Statistical Inference for K Exponential Populations under Joint Progressive Type-I Censored Scheme

In this article, the maximum likelihood estimators of the k independent exponential populations parameters are obtained based on joint progressive type-I censored (JPC-I) scheme. The Bayes estimators are also obtained by considering three different loss functions. The approximate confidence, two Bootstrap confidence and the Bayes credible intervals for the unknown parameters are discussed. A simulated and real data sets are analyzed to illustrate the theoretical results.


Notations pdf
Probability density function cdf Cumulative density function iid Independent and identically distributed  Number of censored stage   Prefixed censoring times, 1 ≤  ≤   ̄ℎ(  ) 1 −  ℎ (  ), Survival functions of the h populations at  .

Introduction
The joint censoring scheme is a very common way in conducting comparative life-tests of products from various units within the same facility.Suppose products are being produced by k different lines under the same conditions, and that k independent samples of sizes  ℎ ,1 ≤ ℎ ≤ are selected from these k lines and placed simultaneously on a life-testing experiment.Then, in order to reduce the cost and the experimental time, the experimenter may choose to terminate the life-testing experiment before complete information on failure times for all experimental units.Data arises from these experiments are called joint censored data.In this situation, the experimenter may be interested in either point or interval estimation of the mean lifetimes of units produced by the different k lines.In the literature, there were four types of joint schemes, namely, joint Type-II, joint progressive Type-II, joint progressive Type-I and joint type-I progressive hybrid censoring schemes.Balakrishnan and Rasouli (2008) developed the likelihood inference for the parameters of two exponential populations under joint Type-II censoring.Shafay 2019) proposed a joint type-I progressive hybrid censoring scheme and investigated the estimation problems in the case of exponential distribution.Recently, Ashour and Abo-Kasem (2017) introduced JPC-I scheme and as a special case, joint Type-I censored scheme.They considered statistical inference for two exponential populations under both JPC-I and joint Type-I censored schemes.Abo-Kasem and Nassar (2019) developed the estimation problems of two Weibull populations with the same shape parameter under JPC-I censoring scheme.They obtained the maximum likelihood estimators (MLEs) and the approximate confidence intervals.They also obtained the Bayes estimates using squared error and LINEX loss functions under the assumption of independent gamma priors.
The JPC-I scheme in comparison to other joint censoring schemes provides an important advantage of the known termination time point of the life-testing experiment.From the experimenter point of view, this makes the JPC-I scheme very appealing for its implementation in practice.In spite of such a practical advantage, most of the inferential studies carried out in the literature focused on joint Type-II and joint progressive Type-II censoring schemes.For this reason, our aim in this paper is to investigate the point and interval estimation of k independent exponential populations under the JPC-I scheme.The rest of the paper is organized as follows: we formulate the problem in Section 2. The maximum likelihood estimation for k exponential populations and the approximate confidence intervals are obtained in Section 3. Section 4 describes the various bootstrap confidence intervals.In Section 5, we obtain the Bayes estimators under squared error, LINEX and general entropy loss functions as well as the Bayesian credible intervals for the parameters.In Section 6, a simulated and real data sets are analyzed.The paper is concluded in Section 7.

Bootstrap Confidence Intervals
In this section, two bootstrap confidence intervals are discussed.The two bootstrap methods are the percentile bootstrap (Boot-p) proposed by Efron (1982), and the bootstrap-t method (Boot-t) proposed by Hall (1988).We can use the following algorithm to obtain the Boot-p and Boot-t intervals,.The Boot-t confidence intervals estimators are obtained according to the following steps: (1-2) Same as the steps 1-2 in (a).where  ̂ ̂ℎ * is the bootstrap variance.

Bayesian Inference
Based on the likelihood function and the independent gamma prior distributions, viz.( ℎ ,  ℎ ) for 1 ≤ ℎ ≤ , with pdf given by where Γ(. )denotes the complete gamma function.Combining ( 2) and ( 4), the joint posterior density of  1 ,  2 , . . .,   given the data is where and .

Numerical Illustration
This section is devoted to illustrate the theoretical result obtained in the previous sections numerically by analyzing a simulated and real data sets.

Example (1): Real data-set
To illustrate the usefulness of the proposed estimators obtained in sections 3, 4 and 5 with real situations, we consider Nelson's data (1982, Ch. 10, Table 4.1) which correspond to breakdown in minutes of an insulating fluid subjected to high voltage stress.These failure times were observed in the form of groups with each group reporting data on 10 insulating fluids.Let us consider the following three groups of samples of failure time data presented in Table (1).Table (2) presents the JPC-I sample data that have been obtained from the three samples in Table (1) with  1 = 1,  2 = 2 and  3 = 3 (in minutes).The generated JPC-I sample size is 23 and presented in Table (2) along with the realized values of other pertinent variables.We obtain the MLEs and Bayes estimates of  1 ,  2 and  3 (with the choice of ( 1 ,  2 ,  3 ,  1 ,  2 ,  3 ) = (1.1,1.4,1.6,1,1,1) as hyper-parameters values, these results are presented in Table (3).Table (4) shows the asymptotic variance covariance matrix of the MLEs.Table (5) presents the 95% approximate, Boot-p, Boot-t (with B =1000 bootstrap samples) and Bayes credible intervals for  1 ,  2 and  3 .From Table 8 and 9, it is observed that the Bayes estimates perform better than MLEs in terms of minimum MSE in most of the cases.Comparing the two schemes, we can see that the estimates  1 and  2 under scheme (2) perform better than those based on scheme (1), while the estimate of  3 is better in scheme (1) than scheme (2).Also, it is noted that the approximate and Bayes credible confidence intervals are not satisfactory compared to the Boot-p and Boot-t confidence intervals for both two schemes.Finally, it can be seen that the Boot-p and Boot-t intervals for  1 and  2 perform better than those based on approximate and Bayes credible intervals in scheme (2) than scheme (1), while Boot-p and Boot-t intervals for  3 performing better in scheme (1) than scheme (2).

Conclusions
In this paper, for k exponential distributions based on the JPC-I scheme the maximum likelihood estimators and the Bayes estimators based on squared error, LINEX and general et al. (2013), Ashour and Abo-Kasem (2014 a,b,c), considered the jointly Type-II censored sample.Rasouli and Balakrishnan (2010) studied the statistical inference of two exponential populations under the joint progressive Type-II censoring.Parsi et al. (2011) and Doostparast et al. (2013) considered the jointly progressive type-II censored sample.Balakrishnan and Feng (2015) generalized the work of Balakrishnan and Rasouli (2008) by considering a jointly Type-II censored sample arising from k independent exponential populations.Also, Balakrishnan et al. (2015) generalized Rasouli and Balakrishnan (2010) work by considerring a jointly progressive Type-II censored sample arising from k independent exponential populations.Abo-Kasem et al. ( the standard normal distribution.

Table 1 :
The failure time data as three groups of insulating fluids

Table 3 :
The MLE and Bayesian estimates of 1 ,  2 and  3

Table 4 :
Estimates of the asymptotic variance covariance matrix of the MLEs based on JPC-I sample.

Table 8 :
The MLEs and Bayes estimates of  1 ,  2 and  3 and MSE's in parentheses using different schemes