Robust Bayesian Analysis of Lifetime Data from Maxwell Distribution

In this paper, we consider robust Bayesian analysis of lifetime data from the Maxwell distribution assuming an ε-contamination class of prior distributions for the parameter. We obtain robust Bayes estimates of the parameter and mean lifetime under squared error and LINEX loss functions in presence of uncensored as well as Type-I progressively hybrid censored lifetime data. A real data set is analysed for numerical illustrations.


Introduction
In Bayesian analysis, the investigator is supposed to posses some subjective a priori information concerning the most probable values of the parameter.In many cases he is able to successfully present his belief about the parameter in form of a single prior density.However, when the belief of investigator cannot be adequately represented in form of a single prior density or there is a possibility of error in prior elicitation, a class of distributions may be used to successfully present the prior belief.In such cases it becomes impossible to proceed with usual Bayesian procedures to make decisions or inferences.The robust Bayesian viewpoint provides a way to deal with such problems and to make decisions that behave satisfactorily when the prior varies over a class of prior distributions.Many authors provided different methods for implementing the robust Bayesian viewpoint.For some literature review one may refer to Box and Tiao (1973), Good (1965Good ( , 1983)), Dempster (1975) and Kadane and Chuang (1978).
A reasonable method for implementing uncertainties in prior elicitation is through the use of ε-contamination class of prior distributions given by Γ = {q : q = (1 − ε)g 0 + εg; g ∈ G} , (1) where ε(0 ≤ ε ≤ 1) is pre-assigned and represents the probability of error in the prior elicitation of the base prior g 0 and g is a distribution from the class G of all possible contaminated distributions.Many authors advocated Bayesian analysis based on the ε-contamination class of prior distributions [See Berger (1982), Berger (1983), Berger andBerliner (1986, 1984) , Sivaganesan and Berger (1987), Chaturvedi (1996) to cite a few].Berger and Berliner (1986) provides a good review of literature and additional motivation for consideration of ML-II procedure of Good (1983) for selecting a prior from an ε-contamination class in a data dependent fashion.According to this procedure, one can select a prior from the considered class by maximizing the predictive density corresponding to the prior.The prior thus obtained is called type-II maximum likelihood prior or ML-II prior in short.The Bayes estimators obtained under ML-II priors are termed as ML-II estimators.Chaturvedi, Pati, and Tomer (2014) carried out robust Bayesian analysis of Weibull distribution by implementing ML-II procedure.
In many real life investigations, for example life testing and reliability, we have to deal with censored data which often arise when life testing experiments are terminated before observing lifetimes of all units on test.In this context, plenty of censoring schemes have been studied and proposed in literature during last few decades.For a general review of literature on censoring schemes one may refer to Lawless (2003) and Balakrishnan and Aggarwala (2000).In this paper we shall consider a very generalized censoring scheme termed as Type-I progressive hybrid censoring scheme(Type-I PHCS ) [Kundu and Joarder (2006) and Childs, Chandrasekar, and Balakrishnan (2008)].This censoring scheme is recent and quite popular in literature [see Tomer and Panwar (2015)].Type-I PHCS is described as follows.Suppose in a life testing experiment, n units are put to test.The maximum duration of the experiment t 0 , the integers are fixed before beginning of the experiment.At the time of first failure X 1 , R 1 units, out of (n − 1) surviving units, are randomly withdrawn from the test.At the time of second failure X 2 , R 2 units, out of remaining (n − R 1 − 1) units, are randomly withdrawn from the test.The process continues till time T = min {X m , t 0 }.In the case when X m > t 0 , we observe the sample X 1:m:n , X 2:m:n , • • • , X d:m:n , where d(≤ m) denotes the number of failures observed before time t 0 , and terminate the experiment at t 0 by withdrawing , whereas if X m < t 0 , we observe X 1:m:n , X 2:m:n , • • • , X m:m:n and the experiment is terminated at X m .The data observed under type-I PHCS is termed as type-I progressively hybrid censored (type-I PHC) data.
The purpose of this article is many fold.We consider robust Bayesian estimation of the parameter and mean lifetime of the Maxwell distribution(MWD) under ε-contamination class of prior distributions in presence of uncensored as well as censored (type-I PHC) lifetime data.Under both types of data, we provide ML-II estimates under symmetric(squared error) and asymmetric(LINEX) loss functions.Rest of the paper is organized as follows.In Section 2, we derive ML-II estimator of the parameter and mean lifetime for uncensored sample.In Section 3, we develop procedure to obtain ML-II estimates for Type-I PHC data.In Section 4, we give a numerical example based on a real data set.Finally, we conclude findings in Section 5.

Estimation with uncensored data
A continuous non-negative random variable(rv) X is said to follow MWD if its probability density function (pdf ) is given by where θ is the unknown parameter.When the lifetime of a device follows the pdf (2), its mean lifetime is ξ = 2 (θ/π) and its reliability function F (t), at a specified mission time t(≥ 0), comes out to be where Γ a (z) = ∞ z u a−1 e −u du.Krishna and Malik (2009) has shown that MWD belongs to the class of increasing failure rate distributions.Therefore, it can be used as a lifetime model in various investigations where age of the device affects it adversely.Krishna and Malik (2012) obtained ML and Bayes estimators of the parameter and reliability function of MWD under Type-II progressive censoring scheme whereas Krishna, Vivekanand, and Kumar (2015) worked out similar problem with randomly censored data.
Suppose that the rv X denotes the lifetime of a device and follows MWD(θ).A random sample of such n independent and identically distributed lifetimes X 1 , X 2 , • • • , X n (denoted by x henceforth) is observed in a certain life testing experiment.The likelihood function of θ, in the light of given sample, comes out to be where T = n i=1 x 2 i .Following our discussion in Section 1, the considered ε-contamination class of prior distributions for θ is given by Γ = {q(θ) : q(θ) = (1 − ε)g 0 (θ|µ 0 ) + εg(θ|µ); g ∈ G} . (5) Here, we take the base prior, a natural conjugate prior [see Chib and Tiwari (1991), Chaturvedi et al. (2014)], given by the pdf where (µ 0 , ν) represents the hyper parameters.The contamination class G is the class of all natural conjugate priors with hyper parameters (µ, ν), given by According to ML-II procedure, discussed in Section 1, we select a prior density from the class Γ by maximizing the predictive density corresponding to q.For this, we first obtain the predictive density corresponding to the base prior g 0 (θ|µ 0 ) as follows.
Similarly, the predictive density for g(θ|µ) comes out to be Now the predictive density corresponding to the generic prior q ∈ Γ is In the ML-II process we choose value of the unknown hyper parameter µ in a data dependent fashion by maximizing the predictive density m(x|q) over the class of all priors q ∈ Γ.Since g 0 is fixed, we have sup and m(x|g) is maximized when we replace µ by its maximum likelihood estimator in g(θ|µ) which is given by μ = max µ 0 , 2νT 3n .
Then we get Thus, the ML-II prior density is given by Following Berger and Berliner (1986), the ML-II posterior density of θ, obtained using ( 4) and ( 10), comes out to be where Similarly, we get which on using (8) and ( 9), comes out to be Remark 2.1.In order to show the feasibility of the ML-II prior, we have Notice that ∂ λ ∂µ 0 is greater than zero if µ 0 < 2νT 3n and equal to zero if µ 0 ≥ 2νT 3n .Thus, if the base prior is not compatible with the data, λ decreases and more weight is provided to the data based part of the ML-II posterior density q * (θ) i.e. ĝ * (θ).As µ 0 → 0, λ → 0 and for µ 0 ≥ 2νT 3n , λ = 1 − ε, which is the maximum possible value of λ.

Estimation under SELF
We derive ML-II estimators of the parameter θ and mean lifetime ξ under squared error loss function (SELF ) along with their posterior variances in the following theorems.
Theorem 1.The ML-II posterior mean and variance of θ are given, respectively, by and Proof.See Appendix.
Theorem 2. The ML-II posterior mean and variance of ξ are given, respectively, by and Proof.The ML-II posterior mean of ξ is and rest part of the Proof is similar to that of Theorem 1.

Estimation under LINEX loss function
In previous section, we used a symmetric loss function SELF for estimation of the unknown parameter θ.This loss function is appropriate for the inferential problems when underestimation and overestimation of the parameter are of equal consequences.However, there may be circumstances when it does not happen.For example, overestimation of average lifetime or reliability of a component of an aircraft may be more serious than its underestimation.In such cases asymmetric loss functions are preferred.Among several asymmetric loss functions [see Calabria and Pulcini (1994)], LINEX loss function introduced by Varian (1975) is quite popular in literature.Zellner (1986) used LINEX loss function for Bayesian estimation of scale parameter.Kim, Jung, and Chung (2011), Doostparast, Ahmadi, and Ahmadi (2013) and Panwar, Tomer, and Kumar (2015) used it for different problems of estimation in presence of censored lifetime data.The expression of the LINEX loss function while estimating the parameter θ by its estimator θ is where ∆ = θ − θ.
Under the LINEX loss function ( 16), the ML-II estimator of θ is given by Here, on using (13), we obtain for µ 0 ≥ 2νT 3n that where H ν (z) is a modified Bessel function of third kind of order ν, [Gradshteyn and Ryzhik (1965), pp.340].
Similarly, for µ 0 < 2νT 3n , we get The expectation of the LINEX loss function for θL with respect to ML-II posterior distribution of θ comes out to be Under the LINEX loss function ( 16), the ML-II estimator of ξ when , The expressions for these are derived in Appendix.The expectation of the LINEX loss function for ξL is

Estimation under type-I PHCS
Suppose that a type-I PHC sample x 1:m:n , x 2:m:n , • • • , x m:m:n (denoted by x henceforth) is obtained by placing n units on a lifetest and following type-I PHCS, described in Section 1. Henceforth, we use notation x i instead of x i:m:n , for brevity.The likelihood function of given observations x [see Tomer and Panwar (2015)] can be written as follows where Therefore, (17) reduces to We proceed with the general case (17).Using (2) and (3), the likelihood (17) becomes The predictive density corresponding to the prior g(θ|µ) on using (18) comes out to be and the predictive density corresponding to the base prior g(θ|µ 0 ) can be obtained from ( 19) when µ = µ 0 .
Here, the value of µ which maximizes the predictive density m(x|g) is where μd is the solution of Then we have We write the ML-II prior density for this case as follows.
On using ( 18) and ( 21), the ML-II posterior density of θ comes out to be where

Estimation under SELF
In presence of type-I PHC data we obtain the ML-II estimator of θ under SELF, on using ( 22), as follows Since the posterior densities g * 0 (θ) and g * (θ) given by expressions ( 23) and ( 24), respectively, do not follow standard distributions, we use M-H algorithm [Metropolis and Ulam (1949)] to evaluate the posterior expectations E g * 0 (θ) and E g * (θ).Similarly, by using M-H algorithm ML-II estimate of mean lifetime can be obtained as follows.
The posterior variances of θ can be obtained from (31) of Appendix on replacing g * 0 by g * 0 and g * by g * and implementing M-H algorithm.Similarly, we evaluate the posterior variance of ξ.

Estimation under LINEX loss function
The expressions for the ML-II estimators of θ and ξ under LINEX loss function can be obtained on using ( 22) as The expectations of LINEX loss function for θL and ξL are respectively given by a(E q * [θ]− θL ) and a(E q * [ξ] − ξL ).Like in Section 3.1, the ML-II estimates and their posterior risks can be obtained using M-H algorithm.
In order to illustrate the ML-II procedure discussed in Section 2, we consider two different base priors IG 1 (2000, 2) and IG 2 (7000, 2) for θ.Then we obtain ML-II estimates of θ as well ξ assuming different values of ε that ranges from 0 to 1.The values of these estimates along with their posterior standard deviations (SDs) under SELF and LINEX loss functions are presented in Table 2.For each loss function, we observe from Table 2 that when ε = 0, the ML-II estimates corresponding to the considered base priors differ significantly but as ε → 1, the estimates under two prior come closer and almost coincide at ε = 1.
Further, to study the behaviour of ML-II estimators in presence of Type-I PHC data, we use expressions that are obtained in Section 3. We consider three Type-I PHC samples which are generated from the original data.These samples are presented in Table 1.With these samples, we obtained the ML-II estimates of θ and mean lifetime ξ under SELF and LINEX loss functions with the same values of hyper-parameters of base priors i.e.IG 1 (2000, 2) and IG 2 (7000, 2).The findings are presented in Tables 3-5 which exhibit same behaviour as we observed in complete sample study.= ({1}*5,1,0,{1}*5), t 0 = 100 17.88 28.92 33.00 41.52 42.12 45.60 48.48 51.84 51.96 84.12 93.12 98.64 Note: a * b=(a, a, a, ..., (b times))

Conclusion
We considered robust Bayesian estimation of the parameter and mean lifetime in the presence of uncensored as well as Type-I PHC lifetime data.In this study, we have shown that εcontamination class of prior distributions can give robust results when the prior belief of the investigator cannot be represented in form of a single prior density or there is a possibility of error in the prior elicitation of unique prior for the parameter.We have illustrated with the help of a real data set that ε-contamination class is a sensible class of priors which may be thought to promote objective thinking by removing the judgment error in prior elicitation process.
Table 2: ML-II estimate of θ and ξ, along with their posterior SDs(in parentheses), under two different base priors for uncensored data.

Table 1 :
Samples obtained under three different Type-I PHCS from the ball bearings data.

Table 3 :
ML-II estimate of θ and ξ, along with their posterior SDs(in parentheses), under two different base priors for S

Table 4 :
ML-II estimate of θ and ξ, along with their posterior SDs(in parentheses), under two different base priors for S 15:23 censoring Scheme.

Table 5 :
ML-II estimate of θ and ξ, along with their posterior SDs(in parentheses), under two different base priors for S 12:23 censoring Scheme.