Estimation using hybrid censored data from a two-parameter distribution with bathtub shape
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 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 items have failed or when a pre-specified terminal time point has been reached. We observe that under this particular life testing situation the realized sample could be one of the following two types.
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 . He constructed a two sided confidence interval for 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 by deriving the exact distribution of its maximum likelihood estimator (MLE). Draper and Guttman (1987) obtained a two sided Bayesian credible interval for using an inverted gamma prior. Gupta and Kundu (1998) compared different confidence intervals of 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, It is assumed that parameters and are unknown. In fact has a bathtub shaped hazard function when and when , it has an increasing hazard function. The case 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 with 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 and 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 and 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 and . 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 denote an ordered sample of size from the distribution as defined in (1) with and being unknown parameters. In this section, we deal with finding MLEs of and when hybrid censored samples are drawn from . Under hybrid censoring, the likelihood function of and is obtained as where and are respectively given by 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 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 and . The loss function is taken to be a squared error and it is defined as, where is an estimate of some parametric function . It is to be noticed that when both the parameters and 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 from the distribution as defined in (1). The desired MLEs of and are obtained using the celebrated EM algorithm. Further, with respect to the loss function , 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 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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