Classical and Bayesian Inference for an Extension of the Exponential Distribution under Progressive Type-II Censored Data with Binomial Removals

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1 J. Stat. Appl. Pro. Lett. 1, No. 3, (2014) 75 Journal of Statistics Applications & Probability Letters An International Journal Classical and Bayesian Inference for an Extension of the Exponential Distribution under Progressive Type-II Censored Data with Binoial Reovals S. K. Singh 1, U. Singh 1, M. Kuar 2, and P. K. Vishwakara 1 1 Departent of Statistics, Banaras Hindu University, Varanasi , India 2 Departent of Statistics and DST-CIMS, Banaras Hindu University, Varanasi , India Received: 27 Apr. 2014, Revised: 2 Jun. 2014, Accepted: 5 Jun Published online: 1 Sep Abstract: Maxiu likelihood and Bayes estiators of the unknown paraeters of an extension of the exponential (EE) distribution have been obtained for Progressive Type-II Censored data with Binoial reovals. Markov Chain Monte Carlo (MCMC) ethod is used to copute the Bayes estiates of the paraeters of interest. The General Entropy Loss Function (GELF) and Squared Error Loss Function (SELF) have been considered for obtaining the Bayes estiators. Coparisons are ade between Bayesian and Maxiu likelihood estiators (MLEs) via Monte Carlo siulation. An exaple is discussed to illustrate its applicability. Keywords: Maxiu likelihood, Bayes estiator, Progressive Type-II Censored data with Binoial reovals 1 Introduction In life testing and reliability probles, the role of hazard rate is very iportant because any phenoenon in real situation are odeled by the probability distribution. In early age, exponential distribution was the ost popular distribution and has been frequently used to analyze the life tie data due to their constant hazard rate and coputational ease. In real situation, constant hazard rate does not occurs coonly but it occurs in onotonic or non onotonic for as for exaple ortality of child with their age distribution and failure of electric products with respect to tie etc, see Nelson [15], Lawless [14] and Barlow and Proschan [13]. Initially gaa and Weibull distribution have been proposed as a generalization of exponential distribution and extensively used for the situation when hazard rate is not constant. But both distributions have their own advantages and disadvantages see Murthy et al. [1]. Considering disadvantages of gaa distribution, Gupta and Kundu [10] proposed a new exponentiated exponential distribution as an alternative to gaa distribution and has any property like gaa distribution with addition to closed for of distribution function and hazard function. For ore details see Gupta and Kundu [10]. In the sae context, Haghighi and Sadeghi [4] proposed EE distribution which is an alternative to gaa, Weibull and exponentiated exponential distribution and having additional iportant feature of an increasing hazard function when their respective probability density functions are onotonically decreasing. However, gaa, Weibull and exponentiated exponential distribution only allow for decreasing or constant hazard rate when their respective probability density functions are onotonically decreasing. The ore applicability of EE distribution has discussed by Haghighi and Sadeghi [4] and Nadarajah and Haghighi [18]. The survival function of EE distribution is given as S(x)=exp[1 (1+λ x) α ], (1) for α > 0, λ > 0 and x > 0. The corresponding cuulative distribution function (cdf), probability density function (pdf) and the quantile function are given as. F(x)=1 exp[1 (1+λ x) α ], (2) Corresponding author e-ail: anustats@gail.co

2 76 S. K. Singh et. al. : Classical and Bayesian Inference for an Extension of the... and f(x)=αλ(1+λ x) α 1 exp[1 (1+λ x) α ], (3) respectively. The hazard function (hrf) is given by Q(p)= 1 { λ (1 log(1 p)) 1/α },0< p<1, h(x)=αλ(1+λ x) α. (4) Now, For α = 1, equation (3) reduced to exponential distribution (see, Nadarajah [18]). Equation (3) has showed the attractive feature of always having the zero ode and yet allowing for increasing, decreasing and constant hrfs. Haghighi and Sadeghi [4] and Nadarajah [18] have been obtained the MLEs for coplete as well as censored case but none of the has paid attention for Bayes analysis under Progressive type-ii censoring with Binoial reovals. But now a days Progressive type-ii censoring with Binoial reovals becoes very popular and practicable in edical and engineering field. In life testing experients, situations do arise when units are lost or reoved fro the experients while they are still alive; i.e, we get censored data fro the experient. The loss of units ay occur due to tie constraints giving type- I censored data. In such censoring schee, experient is terinated at specified tie. Soeties, the experient is terinated after a prefixed nuber of observations due to cost constraints and we get type-ii censored data. Besides these two controlled causes, units ay drop out of the experient randoly due to soe uncontrolled causes. For exaple, consider that a doctor perfor an experient with n cancer patients but after the death of first patient, soe patient leave the experient and go for treatent to other doctor/ hospital. Siilarly, after the second death a few ore leave and so on. Finally the doctor stops taking observation as soon as the predeterined nuber of deaths (say ) are recorded. It ay be assued here that each stage the participating patients ay independently decide to leave the experient and the probability (p) of leaving the experient is sae for all the patients. Thus the nuber of patients who leave the experient at a specified stage will follow Binoial distribution with probability of success (p). The experient is siilar to a life test experient which starts with n units. At the first failure X 1,r 1 (rando) units are reoved randoly fro the reaining (n 1) surviving units. At second failure X 2,r 2 units fro reaining n 2 r 1 units are reoved, and so on; till th failure is observed i.e. at th failure all the reaining r = n r 1 r 2 r 1 units are reoved. Note that, here, is pre-fixed and r, is are rando. Such a censoring echanis is tered as Progressive type-ii censoring with Binoial reovals. If we assue that probability of reovals of a unit at every stage is p for each unit then r i can be considered to follow a Binoial distribution i.e., r i B(n i 1 l=0 r l, p) for,2,3, 1 and with r 0 = 0. For further details, readers are referred to Balakrishnan [2] and Singh et al.[8]. In last few years, the estiation of paraeters of different life tie distribution based on Progressive censored saples have been studied by several authors such as Childs and Balakrishnan [6], Balakrishnan and Kannan [3], Mousa and Jheen [17], Ng et al. [20]. The Progressive type-ii censoring with rando reovals has been considered by Yang et al. [22] for Weibull distribution, Wu and Chang [26] for exponential distribution. Under the Progressive type-ii censoring with rando reovals, Wu and Chang [27] and Yuen and Tse [28] developed the estiation proble for the Pareto distribution and Weibull distribution respectively, when the nuber of units reoved at each failure tie has a discrete unifor distribution, the expected tie of this censoring plan is discussed and copared nuerically. In this paper, we have proposed Bayes estiators for the two paraeter EE based on Progressive type-ii censoring with Binoial reovals. Bayes estiators are obtained under SELF and GELF. Rest of the paper is organized as follows: Section 2, provides the likelihood function. In section 3, MLE and Bayes estiators have been obtained. MCMC ethod is used to copute Bayes estiates of α and λ. The coparison of MLEs and corresponding Bayes estiators are given in section 4. Coparisons are based on siulation studies of risk (average loss over saple space) of the estiators. Section 5, illustrate an exaple by using the real data set. Finally, conclusions are presented in the last section. 2 Likelihood Function Let (X 1,R 1 ),(X 2,R 2 ),(X 3,R 3 ),,(X,R ), denote a Progressive type-ii censored saple with Binoial reovals, where X 1 < X 2 < X 3,,X. With pre-deterined nuber of reovals, say R 1 = r 1,R 2 = r 2,R 3 = r 3,,R = r, the conditional likelihood function can be written as, Cohen [7] L(α;λ ;x R= r)=c f (x i )[S(x i )] r i, (5)

3 J. Stat. Appl. Pro. Lett. 1, No. 3, (2014) / 77 where c = n(n r 1 1)(n r 1 r 2 2)(n r 1 r 2 r 3 3) (n r 1 r 2 r 3,,r +1), and 0 r i (n r 1 r 2 r 3 r i 1 ), for,2,3..., 1. Substituting(1) and(3) into(5), we get L(α,λ ;x R=r)=c αλ(1+λ x i ) α 1 [exp[1 (1+λ x i ) α ]] r i+1 (6) Suppose that an individual unit being reoved fro the test at the i th failure, i = 1,2, ( 1) is independent of the others but with the sae probability p. That is no. R i of the unit reoved at i th failure i = 1,2, ( 1) follows a Binoial distribution with paraeters ( n i 1 l=1 r l, p ) therefore, and for i=2,3,, 1, ( ) n P(R 1 = r 1 ; p)= p r 1 (1 p) n r 1, (7) r 1 P(R; p)=p(r i = r i R i 1 = r i 1, R 1 = r 1 ) ( n i 1 = l=0 r ) l p r i (1 p) n i 1 l=0 r l. r i (8) Now, we further assue that R i is independent of X i for all i. Then using above equations, we can write the full likelihood function as in the following for L(α,λ, p;x,r)=al 1 (α,λ)l 2 (p), (9) where L 1 (α;λ)= αλ(1+λ x i ) α 1 [exp[1 (1+λ x i ) α ]] r i+1, (10) and A= L 2 (p)= p 1 r i (1 p) ( 1)(n ) 1 ( i)r i. (11) c (n )!, does not depend on the paraeters α,λ and p. (n i 1 l=1 r i)! 1 r i! 3 Classical and Bayesian Estiation of Paraeters 3.1 Maxiu Likelihood Estiation The MLE of α and λ are the siultaneous solution of following noral equations and α + ln(1+λ x i ) x i λ +(α 1) α 1+λ x i (1+r i )(1+λ x i ) α ln(1+λ x i )=0 (12) x i (1+r i )(1+λ x i ) α 1 = 0. (13) It ay be noted that (12) and (13) can not be solved siultaneously to provide a nice closed for for the estiators. Therefore, we use fixed point iteration ethod for solving these equations. For details about the proposed ethod readers ay refer Jain et al. [16] and Rao [21].

4 78 S. K. Singh et. al. : Classical and Bayesian Inference for an Extension of the Bayes procedure Since the paraeters α and λ both are unknown, a natural choice for the prior distributions of α and λ are independent gaa distributions as the following fors(14) and(15). g 1 (α)= b 1 a 1 e b 1α α a 1 1 Γ a 1 ; 0<α <, b 1 > 0, a 1 > 0 (14) g 1 (λ)= b 2 a 2 e b 2λ λ a 2 1 Γ a 2 ; 0<λ <, b 2 > 0, a 2 > 0 (15) where a 1,b 1, and a 2,b 2, are chosen to reflect prior knowledge about α and λ. It ay be noted that, the gaa prior g 1 (α) and g 2 (λ) are chosen instead of the exponential prior of α and λ were used by Nassar and Eissa [19], Jung et al. [11] and Singh et al. [9] because the gaa prior is wealthy enough to cover the prior belief of the experienter. Thus the joint prior pdf of α and λ is g(α,λ)=g 1 (α)g 2 (λ) ; α > 0, λ > 0 (16) Cobining the priors given by (14) and (15) with likelihood given by (9), we can easily obtain joint posterior pdf of (α,λ) as π(α,λ x,r)= J 1 J 0 where { } J 1 = α +a1 1 λ +a2 1 e b1α e b 2λ (1+λx i ) α 1 [exp[1 (1+λx i ) α ]] r i+1, (17) and J 0 = 0 0 J 1 dαdλ. Hence, the respective arginal posterior pdfs of α and λ are given by and J π 1 (α x,r)= 1 dλ, (18) 0 J 0 J 1 π 2 (λ x,r)= dα. (19) 0 J 0 Usually the Bayes estiators are obtained under SELF l 1 (φ, ˆφ)= 1 ( φ ˆφ ) 2 ; 1 > 0 (20) Where ˆφ is the estiate of the paraeter φ and the Bayes estiator ˆφ S of φ coes out to be E φ [φ], where E φ denotes the posterior expectation. However, this loss function is syetric loss function and can only be justified, if over estiation and under estiation of equal agnitude are of equal seriousness. A nuber of asyetric loss functions are also available in statistical literature. Let us consider the GELF, proposed by Calabria and Pulcini [5], defined as follows : ( ( ) δ ( ) ) ˆφ ˆφ l 2 (φ, ˆφ)= 2 δ ln 1 ; 2 > 0 (21) φ φ The constant δ, involved in(21), is its shape paraeter. It reflects departure fro syetry. When δ > 0, it considers over estiation (i.e., positive error) to be ore serious than under estiation (i.e., negative error) and converse for δ < 0. The Bayes estiator ˆφ E of φ under GELF is given by, ˆφ E = [E φ (φ δ)] ( δ) 1 (22) provided the posterior expectation exits. It ay be noted here that for δ = 1, the Bayes estiator under loss (21) coincides with the Bayes estiator under SELF l 1. Expressions for the Bayes estiators ˆα E and ˆλ E for α and λ respectively under GELF can be given as and [ ] ( 1 δ) αˆ E = α δ π 1 (α x,r)dα, (23) 0 [ ] ( 1 λˆ E = λ δ δ ) π 1 (λ x,r)dλ, (24) 0

5 J. Stat. Appl. Pro. Lett. 1, No. 3, (2014) / 79 It is to ention here that fro equation (23) and (24), the Bayes estiators ˆα E and ˆλ E are not reducible in nice closed for. Therefore, we use the nuerical techniques for obtaining the estiates. We propose to use the MCMC ethod for obtaining the Bayes estiates of the paraeters. In MCMC technique, Gibbs sapler and Metropolis-Hastings algorith to generate saples fro posterior distributions and copute the Bayes estiates. The Gibbs sapler is best applied on probles where the arginal distributions of the paraeters of interest are difficult to calculate, but the conditional distributions of each paraeter given all the other paraeters and data have nice fors. If conditional distributions of the paraeters have standard fors, then they can be siulated easily. But generating saples fro full conditionals corresponding to joint posterior is not easily anageable, therefore we consider ixing of Metropolis-Hastings for those full conditional in the hybrid sapling i.e., Metropolis step is used to extract saples fro soe of the full conditional to coplete a cycle in Gibbs chain. For ore details about this ethod, see Chib and Greenberg [24], Gelfand and Sith [23] and Gaeran and Lopes[25]. Thus utilizing the concept of Gibbs sapling procedure as entioned above, generate saple fro the posterior density function under the assuption that paraeter α and λ has independent gaa density function with hyper paraeters a 1,b 1, and a 2,b 2, respectively. To corporate this technique we consider full conditional posterior densities of α and λ, π 1 (α λ,x,r) α+a 1 1 e b 1α { } (1+λx i ) α [exp[1 (1+λx i ) α ]] r i+1 (25) and { } π2 (λ α,x,r) λ +a2 1 e b 2λ (1+λx i ) 1 [exp[1 (1+λx i ) α ]] r i+1 (26) respectively. The Gibbs algorith consist the following steps I. Set the initial guess of α and λ say α 0 and λ 0 II. Set III. Generate α i fro π1 (α λ i 1,x,r) and λ i fro π2 (λ α i,x,r) IV. Repeat steps II-III, N ties V. Obtain the Bayes estiates of α and λ under GELF as [ ] 1 [ ] δ 1 1 δ ˆα E = E(α δ data) = [ ] 1 ˆλ E = E(λ δ δ data) = N N 0 N i=n 0 +1 α δ i [ 1 N N 0 N i=n 0 +1 λ δ i ] 1 δ. and Where, (N ) is the burn-in-period of Markov Chain. Substituting δ equal to -1 in step V, we get Bayes estiates of α and λ under SELF. VI. To copute the HPD interval of α and λ, order the MCMC saple of α and λ (say α 1,α 2,α 3,,α N as α [1],α [2],α [3],,α [N] ) and (λ 1,λ 2,λ 3,,λ N as λ [1],λ [2],λ [3],,λ [N] ). Then construct all the 100(1-ψ)% credible intervals of α and λ say ((α [1],α [N(1 ψ)+1] ),,(α [Nψ],α [N] )) and ((λ [1],λ [N(1 ψ)+1] ),, (λ [Nψ],λ [N] )) respectively. Here [x] denotes the largest integer less than or equal to x. Then the HPD interval of α and λ are that interval which has the shortest length. VII. Using the asyptotic norality property of MLEs, we can construct approxiate 100(1-ψ)% confidence intervals for α and λ as ˆα± z ψ/2 ( var( ˆ ˆα)) and ˆλ ± z ψ/2 ( var(ˆλ)) ˆ Where z ψ/2 is the 100(1 ψ/2)% upper percentile of standard noral variate. 4 Siulation Study The estiators ˆα M and ˆλ M denote the MLEs of the paraeters α and λ respectively, while ˆα S and ˆλ S are corresponding Bayes estiators under SELF and ˆα E and ˆλ E are the corresponding Bayes estiators under GELF. We copare the estiators obtained under GELF with corresponding Bayes estiators under SELF and MLEs. The coparisons are based on the siulated risks (average loss over saple space) under GELF and SELF both. Here, ((α L c α U c), (λ L c λ U c)) and ((α L h α U h), (λ L h λ U h)) represent 100(1 ψ)% CI and HPD intervals of α and λ respectively. It ay be entioned here that the exact expressions for the risks can not be obtained as estiators are not found in nice closed for. Therefore, the risks of the estiators are estiated on the basis of Monte-Carlo siulation study of 5000 saples. It ay be noted that the risks of the estiators will depend on values of n,, p, α, λ and δ. Also, the choice of hyper paraeter α and λ can be taken in such a way that if we consider any two independent inforations as prior ean and variance of α and λ are(µ 1 = a 1 b 1,σ 1 = a 1) and(µ b 2 2 = a 2 b 1 2,σ 2 = a 2) respectively, whereas µ b 2 1 and µ 2 are considered as true values of the paraeters 2 α and λ for different confidence in ters of saller and larger variances. In order to consider variation in the values of these, we have obtained the siulated risks for effective saples = 15,18,21 and 27, α = 2 = µ 1 (say, prior ean of α), σ 1 = 1,10 (say, prior variance of α) and δ =±4. Siilarly, these variation is apply on the scale paraeter λ = 3= µ 2 (say, prior ean of λ), σ 2 = 1,10 (say, prior variance of λ) and δ =±4. Figure 1 & 2 shows the risks of an estiators of α and λ for different values of δ under GELF and Figure 3 6 shows the risk of estiators of α and λ for variation of the effective saple size, where the other rest of the paraeters are fixed, which is ention under the Figures. Table 1 & 2 represent the CI, HPD intervals and percentage of coverage probability in all considered situation. It is to be ention here that considered the value of hyper paraeters such as prior ean is taken as guess value

6 80 S. K. Singh et. al. : Classical and Bayesian Inference for an Extension of the... Fig. 1: Risks of Estiators of α and λ under GELF for different values of δ. Table 1: Under saller prior variance σ 1 = 1 and σ 2 = 1 the 95% CI, HPD intervals and % of coverage probability for different saples for fixed n=30,α = 2,λ = 3,a 1 = 4,a 2 = 9,b 1 = 2 and b 2 = 3. α λ α L c α U c α L h α U h % cov.prob λ L c λ U c λ L h λ U h %cov.prob of the paraeters α and λ, when prior variance is sall and large respectively. Fro Table 1 & 2, it is observed that HPD intervals are shorter length than CI and length of the intervals decreases as increent of the effective saple size and also observed that, there is increent in coverage probability of CI and HPD. 5 Real data Analysis For real data illustration, we have taken the following data fro Linhart and Zucchini [12] which shows failure ties of the air conditioning syste of an airplane: 23, 261, 87, 7, 120, 14, 62, 47, 225, 71, 246, 21, 42, 20, 5, 12, 120, 11, 3, 14, 71, 11, 14, 11, 16, 90, 1, 16, 52, 95. We have obtained Kologrov-Sirnov (K-S) statistics, Akaike s inforation criterion (AIC) and Bayesian inforation criterion (BIC) for EE, Weibull, gaa and exponentiated exponential distributions for given data set and the values are suarized in Table 3. Considered criterion, we observed that EE distribution provide better fit than the other three distributions. Hence, EE odel can be considered as an alternative to all three odels. Therefore, we use this data to illustrate the our propose procedures. For this a Progressive type-ii censoring with Binoial reovals are generated fro the given data set under various schees, which are suarized in Table 4. We have obtained the MLEs, Bayes estiates (using non inforative prior), 95% CI and HPD intervals for the paraeters α and λ respectively under SELF and GELF for δ =±4 and value of the hyper paraeters α and λ are taken as a 1 = ,b 1 = and a 2 = ,b 2 = respectively, which are suerized in Table 6 and Table 7. Table 5, shows the MLEs and Bayes estiators of α and λ under SELF, GELF and 95% CI/HPD intervals based on coplete data set. On every censored saple schees the length of HPD intervals are always less than CI.

7 J. Stat. Appl. Pro. Lett. 1, No. 3, (2014) / 81 Fig. 2: Risks of Estiators of α and λ under GELF for different values of δ. Fig. 3: Risks of Estiators of α under GELF for different values of. Table 2: Under larger prior variance σ 1 = 10 and σ 2 = 10 the 95% CI, HPD intervals and % of coverage probability for different saples for fixed n=30,α = 2,λ = 3,a 1 = 0.4,a 2 = 0.9,b 1 = 0.2 and b 2 = 0.3. α λ α L c α U c α L h α U h % cov.prob λ L c λ U c λ L h λ U h %cov.prob E

8 82 S. K. Singh et. al. : Classical and Bayesian Inference for an Extension of the... Fig. 4: Risks of Estiators of α under SELF for different values of. Fig. 5: Risks of Estiators of λ under GELF for different values of. Table 3: Goodness of fit for various data exponentiated exponential Weibull gaa EE Log-likelihood K. S. statistics AIC BIC

9 J. Stat. Appl. Pro. Lett. 1, No. 3, (2014) / 83 Table 4: Failure tie vector Y = (y 2,...,y 30 ) under different PT-II CBR censoring schees S j (n : ) Schee i S 5 (30 : 27) R i y i S 4 (30 : 24) R i y i S 3 (30 : 21) R i y i S 2 (30 : 18) R i y i S 1 (30 : 15) R i y i Table 5: Bayes and ML estiates based on real data set for n = 30; p = 0.5. Paraeter MLE Bayes Estiates(MCMC) Inteval Estiates SELF GELF 95 CI 95 HPD δ = 4 δ = 4 L c U c L h U h α λ Table 6: Bayes and ML estiates, CI and HPD credible intervals for α with fixed n = 30 and p = 0.5 under PT-II CBR. Schee MLE Bayes Estiates(MCMC) Interval Estiates SELF GELF 95% CI 95% HPD δ = 4 δ = 4 α L c α U c α L h α L h S 1 (30 : 15) S 2 (30 : 18) S 3 (30 : 21) S 4 (30 : 24) S 5 (30 : 27) Table 7: Bayes and ML estiates, CI and HPD credible intervals for λ with fixed n = 30 and p = 0.5 under PT-II CBR. Schee MLE Bayes Estiates(MCMC) Interval Estiates SELF GELF 95% CI 95% HPD δ = 4 δ = 4 λ L c λ U c λ L h α L h S 1 (30 : 15) S 2 (30 : 18) S 3 (30 : 21) S 4 (30 : 24) S 5 (30 : 27)

10 84 S. K. Singh et. al. : Classical and Bayesian Inference for an Extension of the... Fig. 6: Risks of Estiators of λ under SELF for different values of. Fig. 7: Probability Plot for real data set exaple. 6 Conclusion After an extensive study of the results of siulation, we ay conclude that in ost of the cases, under both losses, our proposed estiator ˆα E and ˆλ E perfor better than all the considered copetitive estiators of α and λ respectively for δ > 0 (when over estiation is ore serious than under estiation). On the other hand for δ < 0 (when under estiation is ore serious than over estiation) ˆα S and ˆλ E have iniu risk than all the copetitive estiators of α and λ. Therefore, the proposed estiator ˆλ E is recoended for both losses, if under estiation is ore serious than over estiation vice-versa. References [1] Murthy, D. N. P, Xie, M. and Jiang, R., 2004, Weibull Models Wiley, New Jersey. [2] Balakrishnan, N., 2007, Progressive ethodology : An appraisal (with discussion). Test, 16 (2): [3] Balakrishnan, N. and Kannan, N., 2001, Point and Interval Estiation for Paraeters of the Logistic Distribution Based on Progressively Type-II Censored Saples, in Handbook of Statistics N. Balakrishnan and C. R. Rao, volue 20. Eds. Asterda, North- Holand.

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