Parameter Estimation in Marshall-Olkin Exponential Distribution Under Type-I Hybrid Censoring Scheme

Transcription

1 J. Stat. Appl. Po. 3, No. 2, Jounal of Statistics Applications & Pobability An Intenational Jounal Paamete Estimation in Mashall-Olkin Exponential Distibution Unde Type-I Hybid Censoing Scheme Sanjay Kuma Singh, Umesh Singh and Abhimanyu Singh Yadav Depatment of Statistics, DST-CIMS, Banaas Hindu Univesity, Vaanasi , India Received: 18 Jan. 2014, Revised: 12 Ap. 2014, Accepted: 16 Ap Published online: 1 Jul Abstact: The two most popula censoing schemes used in life testing expeiments ae Type-I and Type-II censoing schemes. Hybid censoing scheme is the mixtue of Type-I and Type-II censoing scheme. In this aticle, we conside the estimation of paametes of Mashall-Olkin exponential distibution based on Type-I Hybid censoed data. Both classical and Bayesian methodology have been discussed to estimate the model paametes. In classical set-up maximum likelihood estimatos MLEs of the paametes have been obtained by using Newton-Raphson method and also by using Fishe infomation matix 95% asymptotic confidence intevals ae povided. In Bayesian set-up Lindley s appoximation technique and Makov Chain Monte Calo MCMC technique have been used to compute the Bayes estimatos. Futhe, we have also povided highest posteio density HPD intevals of the paametes based on MCMC samples. To compae the pefomances of the estimatos Monte Calo simulation has pefomed and one data set is analysed fo illustative pupose of the study. Keywods: Hybid Censoing, Maximum likelihood estimato, Bayes estimato, Lindley s appoximation, Makov Chain Monte Calo Technique. 1 INTRODUCTION In Reliability/engineeing o medical sciences, geneally obsevations ae not completely known due to time and cost o inheent stuctue of the situations. Due to this cause censoing of the data can take place natually. Theefoe, the vaious censoing scheme can be classified by the expeimente depending upon the data obtaining pocesses. The two most common censoing schemes in life testing expeiments ae Type-I and Type-II censoing schemes and both censoing scheme have thei own advantages and disadvantages. The Type-I and Type-II censoing schemes, howeve, do not allow emoving suviving items at the times othe than the temination time of the life test. This allowance, howeve, may be desiable when a compomise between educing test time and an expectation of some exteme lifetimes in life test can be sought. Type-I censoing scheme contols the duation of the life test and the efficiency of the test may be too low due to numbe of failue items, wheeas Type-II censoing scheme contols the efficiency of the test but time of the test is uncetain. The mixtue of Type-I and Type-II censoing schemes, named as hybid censoing scheme and this censoing schemes have been widely discussed in the liteatue see Epstin [1]. The hybid censoing scheme is of two types namely Type-I hybid and Type-II hybid censoing scheme. In Type-I hybid censoing, the test is teminated at a time T 1 = minx k:n,t, whee X k:n epesents the failue time of the k th item and T is the pefixed maximum allowable time of the test. In hybid Type-II censoing, the test is teminated at a time T 1 = maxx k:n,t. It is clea that the test have at least k failue items in Type-II hybid censoing scheme whee as in Type-I hybid censoing scheme, the test can neve be eached beyond the time T. Type-I hybid censoing scheme is quite useful in eliability acceptance plan in MIL-STD-781C [10]. Epstein [1] poposed two-sided confidence intevals fo the paamete without any fomal poof. Faibanks et al. [2] modified poposition of Epstein [1] and suggested a simple methodology to obtain confidence intevals. Chen and Bhattachaya [3] have discussed the classical estimation fo the paametes. In Bayesian famewok Dappe and Guttmann [4] used this censoing scheme and poposed the Bayes estimation pocedue using the gamma pio. Seveal authos have discussed about the hybid censoing schemes, see, Gupta and Kundu [9] Ebahimi [5], Coesponding autho asybhu10@gmail.com

2 118 S. K. Singh et. al. : Paamete Estimation in Mashall-Olkin Exponential... Kundu [6], Kundu and Padhan [7], Gupta and Singh [16], Balakishnan N, Kundu D [22], S. Dey, B. Padhan [27], Kundu et al.[22], B. Al-zahani, M. Gindwan [25], Maheshwai et al.[26] and M. K. Rastogi, Y. M. Tipathi [27]. Unde Type-I hybid censoing scheme, we have one of the following two types of censoed data; Case I: {X 1:n < X 2:n < <X k:n } if X k:n < T Case II:{X 1:n < X 2:n < <X m:n } if m,x m+1:n > T In this pape, we have consideed the Type-I hybid censoed lifetime data when lifetime of each expeimental unit follows Mashall-Olkin exponential distibution MOED. The MOED was oiginally poposed by Mashall and Olkin in 1970 see [?]. This distibution is the genealization the exponential distibution. Fistly, G. Sinivasa Rao et al. have used this distibution fo making eliability test plan with sampling point of view see [12]. Since this distibution contain shape and scale paametes and it has vaious shape of hazad ate fo diffeent values of shape paamete α. Theefoe, sometimes it seems to be a good altenative to the gamma distibution, Weibull distibution and othe exponentiated family of distibutions. It has inceasing deceasing hazad function when α > 1α < 1 espectively and has constant hazad ate fo α = 1. Fo othe details about this distibution, we efe A. W. Mashall and I. Olkin [11]. The cumulative distibution function c.d.f. and pobability density function p.d.f of MOED ae given as; Fx,α,λ= 1 e λ x 1 ᾱe λ x ;x 0, α,λ > 0 1 fx,α,λ= αλ e λ x 1 ᾱe λ x 2 ; x 0, α,λ > 0 2 whee α is the shape paamete and λ is the scale paamete of the distibution and ᾱ = 1 α. The aim of this pape is to popose the classical and Bayesian estimation pocedues fo the unknown paametes unde Type-I hybid censoing scheme. It is obseved that the maximum likelihood estimatos can not be obtained diectly, theefoe, iteative pocedue like Newton-Raphson method has been used to solve the non-linea equations. To obtain the Bayes estimatos of the paametes using independent gamma pio fo both shape α and scale λ paametes, it is obseved that, the Bayes estimatos can not be obtained in closed fom. Theefoe, we used the Lindely s appoximation method to obtain the estimatos of the paametes. Futhemoe, we have also constucted 95% asymptotic intevals based on MLEs but we ae unable to constuct 95% highest posteio density HPD intevals using Lindely s method. The othe impotant appoximation like Makov Chain Monte Calo MCMC is used to compute the Bayes estimatos and coesponding highest posteio density HPD cedible intevals of the paametes. We have compaed the pefomances of the classical estimatos with coesponding Bayes estimatos by Monte Calo simulations and poposed pocedue is illustated by one eal data set. The est of the pape is oganized as follows; In section 1, we descibed the model and the available data. The maximum likelihood estimatos and asymptotic confidence intevals ae povided in sections 2. Bayes estimatos ae discussed in subsection of 2. Section 3 povided the illustation of the poposed pocedue by using eal data set. Simulation esults ae pesented in section 4. Finally, conclusions ae given in section 5. 2 ESTIMATION OF THE PARAMETERS 2.1 Classical Infeences fo the paametes In this subsection, we have consideed the maximum likelihood estimation of the paametes. Let us suppose that, X 1:n,X 2:n,,X n:n ae n independent odeed lifetime failue obsevations in pesence of Type-I hybid censoed samples fom MOEDα, λ. Theefoe, in this case the likelihood function fo the consideed cases ae; Case I: Lα,λ= n! n k! αn λ k 1 ᾱe λ x k:n n k e λ [ k x i +n kx k:n ] k 1 ᾱe λ x i 2 3 Case II: n! Lα,λ= n m! αn λ m 1 ᾱe λ T n m e λ [ m ] x i +n mt m 1 ᾱe λ x i 2 4

3 J. Stat. Appl. Po. 3, No. 2, / Thus combined likelihood can be witten as; Lα,λ= n! n! αn λ 1 ᾱe λt n e λ [ ] x i +n t 1 ᾱe λ x i 2 5 whee, and t ae defined as, { k = m f o casei f o caseii and t = { x k:n T f o casei f o caseii Now, the Log likelihood function can be expessed as; ] lnlx α,λ=nlnα+ lnλ λ[ x i +n t 2 ln1 ᾱe λ x n ln1 ᾱe λt 5.1 Theefoe, the MLEs of the paamete α and λ is the simultaneous solution of the following nomal equations. But we obseved that these expessions ae not in closed fom. Theefoe, MLEs can be secued though iteative pocedue. Hee we suggest to use Newton Raphson N-R method. n α 2 e λ x i 1 ᾱe λ x i n e λt 1 ᾱe λt = 0 and [ ] λ x i +n t 2 ᾱx i e λ x i 1 ᾱe λ x i n ᾱte λt 1 ᾱe λt = Inteval Estimation In this subsection, we will find the Fishe infomation matix fo constucting 95% asymptotic confidence inteval fo the paametes based on limiting s-nomal distibution. The Fishe infomation matix can be obtained by using equation 5.1. Thus we have 2 lnl α 2 2 lnl α λ I ˆα, ˆλ= 2 lnl λ α 2 lnl λ 2 ˆα,ˆλ whee, all the coesponding deivatives of the matix ae pesented in subsection of 2.2. The above matix can be inveted to obtain the estimate of the asymptotic vaiance-covaiance matix of the MLEs and diagonal elements of I 1 ˆα, ˆλ povides asymptotic vaiance of α and λ espectively. Then by using lage sample theoy the two sided 1001 β% appoximate confidence inteval fo α can be constucted as ˆα± Z 1 β/2 va ˆα and similaly, fo λ the two sided 1001 β% appoximate confidence inteval can be obtained as ˆλ ± Z 1 β/2 vaˆλ

4 120 S. K. Singh et. al. : Paamete Estimation in Mashall-Olkin Exponential Bayesian Infeences fo the paametes In this section, we have obtained the Bayes estimatos of the paametes α and λ based on Type-I hybid censoed data. In Bayesian analysis, we need to specify pio distibution fo the paametes, theefoe we conside two independent gamma pio such as, gammaa,b as a pio of α and gammac,d consideed as a pio of λ whee a,b,c and d ae the hypepaametes and is non negative. The motivation of consideing these pio is that, it is flexible in natue and mathematical ease. The impotant non-infomative pio is Jeffeys pio. It is to be mentioned hee that the above consideed pio educes in to non-infomative pio by taking the values of hype-paametes ae zeo. Theefoe, the joint pio foα,λ may be taken as ; Then by using equation 5 and the joint posteio is given as πα,λ α a 1 λ c 1 e bα dλ ; a,b,c and d pα,λ x=k 1 α n+a 1 λ +c 1 e [bα+λt0+d] Qα,λPα,λdαdλ Thus, the Bayes estimatos of α and λ unde SELF ae, ˆα B = Eα x,λ=k 1 α n+a λ +c 1 e [bα+λt0+d] Qα,λPα,λdαdλ ˆλ B = Eλ x,α=k 1 α λ α λ α n+a 1 λ +c e [bα+λt0+d] Qα,λPα,λdαdλ and whee; K,T 0,Qα,λ and Pα,λ ae intepeted as K = α n+a 1 λ +c 1 e [bα+λt0+d] Qα,λPα,λdαdλ and α λ T 0= x i +n t,qα,λ=1 ᾱe λt n Pα,λ= 1 ᾱe λ x i 2 Thus the posteio distibution ofα,λ takes a atio of the two integals. We may note that the above equation can not be educed in a closed fom and hence the evaluation of the posteio expectation fo obtaining Bayes estimatos of α andλ will be tedious. To ovecome such difficulties, we popose to use Lindley s appoximation and MCMC method to obtain Bayes estimatos unde squaed eo loss function. Lindely s Appoximation Method Bayes 1 : We conside the Lindley s appoximation technique fo the estimation of the α and λ. Conside that the posteio expectation is expessible in the fom of atio of integal as given below: uα,λe Lα,λ+Gα,λ dα,λ Ix=Eα,λ x= e Lα,λ+Gα,λ dα,λ 6 whee, uα,λ= is a function of α and λ only Lα, λ= Log- likelihood function Gα,λ= Log of joint pio density Accoding to D. V. Lindley [13], if ML estimates of the paametes ae available and n is sufficiently lage then the above atio of the integal can be appoximated as: Ix=u ˆα, ˆλ+ 1 2 [û λ λ + 2û λ ˆp λ ˆσ λ λ +û αλ + 2û α ˆp λ ˆσ αλ +û λ α + 2û λ ˆp α ˆσ λ α +û αα + 2û α ˆp α ˆσ αα ] [û λ ˆσ λ λ + û α ˆσ λ α ˆL λ λ λ ˆσ λ λ + ˆL λ αλ ˆσ λ α + ˆL αλ λ ˆσ αλ + ˆL ααλ ˆσ αα +û λ ˆσ αλ + û α ˆσ αα 7 ˆL αλ λ ˆσ λ λ + ˆL λ αα ˆσ λ α + ˆL αλ α ˆσ αλ + ˆL ααα ˆσ αα ]

5 J. Stat. Appl. Po. 3, No. 2, / whee ˆα and ˆλ ae the MLE of α and λ espectively. ˆp α = û α = u ˆα, ˆλ,û ˆα λ = G ˆα, ˆλ, ˆp ˆα λ = u ˆα, ˆλ u ˆα, ˆλ ˆλ,û αλ = ˆα ˆλ,û u ˆα, ˆλ λ α = ˆλ ˆα,û αα = 2 u ˆα, ˆλ ˆα 2 G ˆα, ˆλ ˆλ, ˆL αα = 2 L ˆα, ˆλ ˆα 2, ˆL λ λ = 2 L ˆα, ˆλ ˆλ 2,û λ λ = 2,u ˆα, ˆλ ˆλ 2, ˆL ααα = 3 L ˆα, ˆλ ˆα 3, ˆL ααλ = 3 L ˆα, ˆλ ˆα ˆα ˆλ ˆL λ λ α = 3 L ˆα, ˆλ ˆλ ˆλ ˆα, ˆL λ αλ = 3 L ˆα, ˆλ ˆλ ˆα ˆλ, ˆL ααλ = 3 L ˆα, ˆλ ˆα ˆα ˆλ, ˆL αλ λ = 3 L ˆα, ˆλ ˆα ˆλ ˆλ, ˆL λ αα = 3 L ˆα, ˆλ ˆλ ˆα ˆα Afte substitution of pα, λ x fom in above equation 6 then this integal must be educes like Lindleys integal, whee: uα,λ=α ] Lα,λ= nlnα+ lnλ λ[ x i +n t 2 ln1 ᾱe λ x i n ln1 ᾱe λt Gα,λ=a 1lnα+c 1lnλ bα+ dλ it may veified that, u α = 1, u αα = u λ λ = u αλ = u λ α = 0, p α = a 1 L α = n α 2 L αα = n α e λ x i n e λt 1 ᾱe λ x i 1 ᾱe λt e 2λ x i α 1 ᾱe λ x 2 + n e 2λt i 1 ᾱe λt 2, b, p λ = c 1 λ d L αλ = L λ α = 2x i e λ x i + 2ᾱx i e 2λ x i 1 ᾱe λ x i 1 ᾱe λ x 2 + n te λt i 1 ᾱe λt + n ᾱte 2λt 1 ᾱe λt 2 L ααα = 2n α 3 4 e 3λ x i L λ = λ x i +n t L λ λ = λ L λ λ λ = 2n λ 3 2 4t 3 n ᾱ 3 e 3λt 1 ᾱe λt 3 1 ᾱe λ x 3 2 n e 3λt i 1 ᾱe λt 3 2 x i ᾱe λ x i x 2 i ᾱe λ x i 1 ᾱe λ x i + 2 x 3 i ᾱe λ x i 2tᾱn e λ x i 1 ᾱe λ x i 1 ᾱe λt 1 ᾱe λ x i 6 L ααλ = L λ αα = 4 x i e 2λ x i 1 ᾱe λ x 2 4 i L αλ λ = L λ λ α = 2 4t 2 n ᾱ 2 e 3λt 1 ᾱe λt 3 x 2 i e λ x i 1 ᾱe λ x i 6 x 2 i ᾱ2 e 2λ x i 1 ᾱe λ x 2 + 2t2 n ᾱe λt i 1 ᾱe λt + 2t2 n ᾱ 2 e 2λt 1 ᾱe λt 2 x 3 i ᾱ2 e 2λ x i 1 ᾱe λ x 2 4 x 3 i ᾱ3 e 3λ x i i 1 ᾱe λ x 3 2t3 n ᾱe λt i 1 ᾱe λt 6t3 n ᾱ 2 e 2λt 1 ᾱe λt 2 x i ᾱe 3λ x i 1 ᾱe λ x 3 4n te 2λt i 1 ᾱe λt 2 4n tᾱe 3λt 1 ᾱe λt 3 x 2 i ᾱe 2λ x i 1 ᾱe λ x i 2 4 If α and λ ae othogonal then σ i j = 0 fo i j and σ i j = 1 L i j x 2 i ᾱ2 e 3λ x i 1 ᾱe λ x 3 2t2 n e λt i 1 ᾱe λt 6t2 n ᾱe 2λt 1 ᾱe λt 2 fo i = j. Afte evaluation of all U-tems, L-tems, and p- tems at the point ˆα, ˆλ and using the above expession, the appoximate Bayes estimato of α unde SELF is, ˆα B = ˆα+ û α pˆ α ˆσ αα û α ˆσ αα ˆσ λ λ ˆL αλ λ + û α ˆσ 2 αα ˆL ααα 8 and similaly the Bayes estimate fo λ unde SELF is, u λ = 1, u αα = u λ λ = u αλ = u λ α = 0 and emaining L-tems and -tems will be same as above, thus we have, ˆλ B = ˆλ + û λ pˆ λ ˆσ λ λ û λ ˆσ 2 ˆL λ λ λ λ λ + û λ ˆσ αα ˆσ λ λ ˆL ααλ 9

6 122 S. K. Singh et. al. : Paamete Estimation in Mashall-Olkin Exponential... Makov Chain Monte Calo method Bayes 2 : The expession of posteio distibution can not be expessed in any standad fom theefoe, numeical technique is needed fo summaising its chaacteistics. In such a situation, the most appopiate MCMC methods namely Gibbs sample and Metopolis-Hastings Algoithm can be effectively used to obtain the Bayes estimates and highest posteio density HPD cedible intevals of the paametes. Fo moe detail about MCMC method see [19], [20], [21] and [28]. Fo implementing the Gibbs algoithm, the full conditional posteio densities of α and λ ae given by; p 1 α x,λ α n+a 1 e bα Qα,λPα,λ 10 and p 2 λ x,α λ +c 1 e λt0+d Qα,λPα,λ 11 whee T 0,Pα,λ,Qα,λ ae defined same as above. The algoithm consist the following steps Set the initial values of α and λ sayα 0,λ 0 Set j=1 Geneate posteio sample fo α and λ fom 10 and 11 espectively. Repeat step 2, fo all j = 1,2,3, N and obtainedα 1,λ 1,α 2,λ 2,,α N,λ N Afte obtaining the posteio samples the Bayes estimate of the paametes unde SELF ae the mean of the posteio samples. Theefoe we have, ˆα B Eα x= 1 N ˆλ B Eλ x= 1 N N α j j=1 N λ j j=1 Afte extacting the posteio samples we can easily constuct the HPD cedible intevals fo α and λ. Theefoe fo this pupose ode α 1,α 2,...,α N as α 1 < α 2 <...<α N and λ 1,λ 2,,λ N as λ 1 < λ 2 <...<λ N. Then 1001 β% cedible intevals of α and λ ae α 1,α [N1 β+1],,α [Nβ],α N and λ 1,λ [N1 β+1],,λ [Nβ],λ N Hee[x] denotes the geatest intege less than o equal to x. Then, the HPD cedible inteval is that inteval which has the shotest length. 3 REAL DATA ANALYSIS In this section, to illustate ou discussed methodology, we have consideed a data set epesenting the failue times of the elease of softwae given in tems of hous with aveage life time be 1000 hous fom the stating of the execution of the softwae. The consideed eal data set have taken fom A. Wood [18]. We have checked the suitability of the consideed distibution to this eal data set ove moe popula life-time models namely exponential, exponentiated exponential and gamma distibution see Singh et al. see [17]. The empiical cdf plot fo consideed eal data set is plotted in Figues 1 and one can easily conclude that the Mashall-Olkin exponential distibution gives bette fits than the above fou competitive distibutions. Fo analysing of this data set unde Type-I hybid censoing scheme,we have consideed six censoing schemes as; Scheme1 : R=8,T = 5.0, Scheme2 : R=10,T = 6.0, Scheme3 : R=12,T = 6.0, Scheme4 : R=14,T = 7.0, Scheme5 : R = 15,T = 8.0 and Scheme6 : R = 16,T = 9.0 Fo all censoing schemes, the maximum likelihood estimates and the Bayes estimates along with 95% confidence intevals CI and HPD cedible intevals ae obtained and pesented in Table 6. Fo obtaining the Bayes estimates fo eal data set, non-infomative pio is consideed.

7 J. Stat. Appl. Po. 3, No. 2, / Fig. 1: The empiical cumulative distibution function ECDF plot fo the consideed eal data set. 4 NUMERICAL COMPARISON In pevious section, we have obtained the estimatos fo the unknown paametes of the consideed model and we notice that the the estimatos ae not in closed fom. Thus, we numeically evaluate the isk mean squae eo values of all estimatos and examined the pefomance of Bayes estimatos with coesponding ML estimatos on the basis of simulated samples fom MOED. Since isk of the estimatos ae the function of n,, T, α, λ and hype paametes a, b, c and d. To study the pefomance of the estimatos, we have geneated a andom sample of size n=30,40 and 50 fo fixed values of α = 3 and λ = 2. In addition, we have consideed the value of hype paamete asa=b=c=d= 0 fo non-infomative pio Pio 0 and a = 18,b = 6,c = 8,d = 4 fo infomative pio Pio 1. In ode to conside hybid censoing scheme, we have chosen diffeent combinations of the censoing paametes [60%, 80%, 90%], T[1.5, 2.5, 3.5]. The simulation esults ae summaised in Tables 1-3. Fom this extensive study, we have obseved the following on the basis of Tables 1-3. The isk of the Bayes estimatos ae smalle than the isk of the maximum likelihood estimatos unde Pio 1. But the isk of Bayes estimatos and ML estimatos ae quite simila unde Pio 0 in all consideed choices of n,r,t see Tables [1-3]. The MSE s of the Bayes as well as ML estimatos ae deceases as pecentage of inceases fo given values of n, α and λ unde both pio i.e Pio 1 and Pio 0. The both consideed appoximation method i.e. Linley s and MCMC woks quite well but in between these two method we do not have any specific tend. The width of the HPD intevals ae much lesse than the length of asymptotic intevals and width of the intevals unde Pio 1 is less than Pio 0 see Table [4-5]. 5 CONCLUSION In this pape, we poposed the classical as well as Bayesian estimation of the paametes of Mashall-Olkin exponential distibution unde Type-I hybid censoed data. The maximum likelihood estimates MLE s of the paametes have been obtained by using N-R method. The Bayes estimatos of the paametes using Lindely s and MCMC method have been discussed. On the basis of compaison of the isk of the estimatos, we obseved that Bayes estimatos pefoms bette than ML estimatos unde both consideed pio. It was also noted that, the length of the HPD intevals ae smalle than the length of asymptotic inteval of the paametes. Fom the discussion mentioned as above, we may conclude that the Bayes estimatos obtained unde Lindley o MCMC method can be ecommended fo thei use.

8 124 S. K. Singh et. al. : Paamete Estimation in Mashall-Olkin Exponential... Table 1: Estimates of the paametes and coesponding mean squae eo MSEs ae coded unde Pio 1 and Pio 0 when T=1.5 n Paamete MLE Pio 1 Pio 0 MSE Bayes1 MSE Bayes2 MSE Bayes1 MSE Bayes2 MSE 18 α λ α λ α λ α λ α λ α λ α λ α λ α λ Table 2: Estimates of the paametes and coesponding mean squae eo MSEs ae coded unde Pio 1 and Pio 0, when T=2.5 n Paamete MLE Pio 1 Pio 0 MSE Bayes1 MSE Bayes2 MSE Bayes1 MSE Bayes2 MSE 18 α λ α λ α λ α λ α λ α λ α λ α λ α λ ACKNOWLEDGEMENTS The authos ae thankful to the edito and the efeees fo thei valuable comments and suggestions egading the impovement of ou manuscipt. The authos ae also thankful to the Depatment of Statistics and DST-CIMS, B.H.U., Vaanasi, fo poviding the computational facilities. The thid autho Abhimanyu Singh Yadav is gateful to UGC, New Delhi,India, fo poviding financial assistance.

9 J. Stat. Appl. Po. 3, No. 2, / Table 3: Estimates of the paametes and coesponding mean squae eo MSEs ae coded unde unde Pio 1 and Pio 0, when T=3.5 n Paamete MLE Pio 1 Pio 0 MSE Bayes1 MSE Bayes2 MSE Bayes1 MSE Bayes2 MSE 18 α λ α λ α λ α λ α λ α λ α λ α λ α λ Table 4: 95% asymptotic and HPD confidence Intevals when infomative Pio Pio 1 is consideed. T n R Asymptotic Inetvals Bayes Intevals α L α U Length λ L λ U Length α L α U Length λ L λ U Length

10 126 S. K. Singh et. al. : Paamete Estimation in Mashall-Olkin Exponential... Table 5: Asymptotic and HPD confidence inteval fo the paametes in the case of non-infomative pio Pio0 T n R Asymptotic Inetvals Bayes Intevals α L α U Length λ L λ U Length α L α U Length λ L λ U Length Table 6: Estimates of the paametes and coesponding inteval estimates based on eal data set Schemes MLE Bayes1 Bayes2 Asymptotic Intevals HPD Intevals α λ α λ α λ α L α U λ L λ L α L λ U λ L λ U I II III IV V VI Refeences [1] Epstein.B, Tuncated life tests in the exponential case, Ann. Math. Statist.,25, ,1954. [2] Faibanks.K, Madson.R, Dyksta.R, A confidence inteval fo an exponential paamete fom a hybid life test, J. Ame. Statist. Assoc., 77, , [3] Chen.S, Bhattachaya.G.K, Exact confidence bounds fo an exponential paamete unde hybid censoing, Comm. Statist. Theo. Meth., 17, , [4] Dape.N, Guttman.I, Bayesian analysis of hybid life tests with exponential failue times, Ann. Inst. Statist. Math., 39, [5] Ebahimi.N, Estimating the paamete of an exponential distibution fom hybid, life test,j. Statist. Plann. Infeence., 23, , [6] Ebahimi.N 1990 Estimating the paamete of an exponential distibution fom hybid life test,j. Statist. Plann. Infeence., 23, [7] Kundu.D, On hybid censoed Weibull distibution, J. Statist. Plann. Infeence.,137, , [8] Kundu.D, Padhan.B, Estimating the paametes of the genealized exponential distibution in pesence of hybid censoing, Comm. Statist. Theo. Meth., 38, , 2009.

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