Research Article Parametric Regression Models Using Reversed...
6
Research Article Parametric Regression Models Using Reversed Hazard Rates Asokan Mulayath Variyath 1 and P. G. Sankaran 2 1 Department of Mathematics and Statistics, Memorial University of Newfoundland, St. John’s, NL, Canada A1C 5S7 2 Department of Statistics, Cochin University of Science and Technology, Cochin, Kerala 682022, India Correspondence should be addressed to P. G. Sankaran; [email protected] Received 19 June 2013; Accepted 5 October 2013; Published 6 January 2014 Academic Editor: Aera avaneswaran Copyright © 2014 A. Mulayath Variyath and P. G. Sankaran. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Proportional hazard regression models are widely used in survival analysis to understand and exploit the relationship between survival time and covariates. For leſt censored survival times, reversed hazard rate functions are more appropriate. In this paper, we develop a parametric proportional hazard rates model using an inverted Weibull distribution. e estimation and construction of confidence intervals for the parameters are discussed. We assess the performance of the proposed procedure based on a large number of Monte Carlo simulations. We illustrate the proposed method using a real case example. 1. Introduction In survival studies, covariates or explanatory variables are usually employed to represent heterogeneity in a population. e main objective in such situations is to understand and exploit the relationship between lifetime and covari- ates. Regres sion models are useful in such contexts to assess the effect of covariates on lifetime. ese models can be for- mulated in many ways and several types are in common use. Parametric regression models for lifetime involve speci- fication for the distribution of a lifetime given a vector of covariates . e most commonly used parametric model is the Weibull regression model, which satisfies the propor- tional relationship between hazard rate functions of the lifetimes of two subjects. e maximum likelihood technique is usually employed to find estimates of the parameters of the model. For more properties and applications of parametric regression models, one should refer to Lawless [1]. In survival studies, there are many occasions where lifetime data are leſt censored. For example, baboons in the Amboseli Reserve, Kenya, sleep in the trees and descend for ageing at certain times of the day. Observers oſten arrive later in the day aſter this descent has occurred and on such days they can only ascertain that the descent took place before a particular time, so that the descent times are leſt censored (see [2]). On such occasions, a reversed hazard rate is more appropriate than a hazard rate to analyze lifetime data due to the fact that estimators of hazard rates are unstable when data are leſt censored. e reversed hazard rate of is defined as () = lim Δ → 0 ( − Δ ≤ | ≤ ) Δ . (1) Introduced by Barlow et al. [3], the function () has been used in various contexts such as the estimation of distribution function under leſt censoring [1], defining a new stochastic order [4], characterization of lifetime distributions [5–7], studying ageing behavior [8, 9], evolving new repair and maintenance strategies [10, 11], the mixed proportional hazards model [12], and stress hybrid hazards model [13]. Recently, Sengupta and Nanda [14] introduced the pro- portional reversed hazards model in a semiparametric setup. In the present work, we introduce a fully parametric regres- sion model that satisfies the proportional reversed hazards property. e inverted Weibull distribution is employed as a lifetime model, which can be extended to any parametric model. A large number of simulation studies indicate that the proposed approach is performing well. e rest of the paper is organized as follows. In Section 2, we introduce a parametric regression model using an inverted Weibull distribution. e proposed model has the property that the reversed hazard rate for the lifetime of pair of sub- jects is proportional. e estimation of the parameters of Hindawi Publishing Corporation Journal of Probability and Statistics Volume 2014, Article ID 645719, 5 pages http://dx.doi.org/10.1155/2014/645719
Transcript of Research Article Parametric Regression Models Using Reversed...
Research ArticleParametric Regression Models Using Reversed Hazard Rates
Asokan Mulayath Variyath1 and P G Sankaran2
1 Department of Mathematics and Statistics Memorial University of Newfoundland St Johnrsquos NL Canada A1C 5S72Department of Statistics Cochin University of Science and Technology Cochin Kerala 682022 India
Correspondence should be addressed to P G Sankaran sankaranpgyahoocom
Received 19 June 2013 Accepted 5 October 2013 Published 6 January 2014
Academic Editor Aera Thavaneswaran
Copyright copy 2014 A Mulayath Variyath and P G SankaranThis 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
Proportional hazard regression models are widely used in survival analysis to understand and exploit the relationship betweensurvival time and covariates For left censored survival times reversed hazard rate functions are more appropriate In this paperwe develop a parametric proportional hazard rates model using an inverted Weibull distribution The estimation and constructionof confidence intervals for the parameters are discussed We assess the performance of the proposed procedure based on a largenumber of Monte Carlo simulations We illustrate the proposed method using a real case example
1 Introduction
In survival studies covariates or explanatory variables areusually employed to represent heterogeneity in a populationThe main objective in such situations is to understandand exploit the relationship between lifetime and covari-ates Regres sion models are useful in such contexts to assessthe effect of covariates on lifetime These models can be for-mulated in many ways and several types are in common useParametric regression models for lifetime involve speci-fication for the distribution of a lifetime 119879 given a vector ofcovariates 119883 The most commonly used parametric modelis the Weibull regression model which satisfies the propor-tional relationship between hazard rate functions of thelifetimes of two subjectsThemaximum likelihood techniqueis usually employed to find estimates of the parameters of themodel For more properties and applications of parametricregression models one should refer to Lawless [1]
In survival studies there are many occasions wherelifetime data are left censored For example baboons in theAmboseli Reserve Kenya sleep in the trees and descend forageing at certain times of the day Observers often arrive laterin the day after this descent has occurred and on such daysthey can only ascertain that the descent took place before aparticular time so that the descent times are left censored(see [2]) On such occasions a reversed hazard rate is more
appropriate than a hazard rate to analyze lifetime data due tothe fact that estimators of hazard rates are unstable when dataare left censored The reversed hazard rate of 119879 is defined as
120582 (119905) = limΔ119905rarr0
119875 (119905 minus Δ119905 le 119879 | 119879 le 119905)
Δ119905 (1)
Introduced by Barlow et al [3] the function 120582(119905) hasbeen used in various contexts such as the estimation ofdistribution function under left censoring [1] defining a newstochastic order [4] characterization of lifetime distributions[5ndash7] studying ageing behavior [8 9] evolving new repairand maintenance strategies [10 11] the mixed proportionalhazards model [12] and stress hybrid hazards model [13]
Recently Sengupta and Nanda [14] introduced the pro-portional reversed hazards model in a semiparametric setupIn the present work we introduce a fully parametric regres-sion model that satisfies the proportional reversed hazardsproperty The inverted Weibull distribution is employed asa lifetime model which can be extended to any parametricmodel A large number of simulation studies indicate that theproposed approach is performing well
The rest of the paper is organized as follows In Section 2we introduce a parametric regressionmodel using an invertedWeibull distribution The proposed model has the propertythat the reversed hazard rate for the lifetime of pair of sub-jects is proportional The estimation of the parameters of
Hindawi Publishing CorporationJournal of Probability and StatisticsVolume 2014 Article ID 645719 5 pageshttpdxdoiorg1011552014645719
2 Journal of Probability and Statistics
the model is discussed in Section 3 Simulation studies areconducted in Section 4 to assess the finite sample behavior ofthe estimatorsThe proposedmodel is applied to real life datain Section 5 to illustrate its utility Finally Section 6 providesthe major conclusions of the study
2 Statistical Model
Let 119879 be a nonnegative random variable representing thelifetime of a subject with the distribution function 119865(119905)Assume that the probability density function of119879119891(119905) existsThe reversed hazard rate of 119879 given in (1) can be written as
120582 (119905) =119891 (119905)
119865 (119905) (2)
Let 119883 be 119901 times 1 vector of auxiliary information which maybe time dependent The proportional reversed hazard (PRH)model is defined by
120582 (119905 | 119883) = 1205820 (119905) 119892 (120573119883) (3)
where 1205820(119905) is the baseline reversed hazard rate 119892(sdot) is a
nonnegative function of 119883 and 120573 119901 times 1 vector of regressionparameters and 120582(119905 | 119909) is the reversed hazard rate of119879 giventhe covariate119883The PRHmodel can be expressed in terms ofthe distribution function as
119865 (119905 | 119883) = (1198650 (119905))119892(120573119883)
(4)
where 119865(119905 | 119883) is the distribution function of 119879 given 119883and 119865
0(119905) is the baseline distribution function in the absence
of covariates It should be noted that for two subjects theratio of reversed hazard rates is independent of the time 119905Semiparametric analysis of themodel (2) is recently discussedin Sengupta andNanda [14] Our objective here is to carry outthe parametric analysis of an inverted Weibull distributionunder left censoring When the lifetime random variablefollows the inverted Weibull distribution the baseline distri-bution function is given by
1198650 (119905) = 119890
minus120574119905120572
119905 gt 0 120572 120574 gt 0 (5)
The baseline reversed hazard rate of 119879 is then obtained as
1205820 (119905) =
120574120572
119905120572+1 (6)
Note that the baseline reversed hazard rate is decreasing as 119905increases In the presence of the covariate119883 we have
120582 (119905 | 119883) =120574120572
119905120572+1119892 (120573119883) (7)
We assume that 119892(120573119883) = 1198901205731015840
119883 so that
119865 (119905 | 119883) = 119890(minus120574119905120572
)1198901205731015840119909
(8)
with
119891 (119905 | 119883) =1205741205721198901205731015840
119909
119905120572+1119890minus(120574119905120572
)1198901205731015840119909
(9)
Suppose that the lifetime random variable 119879 is randomly leftcensored by 119885 In practice one could observe the vectors(119884 120575 119883) where 119884 = max (119879 119885) and 120575 = 119868(119879 = 119884) with 119868(sdot)being the indicator function Let (119910
119894 120575119894 119909119894) 119894 = 1 2 119899 be
iid copies of (119884 120575 119883) Then the likelihood function can bewritten as
119871 (120573 119910) =
119899
prod
119894=1
119891(119910119894| 119909119894)120575119894
119865(119910119894| 119909119894)1minus120575119894
(10)
Under the inverted Weibull distribution assumption thelikelihood function given in (10) is obtained as
119871 (120573 120572 120574 119910) prop
119899
prod
119894=1
(1205741205721198901205731015840
119909119894
119910120572+1
119894
119890minus(120574119910
120572
119894)1198901205731015840119909119894
)
120575119894
(119890(minus120574119910
120572
119894)1198901205731015840119909119894
)
1minus120575119894
(11)
so that the log likelihood function is
119897 (120573 120572 120574 119910) = log 119888 +119899
sum
119894=1
1205751198941205731015840119909119894
+
119899
sum
119894=1
120575119894(log 120574 + log120572) minus (120572 + 1)
119899
sum
119894=1
120575119894log119910119894
minus 120574
119899
sum
119894=1
1198901205731015840119909119894
119910120572
119894
(12)
where 119888 is a real constant independent of 120573 120574 and 120572 Wemaximize (12) to estimate the parameters 120573 120572 and 120574 byequating the partial derivatives with respect to each parame-ter to zero as
120597119897
120597120573=
119899
sum
119894=1
120575119894119909119894minus 120574
119899
sum
119894=1
1198901205731015840
119909119894119909119894
119910120572
119894
= 0
120597119897
120597120572=
119899
sum
119894=1
120575119894
120572minus
119899
sum
119894=1
120575119894log119910119894+ 120574
119899
sum
119894=1
1198901205731015840119909119894 log119910
119894
119910120572
119894
= 0
120597119897
120597120574=
119899
sum
119894=1
120575119894
120574minus
119899
sum
119894=1
1198901205731015840
119909119894
119910120572
119894
= 0
(13)
Since there is no closed form solution available for (13) we usenumerical methods to estimate the parametersThe observedinformation matrix is given by
119868 (120573 120572 120574) =
[[[[[[[[[[[[
[
minus1205972119897
120597120573120597120573
minus1205972119897
120597120573120597120572
minus1205972119897
120597120573120597120574
minus1205972119897
120597120572120597120573
minus1205972119897
1205971205722
minus1205972119897
120597120572120597120574
minus1205972119897
120597120574120597120573
minus1205972119897
120597120574120597120572
minus1205972119897
1205971205742
]]]]]]]]]]]]
]
(14)
Note that the matrix (14) is of order (119901 + 2) times (119901 + 2) Underthe standard regularity conditions the vector of estimates
Journal of Probability and Statistics 3
(120573 120574) is asymptotically (119901 + 2)-variate normal with meanvector (120573 120572 120574) and dispersion matrix 119868lowastminus1 where 119868lowast is theFisher information matrix obtained from 119868 by taking theexpected values of each entry
There are different algorithms available to estimate theparameters by solving the score equations or directly optimiz-ing the likelihood function The Newton-Raphson method isthe most common method used to estimate since it is easyto determine the derivatives of the score equations In thisnumerical iterative method the initial values play a vital roledue to the logarithm function In the simulation studies givenin Section 4 we use the simplex method proposed by Neldarand Mead [15] to estimate the parameters The simplexmethod is a simple method to use to estimate the parametersby maximizing the likelihood function where we do not needthe derivatives of the function to be optimized
3 Testing and Confidence Intervals for 120573
Tests and interval estimates of parameters can be derived bythe likelihood ratio test procedure We are mainly interestedin the regression parameter120573where the parameters 120579 = (120572 120574)are normally considered as nuisance parameters
Let the 119901 vector regression parameter be denoted as 120573 =(1205731 1205732) where 120573
1and 120573
2are of vectors of sizes 119896 and 119901 minus 119896
respectively and 120579 is the other parameter in themodelWe areinterested in testing
1198670 1205731= 1205730
1against 119867
1 1205731= 1205730
1 (15)
where 12057301is the specified regression parameter value To test
1198670 we construct the likelihood ratio statistic
Λlowast= 2119897 (120573
1 1205732 120579) minus 2119897 (120573
0
1 1205732 120579) (16)
where 1205731 1205732 and 120579 are the maximum likelihood estimates
under the full model For a large value of 119899 Λlowast follows the 1205942119896
distribution under the null hypothesisAlternatively we can use the test statistic
Λ1= (1205731minus 1205730
1)119881minus1
11(1205731minus 1205730
1) (17)
where 11988111can be obtained from 119881 = 119868(120573 120579)
minus1 which is par-titioned as
119881 = [1198811111988112
1198812111988122
] (18)
Under the null hypothesis Λ1follows 1205942
119896distribution
Assuming asymptotic normality we can construct the100(1 minus 120572) confidence interval for the individual regressionparameter 120573
119895as
120573119895plusmn 1199111205722
se (120573119895) (19)
where se(120573119895) can be obtained from 119881
11
Another important problem is the selection of importantcovariates in the proportional reversed hazard models Since
we assume a parametric model we can use variable selectionmethods such as the Akaike information criterion (AIC) andthe Bayesian information criterion (BIC) To test adequacy ofthe parametricmodel Cox-Snell residuals can be used whichis explained in Section 5 with a case example
4 Performance Analysis
To assess the performance of the proposed method wecarried out a large number of simulationsWe generated sam-ples of size 100 from an inverted Weibull distribution withdifferent values of parameters as 120572 = (05 1 15 2) and120574 = (05 1 15 2) We considered a single covariate whichis generated from Uniform (01) and a regression parameterassumed to be 120573 = (05 115 2) We developed the censoringmechanism using the random data generated from theexponential distributionwith the parameter 120582We choose thevalue of 120582 such that the percent of censored data is between10 and 20 percent We used the simplex method proposedby Neldar and Mead [15] to estimate the parameters Werepeated the study for 10000 times and computed the meanand standard deviation of the parameter estimatesThe entirestudy was repeated for a sample size of 250 The summary ofthe parameter estimates is given in Table 1
From Table 1 we can see that the mean of the parameterestimates based on 10000 simulation is very close to the trueparameter values and the standard deviation is also smallWhen the sample size increases the standard error of theestimates decreases and bias reduces It should be noted thatthere is a slight positive bias in all cases even though it isnegligibly small Since there are no comparable models basedon reversed hazard rates we did not perform any comparisonstudies
5 An Example
We consider an extract of left censored data from an Aus-tralian twin study given inDuffy et al [16]The data consist ofinformation on the age of appendectomy ofmonozygotic anddizygotic twins There are observations with missing age atonset and therefore the data are left censoredThe individualshaving age at onset of less than 11 are left censored Thecovariate namely Zygocity has values from 1 to 6 This dataset consists of 54 observations of which 15 are left censoredWe use this data to illustrate the utility of the parametricreverse hazard rate model Probability plotting and statisticaltest confirmed the distribution of data as inverted Weibulldistribution We use the simplex method to estimate theparameters Since the parameter values are unknown and toavoid the effect of inappropriate initial values we considerdifferent initial values and choose the estimates which havemaximum likelihood Estimates of the parameters are =
23940 120573 = minus00142 and 120574 = 4440586 The 95 confidenceinterval for 120573 indicates that the regression coefficient corre-sponding to Zygocity is not significantly different than zerothat is the effect of Zygocity is negligible This conclusion isalso verified through the likelihood ratio test statistic value00074 having a 119875 value of 093
4 Journal of Probability and Statistics
Table 1 Averages and standard deviation (in brackets) of parameter estimates
True parameter values Estimate of parameters (119899 = 100) Estimate of parameters (119899 = 250)120572 120573 120574 120573 120574 120573 120574
1 1 1 10228 10189 10351 10080 10127 10096(00924) (04040) (02486) (00556) (02434) (01444)
1 05 1 10199 05104 10287 10075 05046 10115(00906) (03984) (02389) (00564) (02489) (01476)
1 15 1 10196 15321 10302 10083 15113 10121(00877) (04033) (02413) (00547) (02457) (01430)
1 2 1 10200 20348 10313 10090 20178 10113(00862) (04123) (02384) (00535) (02530) (01423)
1 1 05 10234 10260 05107 10085 10090 05049(00977) (04338) (01347) (00607) (02708) (00824)
1 1 15 10199 10191 15590 10081 10064 15225(00860) (03848) (03553) (00539) (02381) (02128)
1 1 2 10200 10223 20800 10081 10104 20301(00843) (03759) (04708) (00534) (02361) (02823)
05 1 1 05099 10277 10252 05040 10088 10109(00459) (03986) (02400) (00285) (02452) (01459)
15 1 1 15308 10156 10372 15142 10081 10121(01330) (04072) (02480) (00814) (02474) (01449)
2 1 1 20460 10229 10329 20160 10129 10100(01794) (04050) (02444) (01111) (02512) (01484)
We use a Cox-Snell residual plot to assess the goodness offit The Cox-Snell residual is defined by
119903119894= minus log119865 (119910
119894) =
120574
119910120572
119894
119890120573119909119894 119894 = 1 119899 (20)
If the model fits the data then the residuals should have astandard exponential distribution so that a hazard plot ofresiduals versus theNelson-Aalen estimator of the cumulativehazard of the residuals will be a straight line with slopeone A plot of Cox-Snell residuals against the Nelson-Aalenestimates of the cumulative hazard rate of residuals is givenin Figure 1 which shows that the fit is reasonably good
6 Conclusions
Proportional reversed hazard rate models are more suitablefor modeling the left censored lifetime data In this paperwe proposed a parametric PRH model assuming that thelifetime data follows an inverted Weibull distribution Theestimation and hypothesis testing of the parameters of themodel have been discussed in detail The performance of theproposedmodel is assessed based on a large number ofMonteCarlo simulations Our simulations results clearly indicatedthat the proposed model is performing well We applied theproposedmodel to a real life example to illustrate the utility ofthe method Recently Bayesian methodology was extensivelyemployed in the analysis of lifetime data The inferenceprocedures of the proposedmodel by selecting an appropriate
00 02 04 06 08 10 12 1400
04
08
12
Residuals
Estim
ated
cum
ulat
ive h
azar
d ra
tes
Figure 1 Plot of cumulative hazard rates of Cox-Snell residualsversus residuals
prior distribution are topics of research to be explored Thepresent work can be easily extended to any location-scalefamilies of distributions
Conflict of Interests
The authors declare that there is no conflict of interest for thispaper
Acknowledgments
Theauthors would like to thank the editor and an anonymousreferee for their valuable comments and suggestions that
Journal of Probability and Statistics 5
substantially improved the overall quality of an earlier versionof this paper The research of Dr Variyath is supported by agrant from the Natural Science and Engineering Council ofCanada
References
[1] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 2003
[2] P K Andersen O Borgan R D Gill and N Keiding StatisticalModels Based on Counting Process Springer New York NYUSA 1993
[3] R E Barlow A W Marshall and P Proschan ldquoProperties ofprobability distributions with monotone hazard raterdquo Annals ofMathematical Statistics vol 34 pp 375ndash389 1963
[4] J Keilson and U Sumitha ldquoUniform stochastic ordering andrelated inequalitiesrdquo Canadian Journal of Statistics vol 15 pp63ndash69 1982
[5] H W Block T H Savits and H Singh ldquoThe reversed hazardrate functionrdquo Probability in the Engineering and InformationalSciences vol 12 no 1 pp 69ndash90 1998
[6] M S Finkelstein ldquoOn the reversed hazard raterdquoReliability Engi-neering and System Safety vol 78 pp 71ndash75 2002
[7] N U Nair P G Sankaran and G Asha ldquoCharacterizations ofdistributions using reliability conceptsrdquo Journal of Applied Sta-tistical Science vol 14 no 3-4 pp 237ndash241 2005
[8] R C Gupta P L Gupta and R D Gupta ldquoModeling failuretimedata by Lehman alternativesrdquoCommunications in Statisticsvol 27 no 4 pp 887ndash904 1998
[9] C-D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006
[10] A W Marshall and I Olkin Life Distributions Springer NewYork NY USA 2007
[11] X Li and M Xu ldquoReversed hazard rate order of equilibriumdistributions and a related aging notionrdquo Statistical Papers vol49 no 4 pp 749ndash767 2008
[12] GHorny ldquoInference inmixed proportional hazardmodels with119870 random effectsrdquo Statistical Papers vol 50 no 3 pp 481ndash4992009
[13] C A V Tojeiro and F Louzada ldquoA general threshold stresshybrid hazard model for lifetime datardquo Statistical Papers vol53 no 4 pp 833ndash848 2012
[14] D Sengupta and A K Nanda ldquoThe proportional reversed haz-ards regression modelrdquo Journal of Statistical Theory and Appli-cations vol 18 no 4 pp 461ndash476 2011
[15] J A Neldar and R Mead ldquoA simplex method for function min-imizationrdquo Computer Journal vol 7 pp 308ndash331 1965
[16] D Duffy N G Martin and J D Mathews ldquoAppendectomy inAustralian twinsrdquo American Journal of Human Genetics vol 47no 3 pp 590ndash592 1990
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2 Journal of Probability and Statistics
the model is discussed in Section 3 Simulation studies areconducted in Section 4 to assess the finite sample behavior ofthe estimatorsThe proposedmodel is applied to real life datain Section 5 to illustrate its utility Finally Section 6 providesthe major conclusions of the study
2 Statistical Model
Let 119879 be a nonnegative random variable representing thelifetime of a subject with the distribution function 119865(119905)Assume that the probability density function of119879119891(119905) existsThe reversed hazard rate of 119879 given in (1) can be written as
120582 (119905) =119891 (119905)
119865 (119905) (2)
Let 119883 be 119901 times 1 vector of auxiliary information which maybe time dependent The proportional reversed hazard (PRH)model is defined by
120582 (119905 | 119883) = 1205820 (119905) 119892 (120573119883) (3)
where 1205820(119905) is the baseline reversed hazard rate 119892(sdot) is a
nonnegative function of 119883 and 120573 119901 times 1 vector of regressionparameters and 120582(119905 | 119909) is the reversed hazard rate of119879 giventhe covariate119883The PRHmodel can be expressed in terms ofthe distribution function as
119865 (119905 | 119883) = (1198650 (119905))119892(120573119883)
(4)
where 119865(119905 | 119883) is the distribution function of 119879 given 119883and 119865
0(119905) is the baseline distribution function in the absence
of covariates It should be noted that for two subjects theratio of reversed hazard rates is independent of the time 119905Semiparametric analysis of themodel (2) is recently discussedin Sengupta andNanda [14] Our objective here is to carry outthe parametric analysis of an inverted Weibull distributionunder left censoring When the lifetime random variablefollows the inverted Weibull distribution the baseline distri-bution function is given by
1198650 (119905) = 119890
minus120574119905120572
119905 gt 0 120572 120574 gt 0 (5)
The baseline reversed hazard rate of 119879 is then obtained as
1205820 (119905) =
120574120572
119905120572+1 (6)
Note that the baseline reversed hazard rate is decreasing as 119905increases In the presence of the covariate119883 we have
120582 (119905 | 119883) =120574120572
119905120572+1119892 (120573119883) (7)
We assume that 119892(120573119883) = 1198901205731015840
119883 so that
119865 (119905 | 119883) = 119890(minus120574119905120572
)1198901205731015840119909
(8)
with
119891 (119905 | 119883) =1205741205721198901205731015840
119909
119905120572+1119890minus(120574119905120572
)1198901205731015840119909
(9)
Suppose that the lifetime random variable 119879 is randomly leftcensored by 119885 In practice one could observe the vectors(119884 120575 119883) where 119884 = max (119879 119885) and 120575 = 119868(119879 = 119884) with 119868(sdot)being the indicator function Let (119910
119894 120575119894 119909119894) 119894 = 1 2 119899 be
iid copies of (119884 120575 119883) Then the likelihood function can bewritten as
119871 (120573 119910) =
119899
prod
119894=1
119891(119910119894| 119909119894)120575119894
119865(119910119894| 119909119894)1minus120575119894
(10)
Under the inverted Weibull distribution assumption thelikelihood function given in (10) is obtained as
119871 (120573 120572 120574 119910) prop
119899
prod
119894=1
(1205741205721198901205731015840
119909119894
119910120572+1
119894
119890minus(120574119910
120572
119894)1198901205731015840119909119894
)
120575119894
(119890(minus120574119910
120572
119894)1198901205731015840119909119894
)
1minus120575119894
(11)
so that the log likelihood function is
119897 (120573 120572 120574 119910) = log 119888 +119899
sum
119894=1
1205751198941205731015840119909119894
+
119899
sum
119894=1
120575119894(log 120574 + log120572) minus (120572 + 1)
119899
sum
119894=1
120575119894log119910119894
minus 120574
119899
sum
119894=1
1198901205731015840119909119894
119910120572
119894
(12)
where 119888 is a real constant independent of 120573 120574 and 120572 Wemaximize (12) to estimate the parameters 120573 120572 and 120574 byequating the partial derivatives with respect to each parame-ter to zero as
120597119897
120597120573=
119899
sum
119894=1
120575119894119909119894minus 120574
119899
sum
119894=1
1198901205731015840
119909119894119909119894
119910120572
119894
= 0
120597119897
120597120572=
119899
sum
119894=1
120575119894
120572minus
119899
sum
119894=1
120575119894log119910119894+ 120574
119899
sum
119894=1
1198901205731015840119909119894 log119910
119894
119910120572
119894
= 0
120597119897
120597120574=
119899
sum
119894=1
120575119894
120574minus
119899
sum
119894=1
1198901205731015840
119909119894
119910120572
119894
= 0
(13)
Since there is no closed form solution available for (13) we usenumerical methods to estimate the parametersThe observedinformation matrix is given by
119868 (120573 120572 120574) =
[[[[[[[[[[[[
[
minus1205972119897
120597120573120597120573
minus1205972119897
120597120573120597120572
minus1205972119897
120597120573120597120574
minus1205972119897
120597120572120597120573
minus1205972119897
1205971205722
minus1205972119897
120597120572120597120574
minus1205972119897
120597120574120597120573
minus1205972119897
120597120574120597120572
minus1205972119897
1205971205742
]]]]]]]]]]]]
]
(14)
Note that the matrix (14) is of order (119901 + 2) times (119901 + 2) Underthe standard regularity conditions the vector of estimates
Journal of Probability and Statistics 3
(120573 120574) is asymptotically (119901 + 2)-variate normal with meanvector (120573 120572 120574) and dispersion matrix 119868lowastminus1 where 119868lowast is theFisher information matrix obtained from 119868 by taking theexpected values of each entry
There are different algorithms available to estimate theparameters by solving the score equations or directly optimiz-ing the likelihood function The Newton-Raphson method isthe most common method used to estimate since it is easyto determine the derivatives of the score equations In thisnumerical iterative method the initial values play a vital roledue to the logarithm function In the simulation studies givenin Section 4 we use the simplex method proposed by Neldarand Mead [15] to estimate the parameters The simplexmethod is a simple method to use to estimate the parametersby maximizing the likelihood function where we do not needthe derivatives of the function to be optimized
3 Testing and Confidence Intervals for 120573
Tests and interval estimates of parameters can be derived bythe likelihood ratio test procedure We are mainly interestedin the regression parameter120573where the parameters 120579 = (120572 120574)are normally considered as nuisance parameters
Let the 119901 vector regression parameter be denoted as 120573 =(1205731 1205732) where 120573
1and 120573
2are of vectors of sizes 119896 and 119901 minus 119896
respectively and 120579 is the other parameter in themodelWe areinterested in testing
1198670 1205731= 1205730
1against 119867
1 1205731= 1205730
1 (15)
where 12057301is the specified regression parameter value To test
1198670 we construct the likelihood ratio statistic
Λlowast= 2119897 (120573
1 1205732 120579) minus 2119897 (120573
0
1 1205732 120579) (16)
where 1205731 1205732 and 120579 are the maximum likelihood estimates
under the full model For a large value of 119899 Λlowast follows the 1205942119896
distribution under the null hypothesisAlternatively we can use the test statistic
Λ1= (1205731minus 1205730
1)119881minus1
11(1205731minus 1205730
1) (17)
where 11988111can be obtained from 119881 = 119868(120573 120579)
minus1 which is par-titioned as
119881 = [1198811111988112
1198812111988122
] (18)
Under the null hypothesis Λ1follows 1205942
119896distribution
Assuming asymptotic normality we can construct the100(1 minus 120572) confidence interval for the individual regressionparameter 120573
119895as
120573119895plusmn 1199111205722
se (120573119895) (19)
where se(120573119895) can be obtained from 119881
11
Another important problem is the selection of importantcovariates in the proportional reversed hazard models Since
we assume a parametric model we can use variable selectionmethods such as the Akaike information criterion (AIC) andthe Bayesian information criterion (BIC) To test adequacy ofthe parametricmodel Cox-Snell residuals can be used whichis explained in Section 5 with a case example
4 Performance Analysis
To assess the performance of the proposed method wecarried out a large number of simulationsWe generated sam-ples of size 100 from an inverted Weibull distribution withdifferent values of parameters as 120572 = (05 1 15 2) and120574 = (05 1 15 2) We considered a single covariate whichis generated from Uniform (01) and a regression parameterassumed to be 120573 = (05 115 2) We developed the censoringmechanism using the random data generated from theexponential distributionwith the parameter 120582We choose thevalue of 120582 such that the percent of censored data is between10 and 20 percent We used the simplex method proposedby Neldar and Mead [15] to estimate the parameters Werepeated the study for 10000 times and computed the meanand standard deviation of the parameter estimatesThe entirestudy was repeated for a sample size of 250 The summary ofthe parameter estimates is given in Table 1
From Table 1 we can see that the mean of the parameterestimates based on 10000 simulation is very close to the trueparameter values and the standard deviation is also smallWhen the sample size increases the standard error of theestimates decreases and bias reduces It should be noted thatthere is a slight positive bias in all cases even though it isnegligibly small Since there are no comparable models basedon reversed hazard rates we did not perform any comparisonstudies
5 An Example
We consider an extract of left censored data from an Aus-tralian twin study given inDuffy et al [16]The data consist ofinformation on the age of appendectomy ofmonozygotic anddizygotic twins There are observations with missing age atonset and therefore the data are left censoredThe individualshaving age at onset of less than 11 are left censored Thecovariate namely Zygocity has values from 1 to 6 This dataset consists of 54 observations of which 15 are left censoredWe use this data to illustrate the utility of the parametricreverse hazard rate model Probability plotting and statisticaltest confirmed the distribution of data as inverted Weibulldistribution We use the simplex method to estimate theparameters Since the parameter values are unknown and toavoid the effect of inappropriate initial values we considerdifferent initial values and choose the estimates which havemaximum likelihood Estimates of the parameters are =
23940 120573 = minus00142 and 120574 = 4440586 The 95 confidenceinterval for 120573 indicates that the regression coefficient corre-sponding to Zygocity is not significantly different than zerothat is the effect of Zygocity is negligible This conclusion isalso verified through the likelihood ratio test statistic value00074 having a 119875 value of 093
4 Journal of Probability and Statistics
Table 1 Averages and standard deviation (in brackets) of parameter estimates
True parameter values Estimate of parameters (119899 = 100) Estimate of parameters (119899 = 250)120572 120573 120574 120573 120574 120573 120574
1 1 1 10228 10189 10351 10080 10127 10096(00924) (04040) (02486) (00556) (02434) (01444)
1 05 1 10199 05104 10287 10075 05046 10115(00906) (03984) (02389) (00564) (02489) (01476)
1 15 1 10196 15321 10302 10083 15113 10121(00877) (04033) (02413) (00547) (02457) (01430)
1 2 1 10200 20348 10313 10090 20178 10113(00862) (04123) (02384) (00535) (02530) (01423)
1 1 05 10234 10260 05107 10085 10090 05049(00977) (04338) (01347) (00607) (02708) (00824)
1 1 15 10199 10191 15590 10081 10064 15225(00860) (03848) (03553) (00539) (02381) (02128)
1 1 2 10200 10223 20800 10081 10104 20301(00843) (03759) (04708) (00534) (02361) (02823)
05 1 1 05099 10277 10252 05040 10088 10109(00459) (03986) (02400) (00285) (02452) (01459)
15 1 1 15308 10156 10372 15142 10081 10121(01330) (04072) (02480) (00814) (02474) (01449)
2 1 1 20460 10229 10329 20160 10129 10100(01794) (04050) (02444) (01111) (02512) (01484)
We use a Cox-Snell residual plot to assess the goodness offit The Cox-Snell residual is defined by
119903119894= minus log119865 (119910
119894) =
120574
119910120572
119894
119890120573119909119894 119894 = 1 119899 (20)
If the model fits the data then the residuals should have astandard exponential distribution so that a hazard plot ofresiduals versus theNelson-Aalen estimator of the cumulativehazard of the residuals will be a straight line with slopeone A plot of Cox-Snell residuals against the Nelson-Aalenestimates of the cumulative hazard rate of residuals is givenin Figure 1 which shows that the fit is reasonably good
6 Conclusions
Proportional reversed hazard rate models are more suitablefor modeling the left censored lifetime data In this paperwe proposed a parametric PRH model assuming that thelifetime data follows an inverted Weibull distribution Theestimation and hypothesis testing of the parameters of themodel have been discussed in detail The performance of theproposedmodel is assessed based on a large number ofMonteCarlo simulations Our simulations results clearly indicatedthat the proposed model is performing well We applied theproposedmodel to a real life example to illustrate the utility ofthe method Recently Bayesian methodology was extensivelyemployed in the analysis of lifetime data The inferenceprocedures of the proposedmodel by selecting an appropriate
00 02 04 06 08 10 12 1400
04
08
12
Residuals
Estim
ated
cum
ulat
ive h
azar
d ra
tes
Figure 1 Plot of cumulative hazard rates of Cox-Snell residualsversus residuals
prior distribution are topics of research to be explored Thepresent work can be easily extended to any location-scalefamilies of distributions
Conflict of Interests
The authors declare that there is no conflict of interest for thispaper
Acknowledgments
Theauthors would like to thank the editor and an anonymousreferee for their valuable comments and suggestions that
Journal of Probability and Statistics 5
substantially improved the overall quality of an earlier versionof this paper The research of Dr Variyath is supported by agrant from the Natural Science and Engineering Council ofCanada
References
[1] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 2003
[2] P K Andersen O Borgan R D Gill and N Keiding StatisticalModels Based on Counting Process Springer New York NYUSA 1993
[3] R E Barlow A W Marshall and P Proschan ldquoProperties ofprobability distributions with monotone hazard raterdquo Annals ofMathematical Statistics vol 34 pp 375ndash389 1963
[4] J Keilson and U Sumitha ldquoUniform stochastic ordering andrelated inequalitiesrdquo Canadian Journal of Statistics vol 15 pp63ndash69 1982
[5] H W Block T H Savits and H Singh ldquoThe reversed hazardrate functionrdquo Probability in the Engineering and InformationalSciences vol 12 no 1 pp 69ndash90 1998
[6] M S Finkelstein ldquoOn the reversed hazard raterdquoReliability Engi-neering and System Safety vol 78 pp 71ndash75 2002
[7] N U Nair P G Sankaran and G Asha ldquoCharacterizations ofdistributions using reliability conceptsrdquo Journal of Applied Sta-tistical Science vol 14 no 3-4 pp 237ndash241 2005
[8] R C Gupta P L Gupta and R D Gupta ldquoModeling failuretimedata by Lehman alternativesrdquoCommunications in Statisticsvol 27 no 4 pp 887ndash904 1998
[9] C-D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006
[10] A W Marshall and I Olkin Life Distributions Springer NewYork NY USA 2007
[11] X Li and M Xu ldquoReversed hazard rate order of equilibriumdistributions and a related aging notionrdquo Statistical Papers vol49 no 4 pp 749ndash767 2008
[12] GHorny ldquoInference inmixed proportional hazardmodels with119870 random effectsrdquo Statistical Papers vol 50 no 3 pp 481ndash4992009
[13] C A V Tojeiro and F Louzada ldquoA general threshold stresshybrid hazard model for lifetime datardquo Statistical Papers vol53 no 4 pp 833ndash848 2012
[14] D Sengupta and A K Nanda ldquoThe proportional reversed haz-ards regression modelrdquo Journal of Statistical Theory and Appli-cations vol 18 no 4 pp 461ndash476 2011
[15] J A Neldar and R Mead ldquoA simplex method for function min-imizationrdquo Computer Journal vol 7 pp 308ndash331 1965
[16] D Duffy N G Martin and J D Mathews ldquoAppendectomy inAustralian twinsrdquo American Journal of Human Genetics vol 47no 3 pp 590ndash592 1990
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
(120573 120574) is asymptotically (119901 + 2)-variate normal with meanvector (120573 120572 120574) and dispersion matrix 119868lowastminus1 where 119868lowast is theFisher information matrix obtained from 119868 by taking theexpected values of each entry
There are different algorithms available to estimate theparameters by solving the score equations or directly optimiz-ing the likelihood function The Newton-Raphson method isthe most common method used to estimate since it is easyto determine the derivatives of the score equations In thisnumerical iterative method the initial values play a vital roledue to the logarithm function In the simulation studies givenin Section 4 we use the simplex method proposed by Neldarand Mead [15] to estimate the parameters The simplexmethod is a simple method to use to estimate the parametersby maximizing the likelihood function where we do not needthe derivatives of the function to be optimized
3 Testing and Confidence Intervals for 120573
Tests and interval estimates of parameters can be derived bythe likelihood ratio test procedure We are mainly interestedin the regression parameter120573where the parameters 120579 = (120572 120574)are normally considered as nuisance parameters
Let the 119901 vector regression parameter be denoted as 120573 =(1205731 1205732) where 120573
1and 120573
2are of vectors of sizes 119896 and 119901 minus 119896
respectively and 120579 is the other parameter in themodelWe areinterested in testing
1198670 1205731= 1205730
1against 119867
1 1205731= 1205730
1 (15)
where 12057301is the specified regression parameter value To test
1198670 we construct the likelihood ratio statistic
Λlowast= 2119897 (120573
1 1205732 120579) minus 2119897 (120573
0
1 1205732 120579) (16)
where 1205731 1205732 and 120579 are the maximum likelihood estimates
under the full model For a large value of 119899 Λlowast follows the 1205942119896
distribution under the null hypothesisAlternatively we can use the test statistic
Λ1= (1205731minus 1205730
1)119881minus1
11(1205731minus 1205730
1) (17)
where 11988111can be obtained from 119881 = 119868(120573 120579)
minus1 which is par-titioned as
119881 = [1198811111988112
1198812111988122
] (18)
Under the null hypothesis Λ1follows 1205942
119896distribution
Assuming asymptotic normality we can construct the100(1 minus 120572) confidence interval for the individual regressionparameter 120573
119895as
120573119895plusmn 1199111205722
se (120573119895) (19)
where se(120573119895) can be obtained from 119881
11
Another important problem is the selection of importantcovariates in the proportional reversed hazard models Since
we assume a parametric model we can use variable selectionmethods such as the Akaike information criterion (AIC) andthe Bayesian information criterion (BIC) To test adequacy ofthe parametricmodel Cox-Snell residuals can be used whichis explained in Section 5 with a case example
4 Performance Analysis
To assess the performance of the proposed method wecarried out a large number of simulationsWe generated sam-ples of size 100 from an inverted Weibull distribution withdifferent values of parameters as 120572 = (05 1 15 2) and120574 = (05 1 15 2) We considered a single covariate whichis generated from Uniform (01) and a regression parameterassumed to be 120573 = (05 115 2) We developed the censoringmechanism using the random data generated from theexponential distributionwith the parameter 120582We choose thevalue of 120582 such that the percent of censored data is between10 and 20 percent We used the simplex method proposedby Neldar and Mead [15] to estimate the parameters Werepeated the study for 10000 times and computed the meanand standard deviation of the parameter estimatesThe entirestudy was repeated for a sample size of 250 The summary ofthe parameter estimates is given in Table 1
From Table 1 we can see that the mean of the parameterestimates based on 10000 simulation is very close to the trueparameter values and the standard deviation is also smallWhen the sample size increases the standard error of theestimates decreases and bias reduces It should be noted thatthere is a slight positive bias in all cases even though it isnegligibly small Since there are no comparable models basedon reversed hazard rates we did not perform any comparisonstudies
5 An Example
We consider an extract of left censored data from an Aus-tralian twin study given inDuffy et al [16]The data consist ofinformation on the age of appendectomy ofmonozygotic anddizygotic twins There are observations with missing age atonset and therefore the data are left censoredThe individualshaving age at onset of less than 11 are left censored Thecovariate namely Zygocity has values from 1 to 6 This dataset consists of 54 observations of which 15 are left censoredWe use this data to illustrate the utility of the parametricreverse hazard rate model Probability plotting and statisticaltest confirmed the distribution of data as inverted Weibulldistribution We use the simplex method to estimate theparameters Since the parameter values are unknown and toavoid the effect of inappropriate initial values we considerdifferent initial values and choose the estimates which havemaximum likelihood Estimates of the parameters are =
23940 120573 = minus00142 and 120574 = 4440586 The 95 confidenceinterval for 120573 indicates that the regression coefficient corre-sponding to Zygocity is not significantly different than zerothat is the effect of Zygocity is negligible This conclusion isalso verified through the likelihood ratio test statistic value00074 having a 119875 value of 093
4 Journal of Probability and Statistics
Table 1 Averages and standard deviation (in brackets) of parameter estimates
True parameter values Estimate of parameters (119899 = 100) Estimate of parameters (119899 = 250)120572 120573 120574 120573 120574 120573 120574
1 1 1 10228 10189 10351 10080 10127 10096(00924) (04040) (02486) (00556) (02434) (01444)
1 05 1 10199 05104 10287 10075 05046 10115(00906) (03984) (02389) (00564) (02489) (01476)
1 15 1 10196 15321 10302 10083 15113 10121(00877) (04033) (02413) (00547) (02457) (01430)
1 2 1 10200 20348 10313 10090 20178 10113(00862) (04123) (02384) (00535) (02530) (01423)
1 1 05 10234 10260 05107 10085 10090 05049(00977) (04338) (01347) (00607) (02708) (00824)
1 1 15 10199 10191 15590 10081 10064 15225(00860) (03848) (03553) (00539) (02381) (02128)
1 1 2 10200 10223 20800 10081 10104 20301(00843) (03759) (04708) (00534) (02361) (02823)
05 1 1 05099 10277 10252 05040 10088 10109(00459) (03986) (02400) (00285) (02452) (01459)
15 1 1 15308 10156 10372 15142 10081 10121(01330) (04072) (02480) (00814) (02474) (01449)
2 1 1 20460 10229 10329 20160 10129 10100(01794) (04050) (02444) (01111) (02512) (01484)
We use a Cox-Snell residual plot to assess the goodness offit The Cox-Snell residual is defined by
119903119894= minus log119865 (119910
119894) =
120574
119910120572
119894
119890120573119909119894 119894 = 1 119899 (20)
If the model fits the data then the residuals should have astandard exponential distribution so that a hazard plot ofresiduals versus theNelson-Aalen estimator of the cumulativehazard of the residuals will be a straight line with slopeone A plot of Cox-Snell residuals against the Nelson-Aalenestimates of the cumulative hazard rate of residuals is givenin Figure 1 which shows that the fit is reasonably good
6 Conclusions
Proportional reversed hazard rate models are more suitablefor modeling the left censored lifetime data In this paperwe proposed a parametric PRH model assuming that thelifetime data follows an inverted Weibull distribution Theestimation and hypothesis testing of the parameters of themodel have been discussed in detail The performance of theproposedmodel is assessed based on a large number ofMonteCarlo simulations Our simulations results clearly indicatedthat the proposed model is performing well We applied theproposedmodel to a real life example to illustrate the utility ofthe method Recently Bayesian methodology was extensivelyemployed in the analysis of lifetime data The inferenceprocedures of the proposedmodel by selecting an appropriate
00 02 04 06 08 10 12 1400
04
08
12
Residuals
Estim
ated
cum
ulat
ive h
azar
d ra
tes
Figure 1 Plot of cumulative hazard rates of Cox-Snell residualsversus residuals
prior distribution are topics of research to be explored Thepresent work can be easily extended to any location-scalefamilies of distributions
Conflict of Interests
The authors declare that there is no conflict of interest for thispaper
Acknowledgments
Theauthors would like to thank the editor and an anonymousreferee for their valuable comments and suggestions that
Journal of Probability and Statistics 5
substantially improved the overall quality of an earlier versionof this paper The research of Dr Variyath is supported by agrant from the Natural Science and Engineering Council ofCanada
References
[1] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 2003
[2] P K Andersen O Borgan R D Gill and N Keiding StatisticalModels Based on Counting Process Springer New York NYUSA 1993
[3] R E Barlow A W Marshall and P Proschan ldquoProperties ofprobability distributions with monotone hazard raterdquo Annals ofMathematical Statistics vol 34 pp 375ndash389 1963
[4] J Keilson and U Sumitha ldquoUniform stochastic ordering andrelated inequalitiesrdquo Canadian Journal of Statistics vol 15 pp63ndash69 1982
[5] H W Block T H Savits and H Singh ldquoThe reversed hazardrate functionrdquo Probability in the Engineering and InformationalSciences vol 12 no 1 pp 69ndash90 1998
[6] M S Finkelstein ldquoOn the reversed hazard raterdquoReliability Engi-neering and System Safety vol 78 pp 71ndash75 2002
[7] N U Nair P G Sankaran and G Asha ldquoCharacterizations ofdistributions using reliability conceptsrdquo Journal of Applied Sta-tistical Science vol 14 no 3-4 pp 237ndash241 2005
[8] R C Gupta P L Gupta and R D Gupta ldquoModeling failuretimedata by Lehman alternativesrdquoCommunications in Statisticsvol 27 no 4 pp 887ndash904 1998
[9] C-D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006
[10] A W Marshall and I Olkin Life Distributions Springer NewYork NY USA 2007
[11] X Li and M Xu ldquoReversed hazard rate order of equilibriumdistributions and a related aging notionrdquo Statistical Papers vol49 no 4 pp 749ndash767 2008
[12] GHorny ldquoInference inmixed proportional hazardmodels with119870 random effectsrdquo Statistical Papers vol 50 no 3 pp 481ndash4992009
[13] C A V Tojeiro and F Louzada ldquoA general threshold stresshybrid hazard model for lifetime datardquo Statistical Papers vol53 no 4 pp 833ndash848 2012
[14] D Sengupta and A K Nanda ldquoThe proportional reversed haz-ards regression modelrdquo Journal of Statistical Theory and Appli-cations vol 18 no 4 pp 461ndash476 2011
[15] J A Neldar and R Mead ldquoA simplex method for function min-imizationrdquo Computer Journal vol 7 pp 308ndash331 1965
[16] D Duffy N G Martin and J D Mathews ldquoAppendectomy inAustralian twinsrdquo American Journal of Human Genetics vol 47no 3 pp 590ndash592 1990
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
Table 1 Averages and standard deviation (in brackets) of parameter estimates
True parameter values Estimate of parameters (119899 = 100) Estimate of parameters (119899 = 250)120572 120573 120574 120573 120574 120573 120574
1 1 1 10228 10189 10351 10080 10127 10096(00924) (04040) (02486) (00556) (02434) (01444)
1 05 1 10199 05104 10287 10075 05046 10115(00906) (03984) (02389) (00564) (02489) (01476)
1 15 1 10196 15321 10302 10083 15113 10121(00877) (04033) (02413) (00547) (02457) (01430)
1 2 1 10200 20348 10313 10090 20178 10113(00862) (04123) (02384) (00535) (02530) (01423)
1 1 05 10234 10260 05107 10085 10090 05049(00977) (04338) (01347) (00607) (02708) (00824)
1 1 15 10199 10191 15590 10081 10064 15225(00860) (03848) (03553) (00539) (02381) (02128)
1 1 2 10200 10223 20800 10081 10104 20301(00843) (03759) (04708) (00534) (02361) (02823)
05 1 1 05099 10277 10252 05040 10088 10109(00459) (03986) (02400) (00285) (02452) (01459)
15 1 1 15308 10156 10372 15142 10081 10121(01330) (04072) (02480) (00814) (02474) (01449)
2 1 1 20460 10229 10329 20160 10129 10100(01794) (04050) (02444) (01111) (02512) (01484)
We use a Cox-Snell residual plot to assess the goodness offit The Cox-Snell residual is defined by
119903119894= minus log119865 (119910
119894) =
120574
119910120572
119894
119890120573119909119894 119894 = 1 119899 (20)
If the model fits the data then the residuals should have astandard exponential distribution so that a hazard plot ofresiduals versus theNelson-Aalen estimator of the cumulativehazard of the residuals will be a straight line with slopeone A plot of Cox-Snell residuals against the Nelson-Aalenestimates of the cumulative hazard rate of residuals is givenin Figure 1 which shows that the fit is reasonably good
6 Conclusions
Proportional reversed hazard rate models are more suitablefor modeling the left censored lifetime data In this paperwe proposed a parametric PRH model assuming that thelifetime data follows an inverted Weibull distribution Theestimation and hypothesis testing of the parameters of themodel have been discussed in detail The performance of theproposedmodel is assessed based on a large number ofMonteCarlo simulations Our simulations results clearly indicatedthat the proposed model is performing well We applied theproposedmodel to a real life example to illustrate the utility ofthe method Recently Bayesian methodology was extensivelyemployed in the analysis of lifetime data The inferenceprocedures of the proposedmodel by selecting an appropriate
00 02 04 06 08 10 12 1400
04
08
12
Residuals
Estim
ated
cum
ulat
ive h
azar
d ra
tes
Figure 1 Plot of cumulative hazard rates of Cox-Snell residualsversus residuals
prior distribution are topics of research to be explored Thepresent work can be easily extended to any location-scalefamilies of distributions
Conflict of Interests
The authors declare that there is no conflict of interest for thispaper
Acknowledgments
Theauthors would like to thank the editor and an anonymousreferee for their valuable comments and suggestions that
Journal of Probability and Statistics 5
substantially improved the overall quality of an earlier versionof this paper The research of Dr Variyath is supported by agrant from the Natural Science and Engineering Council ofCanada
References
[1] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 2003
[2] P K Andersen O Borgan R D Gill and N Keiding StatisticalModels Based on Counting Process Springer New York NYUSA 1993
[3] R E Barlow A W Marshall and P Proschan ldquoProperties ofprobability distributions with monotone hazard raterdquo Annals ofMathematical Statistics vol 34 pp 375ndash389 1963
[4] J Keilson and U Sumitha ldquoUniform stochastic ordering andrelated inequalitiesrdquo Canadian Journal of Statistics vol 15 pp63ndash69 1982
[5] H W Block T H Savits and H Singh ldquoThe reversed hazardrate functionrdquo Probability in the Engineering and InformationalSciences vol 12 no 1 pp 69ndash90 1998
[6] M S Finkelstein ldquoOn the reversed hazard raterdquoReliability Engi-neering and System Safety vol 78 pp 71ndash75 2002
[7] N U Nair P G Sankaran and G Asha ldquoCharacterizations ofdistributions using reliability conceptsrdquo Journal of Applied Sta-tistical Science vol 14 no 3-4 pp 237ndash241 2005
[8] R C Gupta P L Gupta and R D Gupta ldquoModeling failuretimedata by Lehman alternativesrdquoCommunications in Statisticsvol 27 no 4 pp 887ndash904 1998
[9] C-D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006
[10] A W Marshall and I Olkin Life Distributions Springer NewYork NY USA 2007
[11] X Li and M Xu ldquoReversed hazard rate order of equilibriumdistributions and a related aging notionrdquo Statistical Papers vol49 no 4 pp 749ndash767 2008
[12] GHorny ldquoInference inmixed proportional hazardmodels with119870 random effectsrdquo Statistical Papers vol 50 no 3 pp 481ndash4992009
[13] C A V Tojeiro and F Louzada ldquoA general threshold stresshybrid hazard model for lifetime datardquo Statistical Papers vol53 no 4 pp 833ndash848 2012
[14] D Sengupta and A K Nanda ldquoThe proportional reversed haz-ards regression modelrdquo Journal of Statistical Theory and Appli-cations vol 18 no 4 pp 461ndash476 2011
[15] J A Neldar and R Mead ldquoA simplex method for function min-imizationrdquo Computer Journal vol 7 pp 308ndash331 1965
[16] D Duffy N G Martin and J D Mathews ldquoAppendectomy inAustralian twinsrdquo American Journal of Human Genetics vol 47no 3 pp 590ndash592 1990
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
substantially improved the overall quality of an earlier versionof this paper The research of Dr Variyath is supported by agrant from the Natural Science and Engineering Council ofCanada
References
[1] J F Lawless Statistical Models and Methods for Lifetime DataJohn Wiley amp Sons New York NY USA 2003
[2] P K Andersen O Borgan R D Gill and N Keiding StatisticalModels Based on Counting Process Springer New York NYUSA 1993
[3] R E Barlow A W Marshall and P Proschan ldquoProperties ofprobability distributions with monotone hazard raterdquo Annals ofMathematical Statistics vol 34 pp 375ndash389 1963
[4] J Keilson and U Sumitha ldquoUniform stochastic ordering andrelated inequalitiesrdquo Canadian Journal of Statistics vol 15 pp63ndash69 1982
[5] H W Block T H Savits and H Singh ldquoThe reversed hazardrate functionrdquo Probability in the Engineering and InformationalSciences vol 12 no 1 pp 69ndash90 1998
[6] M S Finkelstein ldquoOn the reversed hazard raterdquoReliability Engi-neering and System Safety vol 78 pp 71ndash75 2002
[7] N U Nair P G Sankaran and G Asha ldquoCharacterizations ofdistributions using reliability conceptsrdquo Journal of Applied Sta-tistical Science vol 14 no 3-4 pp 237ndash241 2005
[8] R C Gupta P L Gupta and R D Gupta ldquoModeling failuretimedata by Lehman alternativesrdquoCommunications in Statisticsvol 27 no 4 pp 887ndash904 1998
[9] C-D Lai and M Xie Stochastic Ageing and Dependence forReliability Springer New York NY USA 2006
[10] A W Marshall and I Olkin Life Distributions Springer NewYork NY USA 2007
[11] X Li and M Xu ldquoReversed hazard rate order of equilibriumdistributions and a related aging notionrdquo Statistical Papers vol49 no 4 pp 749ndash767 2008
[12] GHorny ldquoInference inmixed proportional hazardmodels with119870 random effectsrdquo Statistical Papers vol 50 no 3 pp 481ndash4992009
[13] C A V Tojeiro and F Louzada ldquoA general threshold stresshybrid hazard model for lifetime datardquo Statistical Papers vol53 no 4 pp 833ndash848 2012
[14] D Sengupta and A K Nanda ldquoThe proportional reversed haz-ards regression modelrdquo Journal of Statistical Theory and Appli-cations vol 18 no 4 pp 461ndash476 2011
[15] J A Neldar and R Mead ldquoA simplex method for function min-imizationrdquo Computer Journal vol 7 pp 308ndash331 1965
[16] D Duffy N G Martin and J D Mathews ldquoAppendectomy inAustralian twinsrdquo American Journal of Human Genetics vol 47no 3 pp 590ndash592 1990
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
