-able 119885 which has a positive correlation with X (ie 120588
119883119885gt 0)
is known and if 120588119884119883
gt 120588119884119885
gt 0 then it is advisable to estimate119883 by119883 = 119909
1015840(1198851199111015840) which would provide a better estimate of
119883 as compared to 1199091015840
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 939701 6 pageshttpdxdoiorg1011552014939701
2 Journal of Probability and Statistics
The usual chain type ratio and product estimators of 119884under double sampling scheme using two auxiliary variables119883 and 119885 are given respectively by
119910dc119877
= 1199101199091015840
119909
119885
1199111015840
119910dc119875
= 119910119909
1199091015840
1199111015840
119885
(1)
Singh and Choudhury [16] suggested the following expo-nential chain type ratio and product estimators of 119884 underdouble sampling scheme using two auxiliary variables119883 and119885
119910dcRe = 119910 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
119910dcPe = 119910 exp
119909 minus (11990910158401199111015840)119885
119909 + (11990910158401199111015840)119885
(2)
where1199091015840 and 1199111015840 are the samplemeans of119883 and119885 respectivelybased on the first-phase sample of size 119899
1015840 drawn from thepopulation of size 119873 with the help of Simple RandomSampling Without Replacement (SRSWOR) scheme Also 119910and 119909 are the samplemeans of119884 and119883 respectively based onthe second-phase sample of size 119899 drawn from the first-phasesample of size 1198991015840 with the help of SRSWOR scheme
2 Proposed Estimator
It has been theoretically established that in general thelinear regression estimator is more efficient than the ratio(product) estimator except when the regression line of 119884 on119883 passes through the neighborhood of the origin in whichthe efficiencies of these estimators are almost equal Howeverowing to stronger intuitive appeal survey statisticians favourthe use of ratio and product estimators Further we note thatin many practical situations the regression line does not passthrough the neighborhood of the origin In these situationsthe ratio estimator does not perform well as the linearregression estimator Considering this fact Singh and RuizEspejo [18] made an attempt to improve the performance ofthese estimators and suggested the following ratio-producttype estimator for populationmean119884 under double samplingscheme using single auxiliary variable119883
119910119889
RP = 119910[1205721199091015840
119909+ (1 minus 120572)
119909
1199091015840] (3)
where 120572 is a real constantWe propose the following exponential chain ratio-
product type estimator for population mean 119884 under doublesampling scheme using two auxiliary variables119883 and 119885
119910dcRPe = 119910[120572 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
+ (1 minus 120572) exp119909 minus (119909
10158401199111015840)119885
119909 + (11990910158401199111015840)119885
]
(4)
where 120572 is a real constant to be determined such that theMean Square Error (MSE) of the proposed estimator 119910dc
RPe isminimum For 120572 = 1 119910dc
RPe rarr 119910dcRe whereas for 120572 = 0
119910dcRPe rarr 119910
dcPe
Remark It is noted that the proposed estimator in (4) is aspecial case of the class of estimators 119910class = 119910119867(119909 119911
1015840)
proposed by Srivastava [19] where 119867(sdot) is a parametricfunction such that 119867(119909
1015840| 1199041 119885) = 1 and satisfies certain
regularity conditions defined in Srivastava [19]
3 Bias and MSE of the Proposed Estimator
To obtain the Bias and Mean Square Error (MSE) of theproposed estimator 119910dc
RPe we consider
119910 = 119884 (1 + 1198900) 119909 = 119883 (1 + 119890
1)
1199091015840= 119883(1 + 119890
1015840
1) 119911
1015840= 119885 (1 + 119890
1015840
2)
(5)
such that
119864 (1198900) = 119864 (119890
1) = 119864 (119890
1015840
1) = 119864 (119890
1015840
2) = 0 (6)
where |1198900| lt 1 |119890
1| lt 1 |1198901015840
1| lt 1 |1198901015840
2| lt 1
Let 119862119884 119862119883 and 119862
119885be the coefficients of variation of 119884
119883 and 119885 respectively Also let 120588119884119883
120588119884119885 and 120588
119883119885be the
correlation coefficients between119884 and119883119884 and119885 and119883 and119885 respectively Then we have
119864 (1198902
0) = 11989111198622
119884 119864 (119890
2
1) = 11989111198622
119883
119864 (11989010158402
1) = 11989121198622
119883 119864 (119890
10158402
2) = 11989121198622
119885
119864 (11989001198901) = 1198911120588119884119883
119862119884119862119883 119864 (119890
01198901015840
1) = 1198912120588119884119883
119862119884119862119883
119864 (11989001198901015840
2) = 1198912120588119884119885
119862119884119862119885
119864 (11989011198901015840
1) = 11989121198622
119883 119864 (119890
11198901015840
2) = 1198912120588119883119885
119862119883119862119885
119864 (1198901015840
11198901015840
2) = 1198912120588119883119885
119862119883119862119885
(7)
where
1198911= (
1
119899minus
1
119873) 119891
2= (
1
1198991015840minus
1
119873)
1198913= 1198911minus 1198912= (
1
119899minus
1
1198991015840)
1198622
119884=
1198782
119884
1198842 1198622
119883=
1198782
119883
1198832 1198622
119885=
1198782
119885
1198852
120588119884119883
=119878119884119883
119878119884119878119883
120588119884119885
=119878119884119885
119878119884119878119885
120588119883119885
=119878119883119885
119878119883119878119885
1198782
119884=
1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884)2
1198782
119883=
1
(119873 minus 1)
119873
sum
119894=1
(119883119894minus 119883)2
1198782
119885=
1
(119873 minus 1)
119873
sum
119894=1
(119885119894minus 119885)2
Journal of Probability and Statistics 3
119878119884119883
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119883
119894minus 119883)
119878119884119885
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119885
119894minus 119885)
119878119883119885
=1
(119873 minus 1)
119873
sum
119894=1
(119883119894minus 119883) (119885
119894minus 119885)
(8)
Now expressing the estimator 119910dcRPe in terms of 119890
0 1198901 11989010158401
and 1198901015840
2and neglecting the terms of 119890
0 1198901 11989010158401 and 119890
1015840
2involving
degree greater than two we get
119910dcRPe = 119884 [1 + 120572 (119890
1015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
minus120572
2(11989010158402
1minus 11989010158402
2minus 1198902
1) + 1198900
minus1
2(1198901015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
+1
4(11989010158402
1minus 11989010158402
2minus 1198902
1minus 1198901015840
11198901015840
2+ 11989011198901015840
2minus 11989011198901015840
1)
+1
8(11989010158402
1+ 11989010158402
2+ 1198902
1)]
(9)
To the first degree of approximation the Bias and MeanSquare Error (MSE) of the proposed estimator 119910dc
RPe are givenby
119861 (119910dcRPe)
= 119884 [(4120572 minus 1)
8
times 11989131198622
119883+ 11989121198622
119885 minus
(2120572 minus 1)
2
times 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(10)
MSE (119910dcRPe)
= 1198842
[11989111198622
119884+(2120572 minus 1)
2
411989131198622
119883+ 11989121198622
119885
minus (2120572 minus 1) 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(11)
To the first degree of approximation the expressions forBias andMean Square Error (MSE) of the estimators 119910dc
119877 119910dc119875
119910dcRe 119910
dcPe and 119910
119889
RP are respectively given by
119861 (119910dc119877) = 119884 [119891
31198622
119883+ 11989121198622
119885minus 1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885]
119861 (119910dc119875) = 119884 [119891
3120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcRe) = 119884 [
3
811989131198622
119883+ 11989121198622
119885
minus1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcPe) = 119884 [
minus1
811989131198622
119883+ 11989121198622
119885
+1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910119889
RP) = 119884 [12057211989131198622
119883minus (2120572 minus 1) 119891
3120588119884119883
119862119884119862119883]
(12)
MSE (119910dc119877) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
minus21198913120588119884119883
119862119884119862119883minus 21198912120588119884119885
119862119884119862119885]
MSE (119910dc119875) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
+21198913120588119884119883
119862119884119862119883+ 21198912120588119884119885
119862119884119862119885]
MSE (119910dcRe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
minus 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910dcPe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
+ 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910119889
RP) = 1198842
[11989111198622
119884+ 4120572211989131198622
119883
minus 412057211989131198622
119883+ 120588119884119883
119862119884119862119883
+11989131198622
119883+ 2120588119884119883
119862119884119862119883]
(13)
31 OptimumValue of120572 Aswe know120572 is determined so as tominimize theMean Square Error (MSE) of the estimators 119910119889RPand 119910
dcRPe So the optimum values of 120572 for which MSE(119910119889RP)
and MSE(119910dcRPe) are minimum are obtained by using the
following conditions
120597
120597120572MSE (119910
119889
RP) = 0
120597
120597120572MSE (119910
dcRPe) = 0
(14)
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910119889RP is given by
120572opt =1
2[1 + 120588
119884119883
119862119884
119862119883
] (15)
4 Journal of Probability and Statistics
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910dc
RPe is given by
120572opt =1198913(2120588119884119883
119862119884119862119883+ 1198622
119883) + 1198912(2120588119884119885
119862119884119862119885+ 1198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(16)
Substituting the value of 120572 from (15) in (13) we get theminimumMSE of 119910119889RP as
MSE (119910119889
RP)min = 1198842
[11989111198622
119884minus 11989131205882
1198841198831198622
119884] (17)
Substituting the value of 120572 from (16) in (11) we get theminimumMSE of 119910dc
RPe as
MSE (119910dcRPe)min
= 1198842
[11989111198622
119884minus(1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)2
11989131198622
119883+ 11989121198622
119885
]
(18)
4 Efficiency Comparisons
It is well known that the Bias and variance of the usualunbiased estimator 119910 for population mean in SRSWOR are
119861 (119910) = 0 (19)
119881 (119910) = 11989111198782
119884= 11989111198842
1198622
119884 (20)
From (11) (13) and (20) we have
(i) MSE(119910dcRPe) lt 119881(119910) if
120572 lt4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
2 (11989131198622
119883+ 11989121198622
119885)
(21)
(ii) MSE(119910dcRPe) lt MSE(119910dc
119877) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) minus (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(22)
(iii) MSE(119910dcRPe) lt MSE(119910dc
119875) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 3 (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(23)
(iv) MSE(119910dcRPe) lt MSE(119910dc
Re) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)
11989131198622
119883+ 11989121198622
119885
(24)
(v) MSE(119910dcRPe) lt MSE(119910dc
Pe) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
11989131198622
119883+ 11989121198622
119885
(25)
(vi) MSE(119910dcRPe) lt MSE(119910119889RP) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885) + (3119891
31198622
119883minus 11989121198622
119885)
2 (311989131198622
119883minus 11989121198622
119885)
(26)
The range of 120572 provides enough scope for choosing manyestimators that are more efficient than the above consideredestimators
5 Empirical Study
To examine themerits of the proposed estimator of119884 we haveconsidered the following natural population datasets
Population I (source Cochran [20]) is shown as follows
119884 number of ldquoplacebordquo children
119883 number of paralytic polio cases in the ldquoplacebordquogroup
119885 number of paralytic polio cases in the ldquonot inocu-latedrdquo group
119873 = 34 1198991015840 = 15 119899 = 10 119884 = 492 119883 = 259 and119885 = 291
120588119884119883
= 07326 120588119884119885
= 06430 120588119883119885
= 06837 1198622119884
=
10248 1198622119883= 15175and 119862
2
119885= 11492
Population II (source Murthy [21]) is shown as follows
119884 area under wheat in 1964
119883 area under wheat in 1963
119885 cultivated area in 1961
119873 = 34 1198991015840 = 10 119899 = 7 119884 = 19944 119883 = 20889 and119885 = 74759
120588119884119883
= 09801 120588119884119885
= 09043 120588119883119885
= 09097 1198622119884
=
05673 1198622119883= 05191 and 119862
2
119885= 03527
Here we have computed
(i) the Absolute Relative Bias (ARB) of different sug-gested estimators of 119884 using the formula
ARB (sdot) =
10038161003816100381610038161003816100381610038161003816
Bias (sdot)119884
10038161003816100381610038161003816100381610038161003816
(27)
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
2 Journal of Probability and Statistics
The usual chain type ratio and product estimators of 119884under double sampling scheme using two auxiliary variables119883 and 119885 are given respectively by
119910dc119877
= 1199101199091015840
119909
119885
1199111015840
119910dc119875
= 119910119909
1199091015840
1199111015840
119885
(1)
Singh and Choudhury [16] suggested the following expo-nential chain type ratio and product estimators of 119884 underdouble sampling scheme using two auxiliary variables119883 and119885
119910dcRe = 119910 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
119910dcPe = 119910 exp
119909 minus (11990910158401199111015840)119885
119909 + (11990910158401199111015840)119885
(2)
where1199091015840 and 1199111015840 are the samplemeans of119883 and119885 respectivelybased on the first-phase sample of size 119899
1015840 drawn from thepopulation of size 119873 with the help of Simple RandomSampling Without Replacement (SRSWOR) scheme Also 119910and 119909 are the samplemeans of119884 and119883 respectively based onthe second-phase sample of size 119899 drawn from the first-phasesample of size 1198991015840 with the help of SRSWOR scheme
2 Proposed Estimator
It has been theoretically established that in general thelinear regression estimator is more efficient than the ratio(product) estimator except when the regression line of 119884 on119883 passes through the neighborhood of the origin in whichthe efficiencies of these estimators are almost equal Howeverowing to stronger intuitive appeal survey statisticians favourthe use of ratio and product estimators Further we note thatin many practical situations the regression line does not passthrough the neighborhood of the origin In these situationsthe ratio estimator does not perform well as the linearregression estimator Considering this fact Singh and RuizEspejo [18] made an attempt to improve the performance ofthese estimators and suggested the following ratio-producttype estimator for populationmean119884 under double samplingscheme using single auxiliary variable119883
119910119889
RP = 119910[1205721199091015840
119909+ (1 minus 120572)
119909
1199091015840] (3)
where 120572 is a real constantWe propose the following exponential chain ratio-
product type estimator for population mean 119884 under doublesampling scheme using two auxiliary variables119883 and 119885
119910dcRPe = 119910[120572 exp
(11990910158401199111015840)119885 minus 119909
(11990910158401199111015840)119885 + 119909
+ (1 minus 120572) exp119909 minus (119909
10158401199111015840)119885
119909 + (11990910158401199111015840)119885
]
(4)
where 120572 is a real constant to be determined such that theMean Square Error (MSE) of the proposed estimator 119910dc
RPe isminimum For 120572 = 1 119910dc
RPe rarr 119910dcRe whereas for 120572 = 0
119910dcRPe rarr 119910
dcPe
Remark It is noted that the proposed estimator in (4) is aspecial case of the class of estimators 119910class = 119910119867(119909 119911
1015840)
proposed by Srivastava [19] where 119867(sdot) is a parametricfunction such that 119867(119909
1015840| 1199041 119885) = 1 and satisfies certain
regularity conditions defined in Srivastava [19]
3 Bias and MSE of the Proposed Estimator
To obtain the Bias and Mean Square Error (MSE) of theproposed estimator 119910dc
RPe we consider
119910 = 119884 (1 + 1198900) 119909 = 119883 (1 + 119890
1)
1199091015840= 119883(1 + 119890
1015840
1) 119911
1015840= 119885 (1 + 119890
1015840
2)
(5)
such that
119864 (1198900) = 119864 (119890
1) = 119864 (119890
1015840
1) = 119864 (119890
1015840
2) = 0 (6)
where |1198900| lt 1 |119890
1| lt 1 |1198901015840
1| lt 1 |1198901015840
2| lt 1
Let 119862119884 119862119883 and 119862
119885be the coefficients of variation of 119884
119883 and 119885 respectively Also let 120588119884119883
120588119884119885 and 120588
119883119885be the
correlation coefficients between119884 and119883119884 and119885 and119883 and119885 respectively Then we have
119864 (1198902
0) = 11989111198622
119884 119864 (119890
2
1) = 11989111198622
119883
119864 (11989010158402
1) = 11989121198622
119883 119864 (119890
10158402
2) = 11989121198622
119885
119864 (11989001198901) = 1198911120588119884119883
119862119884119862119883 119864 (119890
01198901015840
1) = 1198912120588119884119883
119862119884119862119883
119864 (11989001198901015840
2) = 1198912120588119884119885
119862119884119862119885
119864 (11989011198901015840
1) = 11989121198622
119883 119864 (119890
11198901015840
2) = 1198912120588119883119885
119862119883119862119885
119864 (1198901015840
11198901015840
2) = 1198912120588119883119885
119862119883119862119885
(7)
where
1198911= (
1
119899minus
1
119873) 119891
2= (
1
1198991015840minus
1
119873)
1198913= 1198911minus 1198912= (
1
119899minus
1
1198991015840)
1198622
119884=
1198782
119884
1198842 1198622
119883=
1198782
119883
1198832 1198622
119885=
1198782
119885
1198852
120588119884119883
=119878119884119883
119878119884119878119883
120588119884119885
=119878119884119885
119878119884119878119885
120588119883119885
=119878119883119885
119878119883119878119885
1198782
119884=
1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884)2
1198782
119883=
1
(119873 minus 1)
119873
sum
119894=1
(119883119894minus 119883)2
1198782
119885=
1
(119873 minus 1)
119873
sum
119894=1
(119885119894minus 119885)2
Journal of Probability and Statistics 3
119878119884119883
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119883
119894minus 119883)
119878119884119885
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119885
119894minus 119885)
119878119883119885
=1
(119873 minus 1)
119873
sum
119894=1
(119883119894minus 119883) (119885
119894minus 119885)
(8)
Now expressing the estimator 119910dcRPe in terms of 119890
0 1198901 11989010158401
and 1198901015840
2and neglecting the terms of 119890
0 1198901 11989010158401 and 119890
1015840
2involving
degree greater than two we get
119910dcRPe = 119884 [1 + 120572 (119890
1015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
minus120572
2(11989010158402
1minus 11989010158402
2minus 1198902
1) + 1198900
minus1
2(1198901015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
+1
4(11989010158402
1minus 11989010158402
2minus 1198902
1minus 1198901015840
11198901015840
2+ 11989011198901015840
2minus 11989011198901015840
1)
+1
8(11989010158402
1+ 11989010158402
2+ 1198902
1)]
(9)
To the first degree of approximation the Bias and MeanSquare Error (MSE) of the proposed estimator 119910dc
RPe are givenby
119861 (119910dcRPe)
= 119884 [(4120572 minus 1)
8
times 11989131198622
119883+ 11989121198622
119885 minus
(2120572 minus 1)
2
times 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(10)
MSE (119910dcRPe)
= 1198842
[11989111198622
119884+(2120572 minus 1)
2
411989131198622
119883+ 11989121198622
119885
minus (2120572 minus 1) 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(11)
To the first degree of approximation the expressions forBias andMean Square Error (MSE) of the estimators 119910dc
119877 119910dc119875
119910dcRe 119910
dcPe and 119910
119889
RP are respectively given by
119861 (119910dc119877) = 119884 [119891
31198622
119883+ 11989121198622
119885minus 1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885]
119861 (119910dc119875) = 119884 [119891
3120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcRe) = 119884 [
3
811989131198622
119883+ 11989121198622
119885
minus1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcPe) = 119884 [
minus1
811989131198622
119883+ 11989121198622
119885
+1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910119889
RP) = 119884 [12057211989131198622
119883minus (2120572 minus 1) 119891
3120588119884119883
119862119884119862119883]
(12)
MSE (119910dc119877) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
minus21198913120588119884119883
119862119884119862119883minus 21198912120588119884119885
119862119884119862119885]
MSE (119910dc119875) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
+21198913120588119884119883
119862119884119862119883+ 21198912120588119884119885
119862119884119862119885]
MSE (119910dcRe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
minus 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910dcPe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
+ 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910119889
RP) = 1198842
[11989111198622
119884+ 4120572211989131198622
119883
minus 412057211989131198622
119883+ 120588119884119883
119862119884119862119883
+11989131198622
119883+ 2120588119884119883
119862119884119862119883]
(13)
31 OptimumValue of120572 Aswe know120572 is determined so as tominimize theMean Square Error (MSE) of the estimators 119910119889RPand 119910
dcRPe So the optimum values of 120572 for which MSE(119910119889RP)
and MSE(119910dcRPe) are minimum are obtained by using the
following conditions
120597
120597120572MSE (119910
119889
RP) = 0
120597
120597120572MSE (119910
dcRPe) = 0
(14)
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910119889RP is given by
120572opt =1
2[1 + 120588
119884119883
119862119884
119862119883
] (15)
4 Journal of Probability and Statistics
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910dc
RPe is given by
120572opt =1198913(2120588119884119883
119862119884119862119883+ 1198622
119883) + 1198912(2120588119884119885
119862119884119862119885+ 1198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(16)
Substituting the value of 120572 from (15) in (13) we get theminimumMSE of 119910119889RP as
MSE (119910119889
RP)min = 1198842
[11989111198622
119884minus 11989131205882
1198841198831198622
119884] (17)
Substituting the value of 120572 from (16) in (11) we get theminimumMSE of 119910dc
RPe as
MSE (119910dcRPe)min
= 1198842
[11989111198622
119884minus(1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)2
11989131198622
119883+ 11989121198622
119885
]
(18)
4 Efficiency Comparisons
It is well known that the Bias and variance of the usualunbiased estimator 119910 for population mean in SRSWOR are
119861 (119910) = 0 (19)
119881 (119910) = 11989111198782
119884= 11989111198842
1198622
119884 (20)
From (11) (13) and (20) we have
(i) MSE(119910dcRPe) lt 119881(119910) if
120572 lt4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
2 (11989131198622
119883+ 11989121198622
119885)
(21)
(ii) MSE(119910dcRPe) lt MSE(119910dc
119877) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) minus (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(22)
(iii) MSE(119910dcRPe) lt MSE(119910dc
119875) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 3 (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(23)
(iv) MSE(119910dcRPe) lt MSE(119910dc
Re) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)
11989131198622
119883+ 11989121198622
119885
(24)
(v) MSE(119910dcRPe) lt MSE(119910dc
Pe) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
11989131198622
119883+ 11989121198622
119885
(25)
(vi) MSE(119910dcRPe) lt MSE(119910119889RP) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885) + (3119891
31198622
119883minus 11989121198622
119885)
2 (311989131198622
119883minus 11989121198622
119885)
(26)
The range of 120572 provides enough scope for choosing manyestimators that are more efficient than the above consideredestimators
5 Empirical Study
To examine themerits of the proposed estimator of119884 we haveconsidered the following natural population datasets
Population I (source Cochran [20]) is shown as follows
119884 number of ldquoplacebordquo children
119883 number of paralytic polio cases in the ldquoplacebordquogroup
119885 number of paralytic polio cases in the ldquonot inocu-latedrdquo group
119873 = 34 1198991015840 = 15 119899 = 10 119884 = 492 119883 = 259 and119885 = 291
120588119884119883
= 07326 120588119884119885
= 06430 120588119883119885
= 06837 1198622119884
=
10248 1198622119883= 15175and 119862
2
119885= 11492
Population II (source Murthy [21]) is shown as follows
119884 area under wheat in 1964
119883 area under wheat in 1963
119885 cultivated area in 1961
119873 = 34 1198991015840 = 10 119899 = 7 119884 = 19944 119883 = 20889 and119885 = 74759
120588119884119883
= 09801 120588119884119885
= 09043 120588119883119885
= 09097 1198622119884
=
05673 1198622119883= 05191 and 119862
2
119885= 03527
Here we have computed
(i) the Absolute Relative Bias (ARB) of different sug-gested estimators of 119884 using the formula
ARB (sdot) =
10038161003816100381610038161003816100381610038161003816
Bias (sdot)119884
10038161003816100381610038161003816100381610038161003816
(27)
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 3
119878119884119883
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119883
119894minus 119883)
119878119884119885
=1
(119873 minus 1)
119873
sum
119894=1
(119884119894minus 119884) (119885
119894minus 119885)
119878119883119885
=1
(119873 minus 1)
119873
sum
119894=1
(119883119894minus 119883) (119885
119894minus 119885)
(8)
Now expressing the estimator 119910dcRPe in terms of 119890
0 1198901 11989010158401
and 1198901015840
2and neglecting the terms of 119890
0 1198901 11989010158401 and 119890
1015840
2involving
degree greater than two we get
119910dcRPe = 119884 [1 + 120572 (119890
1015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
minus120572
2(11989010158402
1minus 11989010158402
2minus 1198902
1) + 1198900
minus1
2(1198901015840
1minus 1198901015840
2minus 1198901+ 11989001198901015840
1minus 11989001198901015840
2minus 11989001198901)
+1
4(11989010158402
1minus 11989010158402
2minus 1198902
1minus 1198901015840
11198901015840
2+ 11989011198901015840
2minus 11989011198901015840
1)
+1
8(11989010158402
1+ 11989010158402
2+ 1198902
1)]
(9)
To the first degree of approximation the Bias and MeanSquare Error (MSE) of the proposed estimator 119910dc
RPe are givenby
119861 (119910dcRPe)
= 119884 [(4120572 minus 1)
8
times 11989131198622
119883+ 11989121198622
119885 minus
(2120572 minus 1)
2
times 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(10)
MSE (119910dcRPe)
= 1198842
[11989111198622
119884+(2120572 minus 1)
2
411989131198622
119883+ 11989121198622
119885
minus (2120572 minus 1) 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
(11)
To the first degree of approximation the expressions forBias andMean Square Error (MSE) of the estimators 119910dc
119877 119910dc119875
119910dcRe 119910
dcPe and 119910
119889
RP are respectively given by
119861 (119910dc119877) = 119884 [119891
31198622
119883+ 11989121198622
119885minus 1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885]
119861 (119910dc119875) = 119884 [119891
3120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcRe) = 119884 [
3
811989131198622
119883+ 11989121198622
119885
minus1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910dcPe) = 119884 [
minus1
811989131198622
119883+ 11989121198622
119885
+1
21198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885]
119861 (119910119889
RP) = 119884 [12057211989131198622
119883minus (2120572 minus 1) 119891
3120588119884119883
119862119884119862119883]
(12)
MSE (119910dc119877) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
minus21198913120588119884119883
119862119884119862119883minus 21198912120588119884119885
119862119884119862119885]
MSE (119910dc119875) = 1198842
[11989111198622
119884+ 11989131198622
119883+ 11989121198622
119885
+21198913120588119884119883
119862119884119862119883+ 21198912120588119884119885
119862119884119862119885]
MSE (119910dcRe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
minus 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910dcPe) = 119884
2
[11989111198622
119884+1
411989131198622
119883+ 11989121198622
119885
+ 1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885 ]
MSE (119910119889
RP) = 1198842
[11989111198622
119884+ 4120572211989131198622
119883
minus 412057211989131198622
119883+ 120588119884119883
119862119884119862119883
+11989131198622
119883+ 2120588119884119883
119862119884119862119883]
(13)
31 OptimumValue of120572 Aswe know120572 is determined so as tominimize theMean Square Error (MSE) of the estimators 119910119889RPand 119910
dcRPe So the optimum values of 120572 for which MSE(119910119889RP)
and MSE(119910dcRPe) are minimum are obtained by using the
following conditions
120597
120597120572MSE (119910
119889
RP) = 0
120597
120597120572MSE (119910
dcRPe) = 0
(14)
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910119889RP is given by
120572opt =1
2[1 + 120588
119884119883
119862119884
119862119883
] (15)
4 Journal of Probability and Statistics
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910dc
RPe is given by
120572opt =1198913(2120588119884119883
119862119884119862119883+ 1198622
119883) + 1198912(2120588119884119885
119862119884119862119885+ 1198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(16)
Substituting the value of 120572 from (15) in (13) we get theminimumMSE of 119910119889RP as
MSE (119910119889
RP)min = 1198842
[11989111198622
119884minus 11989131205882
1198841198831198622
119884] (17)
Substituting the value of 120572 from (16) in (11) we get theminimumMSE of 119910dc
RPe as
MSE (119910dcRPe)min
= 1198842
[11989111198622
119884minus(1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)2
11989131198622
119883+ 11989121198622
119885
]
(18)
4 Efficiency Comparisons
It is well known that the Bias and variance of the usualunbiased estimator 119910 for population mean in SRSWOR are
119861 (119910) = 0 (19)
119881 (119910) = 11989111198782
119884= 11989111198842
1198622
119884 (20)
From (11) (13) and (20) we have
(i) MSE(119910dcRPe) lt 119881(119910) if
120572 lt4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
2 (11989131198622
119883+ 11989121198622
119885)
(21)
(ii) MSE(119910dcRPe) lt MSE(119910dc
119877) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) minus (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(22)
(iii) MSE(119910dcRPe) lt MSE(119910dc
119875) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 3 (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(23)
(iv) MSE(119910dcRPe) lt MSE(119910dc
Re) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)
11989131198622
119883+ 11989121198622
119885
(24)
(v) MSE(119910dcRPe) lt MSE(119910dc
Pe) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
11989131198622
119883+ 11989121198622
119885
(25)
(vi) MSE(119910dcRPe) lt MSE(119910119889RP) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885) + (3119891
31198622
119883minus 11989121198622
119885)
2 (311989131198622
119883minus 11989121198622
119885)
(26)
The range of 120572 provides enough scope for choosing manyestimators that are more efficient than the above consideredestimators
5 Empirical Study
To examine themerits of the proposed estimator of119884 we haveconsidered the following natural population datasets
Population I (source Cochran [20]) is shown as follows
119884 number of ldquoplacebordquo children
119883 number of paralytic polio cases in the ldquoplacebordquogroup
119885 number of paralytic polio cases in the ldquonot inocu-latedrdquo group
119873 = 34 1198991015840 = 15 119899 = 10 119884 = 492 119883 = 259 and119885 = 291
120588119884119883
= 07326 120588119884119885
= 06430 120588119883119885
= 06837 1198622119884
=
10248 1198622119883= 15175and 119862
2
119885= 11492
Population II (source Murthy [21]) is shown as follows
119884 area under wheat in 1964
119883 area under wheat in 1963
119885 cultivated area in 1961
119873 = 34 1198991015840 = 10 119899 = 7 119884 = 19944 119883 = 20889 and119885 = 74759
120588119884119883
= 09801 120588119884119885
= 09043 120588119883119885
= 09097 1198622119884
=
05673 1198622119883= 05191 and 119862
2
119885= 03527
Here we have computed
(i) the Absolute Relative Bias (ARB) of different sug-gested estimators of 119884 using the formula
ARB (sdot) =
10038161003816100381610038161003816100381610038161003816
Bias (sdot)119884
10038161003816100381610038161003816100381610038161003816
(27)
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Journal of Probability and Statistics
The optimum value of 120572 which minimizes the MeanSquare Error (MSE) of the estimator 119910dc
RPe is given by
120572opt =1198913(2120588119884119883
119862119884119862119883+ 1198622
119883) + 1198912(2120588119884119885
119862119884119862119885+ 1198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(16)
Substituting the value of 120572 from (15) in (13) we get theminimumMSE of 119910119889RP as
MSE (119910119889
RP)min = 1198842
[11989111198622
119884minus 11989131205882
1198841198831198622
119884] (17)
Substituting the value of 120572 from (16) in (11) we get theminimumMSE of 119910dc
RPe as
MSE (119910dcRPe)min
= 1198842
[11989111198622
119884minus(1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)2
11989131198622
119883+ 11989121198622
119885
]
(18)
4 Efficiency Comparisons
It is well known that the Bias and variance of the usualunbiased estimator 119910 for population mean in SRSWOR are
119861 (119910) = 0 (19)
119881 (119910) = 11989111198782
119884= 11989111198842
1198622
119884 (20)
From (11) (13) and (20) we have
(i) MSE(119910dcRPe) lt 119881(119910) if
120572 lt4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
2 (11989131198622
119883+ 11989121198622
119885)
(21)
(ii) MSE(119910dcRPe) lt MSE(119910dc
119877) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) minus (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(22)
(iii) MSE(119910dcRPe) lt MSE(119910dc
119875) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 3 (119891
31198622
119883+ 11989121198622
119885)
2 (11989131198622
119883+ 11989121198622
119885)
(23)
(iv) MSE(119910dcRPe) lt MSE(119910dc
Re) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885)
11989131198622
119883+ 11989121198622
119885
(24)
(v) MSE(119910dcRPe) lt MSE(119910dc
Pe) if
120572 lt2 (1198913120588119884119883
119862119884119862119883+ 1198912120588119884119885
119862119884119862119885) + 11989131198622
119883+ 11989121198622
119885
11989131198622
119883+ 11989121198622
119885
(25)
(vi) MSE(119910dcRPe) lt MSE(119910119889RP) if
120572 lt
4 (1198913120588119884119883
119862119884119862119883minus 1198912120588119884119885
119862119884119862119885) + (3119891
31198622
119883minus 11989121198622
119885)
2 (311989131198622
119883minus 11989121198622
119885)
(26)
The range of 120572 provides enough scope for choosing manyestimators that are more efficient than the above consideredestimators
5 Empirical Study
To examine themerits of the proposed estimator of119884 we haveconsidered the following natural population datasets
Population I (source Cochran [20]) is shown as follows
119884 number of ldquoplacebordquo children
119883 number of paralytic polio cases in the ldquoplacebordquogroup
119885 number of paralytic polio cases in the ldquonot inocu-latedrdquo group
119873 = 34 1198991015840 = 15 119899 = 10 119884 = 492 119883 = 259 and119885 = 291
120588119884119883
= 07326 120588119884119885
= 06430 120588119883119885
= 06837 1198622119884
=
10248 1198622119883= 15175and 119862
2
119885= 11492
Population II (source Murthy [21]) is shown as follows
119884 area under wheat in 1964
119883 area under wheat in 1963
119885 cultivated area in 1961
119873 = 34 1198991015840 = 10 119899 = 7 119884 = 19944 119883 = 20889 and119885 = 74759
120588119884119883
= 09801 120588119884119885
= 09043 120588119883119885
= 09097 1198622119884
=
05673 1198622119883= 05191 and 119862
2
119885= 03527
Here we have computed
(i) the Absolute Relative Bias (ARB) of different sug-gested estimators of 119884 using the formula
ARB (sdot) =
10038161003816100381610038161003816100381610038161003816
Bias (sdot)119884
10038161003816100381610038161003816100381610038161003816
(27)
(ii) the Percentage Relative Efficiencies (PREs) of differ-ent suggested estimators of 119884 with respect to 119910 usingthe formula
PRE (sdot 119910) =119881 (119910)
MSE (sdot)times 100 (28)
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Journal of Probability and Statistics 5
Table 1 Absolute Relative Bias (ARB) of different estimators of 119884
Estimators Population I Population II119910 00000 00000119910dc119877
00369 00042119910dc119875
00564 00513119910dcRe 00068 00079
119910dcPe 00165 00198
119910119889
RP 00222 00008119910dcRPe 00058 00243
Table 2 Percentage Relative Efficiencies (PREs) of different estima-tors of 119884 with respect to 119910
Estimators Population I Population II119910 100 100119910dc119877
13691 73081119910dc119875
lowast lowast
119910dcRe 18436 25955
119910dcPe lowast lowast
119910119889
RP 13395 15696119910dcRPe 18927 76330lowastData is not applicable
6 Conclusion
It is observed from Table 1 that
(i) for population I
ARB (119910) lt ARB (119910dcRPe) lt ARB (119910
dcRe) lt ARB (119910
dcPe)
lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dc119875)
(29)
(ii) for population II
ARB (119910) lt ARB (119910119889
RP) lt ARB (119910dc119877) lt ARB (119910
dcRe)
lt ARB (119910dcPe) lt ARB (119910
dcRPe) lt ARB (119910
dc119875)
(30)
From Table 2 we see that the Percentage Relative Effi-ciency (PRE) of the proposed estimator 119910dc
RPe for populationsI and II is more as compared to all other existing estimatorsthat is usual unbiased estimator 119910 chain type ratio estimator119910dc119877 chain type product estimator 119910dc
119875 exponential chain type
ratio estimator 119910dcRe exponential chain type product estimator
119910dcPe and ratio-product type estimator 119910119889RPFinally from Tables 1 and 2 we conclude that the
proposed estimator 119910dcRPe (based on two auxiliary variables 119883
and119885) is amore appropriate estimator in comparison to otherexisting estimators as it has appreciable efficiency as well aslower relative bias
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors are grateful to the editor Professor Zhidong Baiand the learned referee for their comments leading to theimprovement of the paper
References
[1] W G Cochran ldquoThe estimation of the yields of the cerealexperiments by sampling for the ratio of grain to total producerdquoThe Journal of Agricultural Science vol 30 pp 262ndash275 1940
[2] D S Robson ldquoApplications of multivariate polykays to the the-ory of unbiased ratio-type estimationrdquo Journal of the AmericanStatistical Association vol 52 pp 511ndash522 1957
[3] M N Murthy ldquoProduct method of estimationrdquo The IndianJournal of Statistics A vol 26 pp 69ndash74 1964
[4] J Neyman ldquoContribution to the theory of sampling humanpopulationsrdquo Journal of American Statistical Association vol 33pp 101ndash116 1938
[5] B V Sukhatme ldquoSome ratio-type estimators in two-phasesamplingrdquo Journal of the American Statistical Association vol57 pp 628ndash632 1962
[6] M A Hidiroglou and C E Sarndal ldquoUse of auxiliary informa-tion for two phase samplingrdquo Survey Methodology vol 24 pp11ndash20 1998
[7] W A Fuller ldquoTwo-phase samplingrdquo in Proceedings of theAnnual Meeting of the Survey Methods Section of the StatisticalSociety of Canada pp 23ndash30 2000
[8] M A Hidiroglou ldquoDouble samplingrdquo Survey Methodology vol27 pp 143ndash154 2000
[9] H P Singh andGKVishwakarma ldquoModified exponential ratioand product estimators for finite population mean in doublesamplingrdquo Austrian Journal of Statistics vol 36 no 3 pp 217ndash225 2007
[10] L N Sahoo G Mishra and S R Nayak ldquoOn two differ-ent classes of estimators in two-phase sampling using multi-auxiliary variablesrdquo Model Assisted Statistics and Applicationsvol 5 no 1 pp 61ndash68 2010
[11] L Chand Some ratio type estimators based on two or moreauxiliary variables [PhD dissertation] Iowa State UniversityAmes Iowa USA 1975
[12] B Kiregyera ldquoA chain ratio-type estimator in finite populationdouble sampling using two auxiliary variablesrdquoMetrika vol 27no 4 pp 217ndash223 1980
[13] G N Singh and L N Upadhyaya ldquoA class of modified chain-type estimators using two auxiliary variables in two phasesamplingrdquoMetron vol 53 no 3-4 pp 117ndash125 1995
[14] B Prasad R S Singh and H P Singh ldquoSome chain ratio-typeestimators for ratio of two populationmeans using two auxiliarycharacters in two phase samplingrdquo Metron vol 54 no 1-2 pp95ndash113 1996
[15] S Singh H P Singh and L N Upadhyaya ldquoChain ratio andregression type estimators for median estimation in surveysamplingrdquo Statistical Papers vol 48 no 1 pp 23ndash46 2007
[16] B K Singh and S Choudhury ldquoExponential chain ratio andproduct type estimators for finite population mean underdouble sampling schemerdquo Global Journal of Science FrontierResearch vol 12 no 6 2012
[17] G K Vishwakarma and R K Gangele ldquoA class of chain ratio-type exponential estimators in double sampling using two
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Journal of Probability and Statistics
auxiliary variatesrdquo Applied Mathematics and Computation vol227 pp 171ndash175 2014
[18] H P Singh andMRuiz Espejo ldquoDouble sampling ratio-productestimator of a finite population mean in sample surveysrdquoJournal of Applied Statistics vol 34 no 1-2 pp 71ndash85 2007
[19] S K Srivastava ldquoA generalized estimator for the mean of afinite population using multi-auxiliary informationrdquo Journal ofAmerican Statistical Association vol 66 pp 404ndash407 1971
[20] W G Cochran Sampling Techniques John Wiley amp Sons NewYork NY USA 1977
[21] M N Murthy Sampling Theory and Methods Statistical Pub-lishing Society Calcutta India 1967
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of