Confidence Intervals For P(X less than Y) In The Exponential Case With Common Location Parameter

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Joural of Moder Applied Statistical Methods Volume Issue Article 7 --3 Cofidece Itervals For P(X less tha Y I he Expoetial Case With Commo Locatio Parameter Ayma Baklizi Yarmouk Uiversity, Irbid,

edu/jmasm Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical heory Commos Recommeded Citatio Baklizi, Ayma (3 "Cofidece Itervals For P(X less tha Y I he

1 Joural of Moder Applied Statistical Methods Volume Issue Article Cofidece Itervals For P(X less tha Y I he Expoetial Case With Commo Locatio Parameter Ayma Baklizi Yarmouk Uiversity, Irbid, Jorda, baklizi@hotmail.com Follow this ad additioal works at: Part of the Applied Statistics Commos, Social ad Behavioral Scieces Commos, ad the Statistical heory Commos Recommeded Citatio Baklizi, Ayma (3 "Cofidece Itervals For P(X less tha Y I he Expoetial Case With Commo Locatio Parameter," Joural of Moder Applied Statistical Methods: Vol. : Iss., Article 7. DOI:.37/jmasm/67645 Available at: his Regular Article is brought to you for free ad ope access by the Ope Access Jourals at DigitalCommos@WayeState. It has bee accepted for iclusio i Joural of Moder Applied Statistical Methods by a authorized editor of DigitalCommos@WayeState.

2 Joural of Moder Applied Statistical Methods Copyright 3 JMASM, Ic. November, 3, Vol., No., /3/$3. Cofidece Itervals For P(X<Y I he Expoetial Case With Commo Locatio Parameter Ayma Baklizi Departmet of Statistics Yarmouk Uiversity Irbid Jorda he problem cosidered is iterval estimatio of the stress - stregth reliability R = P(X<Y where X ad Y have idepedet expoetial distributios with parameters θ ad λ respectively ad a commo locatio parameter µ. Several types of asymptotic, approximate ad bootstrap itervals are ivestigated. Performaces are ivestigated usig simulatio techiques ad compared i terms of attaimet of the omial cofidece level, symmetry of lower ad upper error rates, ad expected legth. Recommedatios cocerig their usage are give. Key words: Bootstrap, expoetial distributio, iterval estimatio, stress-stregth model Itroductio he problem of makig iferece about R = P(X<Y has received a cosiderable attetio i literature. his problem arises aturally i the cotext of mechaical reliability of a system with stregth X ad stress Y. he system fails ay time its stregth is exceeded by the stress applied to it. Aother iterpretatio of R is that it measures the effect of the treatmet whe X is the respose for a cotrol group ad Y refers to the treatmet group. Beg (98 obtaied the (MVUE of R whe X ad Y are idepedet expoetial radom variables with uequal scale ad uequal locatio parameters. Gupta ad Gupta (988 obtaied the maximum likelihood estimator (MLE, the MVUE, ad a Bayes estimator of R i case of differet locatio parameters ad a commo scale parameter. Various other versios of this problem have bee discussed i literature, see Johso et al. (994. Ayma Baklizi is a Assistat Professor of Applied Statistics. His research iterests are i accelerated life tests ad cesored data. baklizi@hotmail.com. he problem of developig cofidece itervals for the stress - stregth probability has received relatively little attetio; Halperi (987 ad Hamdy (995 developed distributio free cofidece itervals, while Bai ad Hog (99 discussed poit ad iterval estimatio of i the case of two idepedet expoetials with commo locatio parameter, they derived two types of approximate itervals but did ot study their fiite sample properties ad did ot give a idea about how do they compare with each other. I this article, for the same problem cosidered by Bai ad Hog (99, we shall ivestigate ad compare the performace of the two itervals of Bai ad Hog together with some other types of cofidece itervals like itervals based o the trasformed maximum likelihood estimator, the likelihood ratio statistic ad itervals based o the bootstrap (Efro & ibshirai, 993. he model ad maximum likelihood estimatio of its parameters will be preseted i sectio. he o-bootstrap cofidece itervals will be preseted i sectio 3, while bootstrap itervals are discussed i sectio 4. A Mote Carlo study desiged to ivestigate ad compare the itervals is described i sectio 5. Results ad coclusios are give i the fial sectio. 34

3 34 CONFIDENCE INERVALS FOR P(X<Y he Model ad Maximum Likelihood Estimatio I this study, X ad Y are idepedetly expoetially distributed radom variables with scale parameters θ ad λ respectively ad a commo locatio parameter µ, that is θ ( x µ f X ( x, θ, µ = θe, x µ ; λ( y µ f ( y, λ, µ = λe, y µ. Y Let X,..., X be a radom sample for X ad Y,..., Y be a radom sample for Y. he parameter R we wat to estimate is θ R = p( X < Y =. θ + λ he likelihood fuctio ca be writte as L ( θ, λ, µ = θ λ exp θ( x µ λ( y µ i i I( z µ where z ( x,, x, y, y = ad ( mi, I. idicates the usual idicator fuctio. he maximum likelihood estimators of θ, λ, ad µ are give by (Ghosh & Razmpour, 984 µˆ = z, ˆ θ =, ad ˆ λ =, where ( x i z ad ( y i z = =. he maximum likelihood estimator of R is therefore ˆ R =. Now we will describe the + various itervals uder study. Cofidece Itervals for R Exact cofidece itervals that are coveiet to use for R are ot available ad hece approximate methods that exist i a simple closed form are eeded. I this sectio ad the followig sectio we shall develop various types of itervals for the stress stregth reliability (R. Itervals Based o the Asymptotic Normality of the MLE (AN Itervals Bai ad Hog (99 showed that if = + such that γ, < γ <. R N, σ where he ( ( ( ( γ R R σ =. his fact ca be used to γ costruct approximate cofidece itervals for R. he itervals are of the form ( ± z ( ( α, where z α is the α -quatile of the stadard ormal distributio. Itervals Based o the Asymptotic Normality of the rasformed MLE (RAN Itervals Whe the maximum likelihood estimator of the parameter of iterest has its rage i oly a part of the real lie, a mootoe trasformatio of this parameter with cotiuous derivatives ad rage i the etire real lie will geerally be better approximated by a asymptotic ormal distributio as suggested by may authors icludig Lawless (98 ad Nelso (98. Let K ( R be a mootoe fuctio of R ad let K ' ( R be the first derivative, the by applyig the delta method (Serflig, 98 we get ( K ( R ˆ ' K( R N (, K ( R V. Usig this, a α cofidece iterval for R may be obtaied as K K ( K z gk Vˆ α, K + zk Vˆ (. A appropriate trasform is the ta (Jeg & Meeker, 3. Usig this trasform a α cofidece iterval for R is give by

4 AYMAN BAKLIZI 343 ta ta z ± α ( ( + ( (. ( Bai ad Hog s Itervals (BH itervals Ghosh ad Razmpour (984 showed that (,, Z is a complete sufficiet for ( θ, λ, µ ad that the joit probability desity fuctio of (, which is idepedet of Z is g ( t, t θ λ θ exp t = t t t + λ Γ( Γ( Γ( Γ( ( θt λt t., > + t, Usig stadard trasformatio techiques, it ca be show that the probability desity fuctio of the radom variable θ U = is give by (Bai ad Hog, θ + λ 99 g + ( u, π,, = πb( u,, ( π b( u,,, u. θ where π = ad θ + λ b ( u, r, s Γ = Γ ( r + s (( r Γ s u r s ( u, u. is the beta probability desity fuctio with parameters r ad s. Bai ad Hog (99 showed that a approximate α iterval for R is of the form k α t, α ( k + ( + α t kα t k α t k α t where t ad t are the observed values of ad respectively, ad k α is such that G ( kα, ˆ, π, = α. Here πˆ is a estimator of π obtaied by substitutig the maximum likelihood estimators of θ ad λ i the formula of π, ad G is the distributio fuctio of mixed beta radom variable U. Itervals Based o the Likelihood Ratio Statistic (LR Itervals he likelihood fuctio of ( θ, λ, µ is give by L ( θ, λ, µ θ = θ λ exp ( x µ λ ( y µ I ( z µ. i he likelihood ratio statistic for testig H : R = R is defied as (Bardorff-Nielse ad Cox, 994 W = ( l( Ω l( ϖ, where l( Ω is the log-likelihood fuctio evaluated at the values of the urestricted maximum likelihood estimator of ( θ, λ, µ. While l( ϖ is the log-likelihood fuctio evaluated at the values of the restricted maximum likelihood estimator uder the ull hypothesis. Recall that the urestricted maximum likelihood estimators are µˆ = z, ˆ θ =, ad ˆ λ =, where ( x i z ad ( y i z = i k = t. Uder the ull hypothesis H : R = R we fid readily R that λ = θ ad thus the maximum R likelihood estimator of θ is ~ + θ =. Substitutig i the formula R + R

5 344 CONFIDENCE INERVALS FOR P(X<Y of the likelihood ratio statistic ad simplifyig we get W ( R l + l R l + R = + l R + R he distributio of ( ( +. W R is χ (Bardorff- Nielse ad Cox, 994. he bouds of likelihood ratio cofidece itervals with ( α omial coverage probability are the two roots of W ( R = χα,, where χ α, is the upper α quatile of the chi square distributio with oe degree of freedom. Parametric Bootstrap Itervals he followig methods of derivig cofidece itervals are based o the Bootstrap approach (Efro & ibshirai, 993. hey are computer itesive methods based o resamplig with replacemet from the origial data ad the usig these Bootstrap samples to study the behaviour of estimators ad tests. Whe the parametric form of the distributio from which the data are geerated is kow except for some ukow parameters, we geerate from this distributio after its parameters are replaced by their estimates. he advatage of bootstrap methods is their wide applicability ad remarkable accuracy, especially i situatios where the traditioal methods do ot work. here are several Bootstrap based itervals discussed i the literature (Efro ad ibshirai, 993, the most commo oes are the bootstrap t iterval, the percetile iterval ad the bias corrected ad accelerated ( BC iterval. a he Bootstrap t Iterval Based o the MLE (BS Itervals Let be the maximum likelihood estimator of R ad let ˆR be the maximum likelihood estimator calculated from the bootstrap sample. Let z α be the α quatile of the bootstrap distributio of where ˆ ( R ˆ Z ( Vˆ =, V is estimated variace of calculated from the bootstrap sample. he bootstrap-t iterval is give by z Vˆ, z Vˆ α + α where z α is determied by simulatio. he Bootstrap t Iterval Based o the rasformed MLE (RBS Itervals Let be the maximum likelihood estimator of R ad let ˆR be the maximum likelihood estimator calculated from the bootstrap sample. Let z α be the α quatile of the bootstrap distributio of Q where ˆ ( R ˆ ( ta ta ˆ R ( Vˆ =, + V is estimated variace of calculated from the bootstrap sample. he bootstrap-t iterval is give by where q ˆ R + q α q ad α V α ( ( +, ( ( + V α bootstrap distributio of simulatio. q are the quatiles of the Q determied by

6 AYMAN BAKLIZI 345 he Percetile Iterval (PRC Iterval Here we simulate the bootstrap distributio of ˆR by resamplig repeatedly from the parametric model of the origial data ad calculatig, i i =,, B where B is the umber of bootstrap samples. Let Ĥ be the cumulative distributio fuctio of ˆR, the the α iterval is give by ˆ α, ˆ α H H. he Bias Corrected ad Accelerated Iterval ( BCa Iterval he bias corrected ad accelerated iterval is calculated also usig the percetiles of the bootstrap distributio of ˆR, but ot ecessarily idetical with the percetile iterval described i the previous subsectio. he percetiles deped o two umbers â ad ẑ called the acceleratio ad the bias correctio. he α iterval is give by ˆ ˆ G α, G α where ( ( ( zˆ + zα ( α = Φ z ˆ +, aˆ zˆ + zα zˆ + z α ( α = Φ z ˆ +, aˆ zˆ + z α Φ (. is the stadard ormal cumulative distributio fuctio, zα is the α quatile of the stadard ormal distributio. he values of â ad ẑ are calculated as follows; a ˆ = 6 (. ( i (. ( i 3 3 is the maximum likelihood estimator of R usig the origial data excludig the i-th observatio ad where ( i ( i (. =. he value of ẑ is give by zˆ { } # < B = Φ. Small Sample Performace of the Itervals For the cofidece itervals with omial cofidece coefficiet ( α, we use the criterio of attaimet of lower ad upper error probabilities which are both equal to α. Attaimet of lower ad upper omial error probabilities is importat because otherwise we will use a iterval with ukow error probabilities ad our coclusios therefore are imprecise ad ca be misleadig. Attaimet of omial error probabilities (assumed equal meas that if the iterval fails to cotai the true value of the parameter, it is equally likely to be above as to be below the true value. Users of two sided cofidece itervals expect the lower ad upper error probabilities to be symmetric because they are usig symmetric percetiles of the approximatig distributios to form their cofidece itervals. However, symmetry of error probabilities may ot occur due to the skewess of the actual samplig distributio Jeigs (987. Aother criterio for comparig cofidece itervals is their expected legths, obviously the shortest cofidece iterval amog itervals havig the same cofidece level is the best. We have simulated the expected legths of the three cosidered itervals. A simulatio study is coducted to ivestigate the performace of the itervals. he idices of our simulatios are:

7 346 CONFIDENCE INERVALS FOR P(X<Y (, = (,, (,,( 3,3, ( 4,4, (,4, ( 4,, (,4, ( 4, R : he true value of R=p(X<Y ad is take to be.5,.7,.9,.95. For each combiatio of, ad R, samples were geerated for X takig θ =, µ =, ad samples for Y with λ =, µ =. he itervals are calculated, R we used B = for bootstrap calculatios. he followig quatities are simulated for each iterval usig the results of the samples; the expected width of the iterval (W: he average of the widths of the itervals. Lower error rates (L: he fractio of itervals that fall etirely above the true parameter. Upper error rates (U: he fractio of itervals that fall etirely below the true parameter. otal error rates (: he fractio of itervals that did ot cotai the true parameter value. able : Simulated error rates ad expected legths of the itervals (, R AN RAN BH LR BS RBS PRC BCa (,.5 L U W L U W L U W L U W (,.5 L U W L U W L U W L U W (3, 3.5 L

8 AYMAN BAKLIZI 347 (, R AN RAN BH LR BS RBS PRC BCa U W L U W L U W L U W (4, 4.5 L U W L U W L U W L U W (, 4.5 L U W L U W L U W L U W

9 348 CONFIDENCE INERVALS FOR P(X<Y (, 4.5 L U W L U W L U W L U W (4,.5 L U W L U W L U W L U W (4,.5 L U W L U W L U W L U W

10 AYMAN BAKLIZI 349 Coclusio Our simulatios idicate that the performace of itervals based o asymptotic ormality (AN itervals are ot satisfactory eve for relatively large samples, they are quite ati-coservative i the sese that their coverage probabilities are ofte higher tha the omial cofidece level. Also they are quite asymmetric, especially for values of R far from.5. he performace of the itervals based o the trasformed maximum likelihood estimator (RAN itervals is about similar to that of AN itervals, but their aticoservativeess ad asymmetry beig slightly less severe tha AN itervals. Cocerig Bai ad Hog (BH itervals, they ofte attai the omial sizes but are asymmetric for values of R away from.5. O the other had, the Likelihood ratio (LR itervals attai the omial size ad are almost symmetric eve for small sample sizes. For the Bootstrap itervals, it appears that the bootstrap t itervals (BS ad (RBS are symmetric but ted to be coservative for small sample sizes, while the percetile iterval (PRC attais the omial level but teds to be asymmetric for values of R far from.5. he bias corrected ad accelerated iterval appear to be the best iterval based o the bootstrap priciple, they attai the omial level ad are symmetric i almost all situatios cosidered. With regard to iterval widths, our simulatio results suggest that all itervals have about equal performace. No itervals appear to be uiformly shorter or loger tha the others. Overall, the (BCa iterval appears to have the best performace accordig to the criteria of attaimet of coverage probability, symmetry ad expected legth followed by the (LR itervals. Although the other itervals (especially AN itervals are ati-coservative ad sometimes extremely asymmetric, which limit their usefuless, especially whe lower or upper cofidece bouds are desired. Refereces Bai, D. S. (99 Estimatio of p(x<y i the expoetial case with commo locatio parameter. Commuicatios i Statistics, (: Bardorff-Nielse, O., & Cox, D. R. (994. Iferece ad asymptotics. NY: Chapma ad Hall. Beg, M. A. (98. Estimatio of p(y<x for trucatio parameter distributios. Commuicatios i Statistics, 9, Efro, B., & ibshirai, R. (993. A itroductio to the bootstrap. New York: Chapma ad Hall. Ghosh, M., & Razmpour, A. (984 Estimatio of the commo locatio parameter of several expoetials. Sakhya, A46 : Gupta R. D., & Gupta R. C. (988. Estimatio of p ( Yp > Max( Y, Y,, Yp i the expoetial case. Commuicatios i Statistics, A7, Jeigs, D. (987. How do we judge cofidece itervals adequacy? he America Statisticia, 4(4, Johso, N., Kotz, S., & Balakrisha, N. (994. Cotiuous uivariate distributios. (Vol.. New York: Wiley.

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