Estimation using hybrid censored data from a two-parameter distribution with bathtub shape

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Abstract

The problem of estimating unknown parameters of a two-parameter distribution with bathtub shape is considered under the assumption that samples are hybrid censored. The maximum likelihood estimates are obtained using an EM algorithm. The Fisher information matrix is obtained as well and the asymptotic confidence intervals are constructed. Further, two bootstrap interval estimates are also proposed for the unknown parameters. Bayes estimates are evaluated under squared error loss function. Approximate explicit expressions for these estimates are derived using the Lindley method as well as using the Tierney and Kadane method. An importance sampling scheme is then proposed to generate Markov Chain Monte Carlo samples which have been used to compute approximate Bayes estimates and credible intervals for the unknowns. A numerical study is performed to compare the proposed estimates. Finally, two data sets are analyzed for illustrative purposes.

Introduction

The importance of censoring in life testing and reliability investigations is quite inevitable as often it is not desirable to obtain lifetimes of all experimental units placed on a test. Observations obtained from such experiments are known to be censored. Several censoring schemes have been proposed in the literature and the most common ones are Type I and Type II censoring schemes. We refer to Lawless (1982) for a detailed discussion on these two schemes. A mixture of Type I and Type II censoring schemes is known as a hybrid censoring scheme. Observations from this scheme are drawn in the following manner. Suppose that n units having a common lifetime distribution are placed on a life testing experiment. Then the test is terminated either when a pre-assigned number of r(n) items have failed or when a pre-specified terminal time point T has been reached. We observe that under this particular life testing situation the realized sample could be one of the following two types. Case I:{X1:n,X2:n,,Xr:n},if  Xr:n<TCase II:{X1:n,X2:n,,Xm:n},if  m<r,Xm+1:n>T.

Epstein (1954) introduced a hybrid censoring scheme and applied it to the case when lifetimes of the experimental units follow an exponential distribution with unknown mean θ. He constructed a two sided confidence interval for θ using this scheme. Since its introduction, several researchers have analyzed hybrid censored data arising from various parametric distributions and obtained many interesting inferential results. Chen and Bhattacharyya (1988) proposed a one sided confidence interval for the exponential mean θ by deriving the exact distribution of its maximum likelihood estimator (MLE). Draper and Guttman (1987) obtained a two sided Bayesian credible interval for θ using an inverted gamma prior. Gupta and Kundu (1998) compared different confidence intervals of θ with those of Bayesian credible intervals and critically analyzed proposed methods through a numerical study. Kundu (2007) made an excellent study on a hybrid censored Weibull distribution and obtained many interesting results. Dube et al. (2011) analyzed a hybrid censored two-parameter lognormal distribution. Authors obtained MLEs of unknown parameters using EM algorithm and then compared them with corresponding approximate maximum likelihood estimates (AMLEs) numerically. Some specific comments are also given based on the findings in the paper. For some more relevant works on hybrid censoring, one may refer to Ebrahimi (1990), Banerjee and Kundu (2008), Kundu and Pradhan (2009a).

Chen (2000) proposed a two-parameter distribution with bathtub shape or increasing failure rate function. In this article, we consider the estimation of unknown parameters of this distribution when samples are hybrid censored. The probability density function (PDF) and cumulative distribution function (CDF) are respectively of the form, fX(x;α,β)=αβxβ1exp{α(1exβ)+xβ},x>0,α>0,β>0,FX(x;α,β)=1exp{α(1exβ)},x>0. It is assumed that parameters β and α are unknown. In fact fX(x;α,β) has a bathtub shaped hazard function when β<1 and when β1, it has an increasing hazard function. The case α=1 corresponds to the exponential power distribution. The lifetime distributions with bathtub shaped hazard rate functions have attracted the attention of many researchers as the lifetimes of various industrial items including electrical and mechanical products, as well as survival times of various biological entities exhibit such characteristics (see for instance, Rajarshi and Rajarshi, 1988, Wu, 2008). Many authors have investigated various parametric distributions with bathtub shaped hazard functions. Wu et al. (2004) considered estimation of the shape parameter β with α known and obtained optimal estimates under doubly Type II censored samples. Based on Type II censored samples, Chen (2000) constructed exact confidence intervals for the shape parameter and also obtained exact confidence regions for both the unknown parameters. Wu (2008) obtained MLEs of α and β under progressively Type II censored samples and also provided exact confidence intervals and confidence regions for these parameters. For some more works on such type of distributions, we refer to Rajarshi and Rajarshi (1988), Gurvich et al. (1997), Wu et al. (2011) and Sarhan et al. (2012).

The rest of the paper is presented as follows. Section  2 deals with computing MLEs for α and β using the EM algorithm. For the purpose of constructing approximate interval estimates, the Fisher information matrix is also computed in this section. Bootstrap confidence intervals are discussed in Section  3. In Section  4, Bayes estimates are obtained using Lindley method as well as using Tierney and Kadane method. Importance sampling scheme is also proposed to compute the approximate Bayes estimates of α and β. Further, credible intervals are constructed for the unknowns using the samples generated from this scheme. In Section  5, a numerical study is performed between proposed estimates in terms of their mean square error and bias values and two data sets are analyzed for the purpose of illustration in Section  6. Different interval estimates are also compared in these two sections. Finally, a conclusion is presented in Section  7 and some possible extensions are discussed.

Section snippets

Maximum likelihood estimation

Let X1:n<X2:n<<Xn:n denote an ordered sample of size n from the distribution as defined in (1) with α and β being unknown parameters. In this section, we deal with finding MLEs of α and β when hybrid censored samples are drawn from fX(x;α,β). Under hybrid censoring, the likelihood function of α and β is obtained as L(α,β)αdβdi=1dxi:n(β1)eα(1exi:nβ)+xi:nβeα(nd)(1ecβ), where d and c are respectively given by d={r,for Case I ,m,for Case II ,andc={xr:n,for Case I ,T,for Case II . We note

Bootstrap confidence intervals

In this section, we use the bootstrap technique to construct different confidence intervals for the unknown parameters. Work of Efron (1982) led the exponential growth of the bootstrap method in various inferential problems. He proposed bootstrap percentile (Boot-p) intervals as an alternative to the standard normal theory. Several other aspects of this method are also discussed in Efron and Tibshirani (1986). Hall (1988) proposed yet another approach to construct confidence intervals and it is

Bayesian estimation

Suppose that X1:n,X2:n,,Xd:n is a hybrid censored sample taken from a two-parameter distribution with bathtub shape as defined in (1). Based on such a sample, in this section we apply the Bayesian approach to compute estimates for α and β. The loss function is taken to be a squared error and it is defined as, LBS(g(θ),gˆ(θ))=(gˆ(θ)g(θ))2, where gˆ(θ) is an estimate of some parametric function g(θ). It is to be noticed that when both the parameters α and β are unknown then there does not exist

Simulation results

In this section, we perform a Monte Carlo simulation study to compare proposed estimates in terms of their mean square error (MSE) and bias values. We computed these values based on the generation of 5000 random samples of size n from the fX(x;α,β) distribution as defined in (1). The desired MLEs of α and β are obtained using the celebrated EM algorithm. Further, with respect to the loss function LBS, different approximate Bayes estimates are also obtained. Precisely, we derived these estimates

Data analysis

Two examples are presented in this section for the purpose of illustration. First, we treat a real data set.

Example 1 Real Data

We consider the data presented in Aarset (1987) which represents the times to failure of n=50 units put on a life-test and is listed below as

For computational convenience we divided each data point by 200. Aarset showed that hazard rate of this data is bathtub-shaped. In fact, we tried to fit four different distributions including the one given in (1). The other three are, namely, the

Conclusions

The problem of estimating unknown parameters of a two-parameter distribution with bathtub shape is discussed in this paper under the assumption that samples are hybrid censored. We obtained MLEs for the unknown parameters and constructed asymptotic confidence intervals using the Fisher information matrix. Bootstrap methods are also used to obtain desired confidence intervals. Different Bayes estimates are studied under squared error loss function with respect to informative as well

Acknowledgments

The authors are thankful to Prof. D. Kundu and Dr. B. Pradhan for many helpful discussions during this research project. They also thank the Associate Editor and an anonymous referee for their useful suggestions which led to significant improvements in this paper.

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