A simple test procedure is proposed for testing the null hypothesis that two life distributions are equal

against the alternative that one is more new better than used than the other. The proposed test is based on a sub

sample of size two from each sample. The properties of the test procedure such as asymptotic normality,

consistency etc., are established. The performance of the test procedure is evaluated by finding Pitman

asymptotic relative efficiencies(ARE) for different distributions in comparison with the test procedure due to

Hollander and Proschan(1986). A simple two-sample test statistic for testing the null hypothesis of equality

of two life distributions against the alternative that one life distribution possesses more of New better than used

property than does another life distribution at a specified age. The newly proposed test is based on a U-statistic

whose kernel is based on minima of a sub sample of size two. The asymptotic normality of the proposed test

is established. The Pitman ARE of the proposed test relative to the test proposed by Lim, Kim and Park(2005)

are computed by considering the alternative distributions with location and scale parameters. The Pitman AREs

show that the performance of the tests proposed here is better as compared to its competitors

Nagaraj, V. Gudaganavar.

Pandit, P.V.

ePrints@Bangalore University

Research Journal of Mathematics and Statistics 1(1): 9-13, 2009ISSN: 2040-7505© Maxwell Scientific Organization, 2009Submitted Date: June 08, 2009 Accepted Date: August 05, 2009 Published Date: September 23, 2009Corresponding Author: Parameshwar V. Pandit, Department of Statistics, Bangalore University, Bangalore -560 056, India 9On Tests for Detecting More Nbu-ness Property of Life Distributions1Parameshwar V. Pandit and 2Nagaraj V. Gudaganavar1Department of Statistics, Bangalore University, Bangalore -560 056, India2Department of Statistics, Anjuman Arts,Science and Commerce College, Dharwad-580001, IndiaAbstract: A simple test procedure is proposed for testing the null hypothesis that two life distributions are equalagainst the alternative that one is more new better than used than the other. The proposed test is based on a subsample of size two from each sample. The properties of the test procedure such as asymptotic normality,consistency etc., are established. The performance of the test procedure is evaluated by finding Pitmanasymptotic relative efficiencies(ARE) for different distributions in comparison with the test procedure due toHollander and Proschan(1986).  A simple two-sample test statistic for testing the  null hypothesis of equalityof two life distributions against the alternative that one life distribution possesses more of New better than usedproperty than does another life distribution at a specified age. The newly proposed test is based on a U-statisticwhose kernel is based on minima of a sub sample of size two. The asymptotic normality of the proposed testis established. The Pitman ARE of the proposed test relative to the test proposed by Lim, Kim and Park(2005)are computed by considering the alternative distributions with location and scale parameters. The Pitman AREsshow that the performance of the tests proposed here is better as compared to its competitors.Key words: NBU, NBU-t0, positive ageing, pitman ARE, Two-sample test, U-statistic INTRODUCTIONIn reliability and survival analysis, the study of lifetime of units whose distribution possesses positive ageingplays vital role. The classes of life distribution possessingpositive ageing were discussed by Bryson andSiddiqui(1969) and then in a book by Barlow andProschan (1981). The problem of testing exponentialityagainst monotone failure rate were considered byProschan and Pyke(1967), Barlow and Proschan (1969)among others. An important class of life distributionpossessing positive ageing is the class of new better thanused and its extensions. The NBU class of distributionsplays an important role in maintenance policy of a system,which is a subclass of NBU-t0 class of distributions. TheNBU-ness or NBU-t0 property of a life distribution is ofinterest in reliability theory. Testing exponentialityagainst the new better than used alternatives are discussedby Hollander and Proschan (1972), Koul (1977),Kumazawa (1983), Ahmad (1994, 2004) among others.Many authors proposed a number of tests for detectingsuch a property ; for example, Hollander and Proschan(1972) and Hollander et al.(1986a). Later, Ebrahimi andHabibullah(1990) extended the results of  Hollander etal.(1986a) to propose a notion of new better than used oforder k and developed a class of tests for detecting NBU-t0 property. Recently Pandit(2004,2005) and Pandit andAnuradha(2007) proposed test procedures for detectingNBU-t0 alternatives.Hollander, Park and Proschan (1986b) developed atest procedure for testing the null hypothesis that two lifedistributions F and G are equal versus the alternativehypothesis that F is ‘more NBU than’ G. Pandit andGudaganavar (2007) proposed  test procedures fordetecting more NBU and NBU-t0 alternatives. Recently,Lim, Kim and Park(2005) developed a class of testprocedures for testing the null hypothesis that two lifedistributions F and G are equal against the  alternative thatF is ‘more NBU-t0 than’ G .  In material and methods section we propose two newtest procedures one for testing equality of twodistributions against the alternative that one distributionpossesses more NBU property than another distributionand another for testing equality of two distributionsagainst the alternative that one distribution possessesmore NBU property than another distribution at specifiedage.MATERIALS AND METHODSA life is represented by a non-negative randomvariable X with distribution function F and survivalfunction  =1 - F.  A distribution F is said to be newbetter than (NBU) used  if  (x+t) #  (x)  (t), forall x, t $ 0.  Hollander, Park and Proschan (1986)introduced the larger class of life distributions called newbetter than used of age to defined as follows.Definition 1: The distribution F is said to be new betterthan used of age to, (NBU-to), if for all x³0.  (1.1)   The concept of new better than used of specified agecan be interpreted as ‘a used item of age t0 hasstochastically  less  remaining  life than does a new item.Res. J. Math. Stat., 1(1): 9-13, 200910The NBU-t0 class is a larger class of life distributions thanNBU and the boundary members of NBU-t0 class includenot only exponential distribution but also other lifedistributions, such as one truncated at t0 or one withperiodic failure rate of period t0. We develop a simple and efficient test for testing thenull hypothesis that two life distributions F and G areequal against the alternative that F is ‘more NBU than’ Gin the proposed two-sample more NBU test subsection.The proposed test is based on U-statistics whose kernelsare based on the sub samples drawn from the twosamples. We present a new test procedure for testingagainst the alternative that F is ‘more NBU at specifiedage than’ G, in the proposed two-sample more NBU-t0test subsection.The Proposed two-sample more NBU test: Let X 1,X2,…,Xm  and Y1, Y2,…,Yn denote two random samplesfrom continuous life distributions F and G, respectively.We want to develop test statistic for testing the nullhypothesis H0: F = G (the common distribution is not specified)versus the alternative hypothesisH1:  F is ‘more NBU’ than G.Consider the parameter,where  and           .Here, )(F) and )(G) can be considered as the measure ofdegree of the NBU-ness. Ahmed (2004) test used thismeasure as basis for their test statistic.  If F (G) belongsto NBU, then )(F) > 0 ()(G) > 0) and )(F,G) can betaken as a measure by which F is ‘more NBU-t0’ than G.Under H0, ( to (F,G) and it is strictly greater than zerounder H 1.An unbiased estimator for )(F,G), which is a U-statisticis defined as  whereandHere  are  the sample means of x-sample and y-sample respectively. The asymptotic normality of the testUm,n  is presented in the following theorem.T he or em :  T h e  a s y m p t o t ic  d i s tr i b ut i o n  o f is normal with mean zero andvariance given by where and where andProof: Proof follows from Hoeffding(1948)).A consistent estimator,  for F2(Um,n)  is obtainedby replacing  by , the empirical survivalfunction of F and , by  the empirical survivalfunction of G. Hence, approximate "-level test rejects H0in favour of H1 , if, where Z" is the upper" - percentile point of standard normal distribution.Since, )(F,G) > 0 under H1 and the asymptotic normalityof Um,n  , the test based on Um,n  is consistent against thealternative F is ‘more NBU than’ G.The Proposed two-sample more NBU-t0 test: Let X 1,X2,…,Xm  and Y1, Y2,…,Yn denote two random samplesfrom continuous life distributions F and G, respectively.We want to develop test statistic for testing the nullhypothesis H0: F = G (the common distribution is not specified)versus the alternative hypothesisH1: F is ‘more NBU-t0’ than G , based on the two random samples. Our statistic is motivated by considering the conceptof new better than used of specified age t0. Consider the parameter,where Res. J. Math. Stat., 1(1): 9-13, 200911and .Here ( t0(F) can ( t0(G) be considered as the measure ofdegree of the NBU-ness at age t0. Ahmed (2004) test usedthis measure as basis for their test statistic.  If F (G)belongs to NBU-t0 class, then ( t0(F) > 0 (( t0 (G) > 0) and( t0(F,G) can be taken as a measure by which F is ‘moreNBU-t0’ than G. ( t0(F,G)=0 Under H 0,  and it is strictlygreater than zero under H 1.An unbiased estimator for ( t0(F,G), which is a U-statistic is defined aswhereand Here .The asymptotic normality of the test V t0 is presented in thefollowing theorem.T he or em :  T h e  a s y m p t o t i c  d i s t r i b u ti o n  of is normal with mean zero andvariance given by where andProof: Proof follows from Hoeffding(1948)).A consistent estimator,  for  isobtained by replacing  by , the empiricalsurvival function of F and  by  , theempirical survival function of G. Hence, approximate a-level test rejects H0 in favour of H 1, if ,where  is the upper - percentile point of standard normaldistribution.  Since, ((F,G) > 0 under H1 and theasympto tic normality of  , the test based on  isconsistent against the alternative F is ‘more NBU-t0 than’G.RESULTS AND DISCUSSIONSThe performance of the test procedure, Um ,ndeveloped for testing whether F is ‘more NBU’ than G isstudied in subsection “Asymptotic Relative Efficiency ofUm,n” and that of  developed for testing whether F is‘more NBU at specified age’ than G is presented insubsection “Asymptotic Relative Efficiency of .”Asymptotic Relative Efficiency of Um ,n: We study theasymptotic relative efficiency of Um,n , relative to the Tm,ntest of Hollander, Park and Proschan(1986b) for the twopairs of distributions (F i,q,G). Here, we assume that G isan exponential distribution with mean one. We considerF1,2 as W eibull distribution and F2,2 as Linear failure ratedistribution as defined below:1. Weibull Distribution:2. Linear Failure Rate DistributionThe Pitman asymptotic efficacy isThe ARE’s of the proposed tests with respect to theTm ,n test of Hollander, Park and Proschan(1986b) forWeibull distribution with parameter 2, F1,q and Linearfailure rate distribution are respectively 1.8432 and2.6923. Next, The Pitman efficacies of two sample test basedon Um,n  proposed in this chapter is determined byspecifying a common distribution with parameter q in thenull hypothesis and by considering sequence ofalternatives (F2NN, F2) where, , a>0 be aconstant. The sequence of distribution considered here are(F1,2NN, F1,2), (F2,2NN, F2) and (F3 ,2NN, F2), where F1,2, F2,2and F3,2 are given by1) Weibull DistributionRes. J. Math. Stat., 1(1): 9-13, 200912and2) Linear Failure Rate Distributionand3) Makeham DistributionandIt is to be noted that as Nv4, the sequence of alternativesconverges to the null hypothesis. The efficacy of the Um,ntest is given bywhere  (Um,n) is null asymptotic standard deviationof Um,n  and We present the asymptotic relativeefficiencies(ARE) of Um,n  relative to Tm,n  in Table 1.Asymptotic Relative Efficiency of : We study theasymptotic relative efficiency of , relative to the test of Lim,Kim and Park (2005) for the three pairs ofdistributions (F i,q,G). Here, we assume that G is anexponential distribution with mean one.  Here we considerthe following distributions for F which are given asfollows:1. Exponential Distribution with location parameter:2. Linear Failure Rate Distribution3. NBU-t0 Distribution:The Pitman asymptotic efficacy isThe ARE’s of the proposed tests with respect to the testproposed by Lim, Kim and Park(2005) for exponentialdistribution with location parameter, F1,q for values of 0.1,0.3, 0.5 are respectively 3.8109, 6.6444 and 21.4096. Thetable1 gives the asymptotic relative efficiency (ARE) of relative to the statistic due to Lim,Kim and Park(2005) , for different pairs of distributions (F i,q,G) .From Table 2, it is evident that the newly proposedtest is better linear failure rate distribution and theperformance of the newly proposed test is better thanNBU-t0 distribution for t0, the specified age is above 0.5.It is expected that a test should be better for NBU –t0distribution when the unit has survived for a fixed periodof time.Tab le 1: AR E  of U m,n  w.r .t. Tm,n  2 (F1 ,2NN , F1 ,2) (F2 ,2NN , F2 ,2) (F3 ,2NN , F3 ,2)2 0.4393 0.3396 0.29413 2.5386 4.4836 0.51904 1.1465 5.7197 0.66205 2.6801 6.8133 0.76556 1.9453 7.8100 0.84577 2.2138 8.7193 0.90588 2.9478 9.5681 0.95419 3.90961 0.3634 0.996410 3.51951 1.1166 1.0311Table 2: AR E  of   w.r .t. T 0           -----------------------------------------------------------------------------Distributions 0.1 0.3 0.5 0.7 0.9 1.0F2,q 3.7462 3.3088 2.9507 2.6575 2.4175 2.3143F3,q 0.0445 0.5258 2.1156 7.0143 16.1347 20.1794Res. J. Math. Stat., 1(1): 9-13, 200913CONCLUSIONSFrom the above discussion, we conclude as:(1) The asymptotic relative efficiency of the proposedtest, Um,n  with respect to the test due to Hollander,Park and Proschan (1986b) is computed for two pairsof distributions (Fq,G) with G as exponential withmean one and Fq as Weibull and Linear failure ratedistributions.(2) It is observed that the proposed test, Um,n  , performsbetter for the alternatives considered Fq is eitherweibull or linear failure rate distribution when it isthe comparison with exponential distribution.(3) We have studied asymptotic performance of thenewly proposed test, Um,n  , when the common nulldistribution is Weibull, Makeham and Linear failurerate distribution.(4) It is observed from table 1, that the performance ofthe new test, Um,n  , is better than the test due toHollander, Park and Proschan (1986b) Tm,n  when thecommon null distribution is Weibull, Makeham andLinear failure rate distribution.  (5) Hence if the data under consideration is exactlyNBU, the new test proposed would be a betterchoice.(6) The Asymptotic relative efficiencies of the proposedtest, , with respect to the test due to Lim,Kim andPark(2005) is computed for three pairs ofdistributions (Fq,G) with G is exponential withmeanone and  Fq as exponential with locationparameter q, Makeham distribution with parameter qand NBU-t0 distribution.(7) It is observed that the proposed test, , performs betterfor the alternatives considered Fq is either exponentialor Makeham for all the ages. (8) Hence, if the data under consideration is exactlyNBU-t0 , the new test  would be a better choice whenthe age t0 is greater than 0.3.(9) Similar studies about the Pitman ARE comparisonshave been made by taking common distributions asWeibull, Makeham and NBU-t0 distributions andfound that new test is better choice. REFERENCESAhmad, I.A., 1994. A class of statistics useful in testingincreasing failure rate and new better than used lifedistributions. J.Stat. Plan. Inf., 41:141-149.Ahmad, I.A., 2004. A simple and more efficient newapproach to life testing. Comm. Stat. Theory andMethods, 33(9): 2199-2215.Barlow, R.E. and F. Proschan, 1969. A note on tests ofmonotone failure rate based on incomplete data.Ann. Math. Stat., 40: 595-600.Barlow,R.E. and F. Prosechan, 1981. Statistical Theory ofReliability and life testing.  To Begin with SilverSpring ND.Bryson,M.C. and  M.M. Siddiqui, 1969. Some criteria foraging. J.Am. Stat. Assoc., 64: 1472-1483.Ebrahimi, N. and M. Habbibullah, 1990. Testing whetherthe survival distribution is new better than used ofspecified age. Biometrika, 77: 212-215.Hoeffding, W., 1948. A class of statistics withasymptotically normal distribution. Ann. M ath. Stat.,19: 293-325.Hollander, M. and F. Proschan, 1972. Testing whethernew  is  better  than  used.  Ann. Math. Stat., 43:1136-1146.Hollander, R.M ., D.H . Park and F . Proschan, 1986a. Aclass of life distributions for aging. J.Am. Stat.Assoc., 81: 91-95.Hollander, R.M ., D.H. Park and F. Proschan, 1986b.Testing whether F is “more NBU” than isG.Microelectron. Reliab. 26(1): 39-44.Koul, H.L., 1977. A test for new better than used.Commun. Stat. A- Theo.  Meth. 6: 563-674.Kumazawa, Y., 1983. Testing for new is better than used.Comm. Stat. Theo. M ath., 12: 311-321.   Lim,J.H., D.K. Kim and D.H. Park, 2005. Tests fordetecting more NBU-ness at specific age. Australianand Newziland. J.Stat.,  47(3): 329-337.Proschan, F. and R. Pyke, 1967. Tests for monotonefailure rate. In Proceedings of the Fifth BerkeleySymposium on M athem atical Sta tistics andProbability, 3:  293-312.Pandit,P.V., 2004. A note on testing exponentialityagainst new better than used of specified age. Econ.Qual. Cont., 19(2): 205-213.Pandit,P.V., 2005. Testing new is better than used of anunknown specified age. Far East J. Theor. Stat.,17(2): 197-206.Pandit, P.V. and M.P. Anuradha, 2007. On testingexponentiality against new better than used ofspecified age. Stat. Methodol., 4: 13-21.Pandit, P.V. and N.V. Gudaganavar, 2007. Tests fordetecting more positive ageing property of lifedistributions. Int. J. Agr. Stat. Sci., 3(1): 189-195.

On Tests for Detecting More Nbu-ness Property of Life Distributions

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On Tests for Detecting More Nbu-ness Property of Life Distributions

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