Estimation of Parameters of Generalized Inverted Exponential Distribution for Progressive Type-II Censored Sample with Binomial Removals

Abstract

We obtained the maximum likelihood and Bayes estimators of the parameters of the generalized inverted exponential distribution in case of the progressive type-II censoring scheme with binomial removals. Bayesian estimation procedure has been discussed under the consideration of the square error and general entropy loss functions while the model parameters follow the gamma prior distributions. The performances of the maximum likelihood and Bayes estimators are compared in terms of their risks through the simulation study. Further, we have also derived the expression of the expected experiment time to get a progressively censored sample with binomial removals, consisting of specified number of observations from generalized inverted exponential distribution. An illustrative example based on a real data set has also been given.

1. Introduction

The one parameter exponential distribution is the simplest and the most widely discussed distribution in the context of life testing. This distribution plays an important role in the development to the theory, that is, any new theory developed can be easily illustrated by the exponential distribution due its mathematical tractability; see Barlow and Proschan [1] and Leemis [2]. But its applicability is restricted to a constant hazard rate because hardly any item/system can be seen which has time independent hazard rate. Therefore, the number of generalizations of the exponential distribution has been proposed in earlier literature where the exponential distribution is not suitable to the real problem. For example, the gamma (sum of independent exponential variates) and Weibull (power transformed distribution) distributions are the most popular generalizations of the exponential distribution. Most of the generalizations of the exponential distribution possess the constant, nonincreasing, nondecreasing and bathtub hazard rates.

But in practical real problems, there may be a situation where the data shows the inverted bathtub hazard rate (initially increases and then decreases, i.e., unimodal). Let us take an example, in the course of study of breast cancer data, we observed that the mortality increases initially, reaches to a peak after some time, and then declines slowly, that is, associated hazard rate is inverted bathtub or particularly unimodal. For such types of data, another extension of the exponential distribution has been proposed in statistical literature. That is known as one parameter inverse exponential or one parameter inverted exponential distribution (IED) which possess the inverted bathtub hazard rate. Many authors have proposed the use of IED in survival analysis; see Lin et al. [3], and Singh et al. [4]. Abouammoh and Alshingiti [5] have proposed two parameters generalization of IED called as generalized inverted exponential distribution (GIED) and they have showed that GIED is better than IED for real data set on the basis of likelihood ratio test and 𝐾-𝑆 statistics. They have also discussed the maximum likelihood and least square methods for the estimation of the unknown parameters of GIED. Krishna and Kumar [6] have studied the reliability estimation based on progressive type-II censored sample under classical setup.

In life testing experiments, situations do arise that the units under study are lost or removed from the experiments while they are still alive; that is, generally we get censored data from the life testing experiments. The loss of units may occur due to time constraints, giving type-I censored data. In type-I censoring scheme, the number of observations is random as the experiment is terminated at a prespecified time. Sometimes, the experiment has to be terminated after a prefixed number of observations and the data, thus obtained, is referred as type-II censored data. Besides these, there are many uncontrolled causes, resulting to loss of intermediate observations; see Balakrishnan [7]. One such censoring procedure named as progressive type-II censoring scheme can be described notationally as follows. Let the lifetimes of 𝑛 identical units/items be studied. At the first failure 𝑥₁, called first stage, 𝑟₁ units are removed from the remaining (𝑛−1) surviving units. At second failure 𝑥₂, called second stage 𝑟₂ units from remaining 𝑛−2, −𝑟₁ units are removed, and so on, till 𝑚th failure is observed; that is, at 𝑚th stage all the remaining (𝑛−𝑅₁−⋯−𝑅_𝑚−1−𝑚+1) units are removed.It may be mentioned here that the number of units 𝑟_𝑖 dropped out from the test at 𝑖th stage should be less than 𝑛−𝑚−∑^𝑖−1_𝑙=0𝑟_𝑙 in order to insure the availability of 𝑚 observations. In many practical situations, 𝑟_𝑖’s may be random and cannot be predetermined, for example, in clinical trials. Considering 𝑟_𝑖 to be random, Yuen and Tse [8] have discussed progressive censoring scheme with binomial removals. They assumed that the number of random removals at the stage 𝑟_𝑖 is random and follows binomial distribution with probability 𝑝. It may be noted here that in clinical trials, the assumption that 𝑟_𝑖’s are less than 𝑛−𝑚−∑^𝑖−1_𝑙=0𝑟_𝑙 looks unrealistic, but, in life testing experiments, it should not pose any problem as it is used to decide the value of 𝑟_𝑖’s only. Thus, 𝑟₁ (at 1th stage) is to be considered to follow the binomial distribution with parameter 𝑛−𝑚 and 𝑝, that is, binomial (𝑛−𝑚,𝑝), and in the same way, 𝑟₂ (at 2th stage) follows binomial (𝑛−𝑚−𝑟₁,𝑝). In general, the number of units removed at 𝑖th stage 𝑟_𝑖 follows the binomial distribution with parameter 𝑛−𝑚−∑^𝑖−1_𝑙=0𝑟_𝑙 and 𝑝 for 𝑖=1,2,3,…,𝑚−1.

For further details on progressive censoring and its further development, readers may be referred to Balakrishnan [7]. The estimation of parameters of several lifetime distributions based on progressive censored samples has been studied by many authors; see Childs and Balakrishnan [9], Balakrishnan and Kannan [10], Mousa and Jaheen [11], and Ng et al. [12]. The progressive type-II censoring with binomial removal has been considered by Tse et al. [13] for Weibull distribution, Wu and Chang [14] for exponential distribution. Under the progressive type-II censoring with random removals, Wu and Chang [15], Yuen and Tse [8], and Singh et al. [16] developed the estimation problem for the Pareto distribution, Weibull distribution, and exponentiated Pareto distribution, respectively.

The objective of this paper is to obtain the MLEs and Bayes estimators of the unknown parameters of GIED under symmetric and asymmetric loss functions and compare the performances of the competing estimators. Further, we have also investigated the total experiment time of experiment on the basis of numerical study. The rest of the paper has been organized in the following section. Section 2 provides a brief discussion about the progressive type-II censoring scheme with binomial removals. In the next section, we have obtained MLEs and Bayes estimators of the model parameters. The expression for the expected experiment time for progressive type-II censored data with binomial removals has been derived in Section 4. The algorithm for simulating the progressive type-II censored data with binomial removal has been described in Section 5. The comparison study of the MLEs and Bayes estimators has been given in Section 6. In Section 7, the methodology is illustrated through a real data set. Finally, the conclusions have been provided in the last section.

2. The Model

The GIED (𝜆,𝛼) with cumulative density function (cdf) is expressed as follows 𝐹(𝑥)=1−1−𝑒^{−𝜆/𝑥𝛼};𝑥≥0,𝜆>0,𝛼>0, (1) and the probability density function (pdf) is given by: 𝑓(𝑥)=𝛼𝜆𝑥²𝑒^−𝜆/𝑥1−𝑒^{−𝜆/𝑥𝛼−1};𝑥≥0,𝜆>0,𝛼>0. (2) The survival function of 𝑥 is 𝑆(𝑥)=1−𝑒^{−𝜆/𝑥𝛼};𝑥≥0,𝛼>0,𝜆>0. (3) Let (𝑋₁,𝑅₁),(𝑋₂,𝑅₂),(𝑋₃,𝑅₃),…,(𝑋_𝑚,𝑅_𝑚) be the progressive type-II censored sample, where 𝑅_𝑖’s are random removals. For fixed values of 𝑅_𝑖’s, say 𝑅₁=𝑟₁,𝑅₂=𝑟₂,𝑅₃=𝑟₃,…,𝑅_𝑚=𝑟_𝑚, the conditional likelihood function can be written as 𝐿(𝛼,𝜆;𝑥∣𝑅=𝑟)=𝑐^∗𝑚_𝑖=1𝑓𝑥_𝑖𝑆𝑥_𝑖^𝑟𝑖, (4) where 𝑐^∗=𝑛(𝑛−𝑟₁−1)(𝑛−𝑟₁−𝑟₂−2)(𝑛−𝑟₁−𝑟₂−𝑟₃−3)⋯(𝑛−𝑟₁−𝑟₂−𝑟₃,…,𝑟_𝑚−𝑚+1) and 0≤𝑟_𝑖≤(𝑛−𝑚−𝑟₁−𝑟₂−𝑟₃,…,𝑟_𝑖−1) for 𝑖=1,2,3,…,𝑚 and 𝑟₀=0. Substituting (2) and (3) into (4), we get 𝐿(𝛼,𝜆;𝑥∣𝑅=𝑟)=𝑐^∗𝑚_𝑖=1𝛼𝜆𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖𝛼−1}×1−𝑒^{−𝜆/𝑥𝑖𝛼𝑟𝑖}. (5)

As mentioned earlier, individual units removed from the test at 𝑖th stage, 𝑖=1,2,…,𝑚 independent of each other and the probability of removal is same 𝑝 for all. Therefore, the number 𝑅_𝑖 of the unit removed at 𝑖th stage 1=1,2,3,…,𝑚 follows binomial distribution with parameters (𝑛−𝑚−∑^𝑖−1_𝑙=1𝑟_𝑙,𝑝); that is, 𝑃𝑅₁=𝑟₁=𝑟;𝑝𝑛−𝑚₁𝑝^𝑟11−𝑝^{𝑛−𝑚−𝑟1}, (6) and for 𝑖=2,3,…,𝑚−1, 𝑃𝑅_𝑖=𝑟_𝑖∣𝑅_𝑖−1=𝑟_𝑖−1,…𝑅₁=𝑟₁=𝑛−𝑚−^𝑖−1_𝑙=0𝑟_𝑙𝑟_𝑖𝑝^𝑟𝑖1−𝑝^{𝑛−𝑚−∑𝑖−1𝑙=0𝑟𝑙}. (7) Now, we further assume that 𝑅_𝑖’s are independent of 𝑋_𝑖’s for all 𝑖. Then, the full likelihood function takes the following form: 𝐿𝛼,𝜆,𝑝;𝑥=𝐿(𝛼,𝜆;𝑥∣𝑅=𝑟)𝑃𝑅=𝑟;𝑝, (8) where 𝑃𝑅𝑅=𝑟;𝑝=𝑃₁=𝑟₁𝑃𝑅₂=𝑟₂∣𝑅₁=𝑟₁×𝑃𝑅₃=𝑟₃∣𝑅₂=𝑟₂,𝑅₁=𝑟₁⋯𝑃𝑅_𝑚−1=𝑟_𝑚−1∣𝑅_𝑚−2=𝑟_𝑚−2,…𝑅₁=𝑟₁. (9) Substituting (6) and (7) into (9), we get 𝑃=()𝑅=𝑟;𝑝𝑛−𝑚!𝑝^{∑𝑚−1𝑖=1𝑟𝑖}1−𝑝^{(𝑚−1)(𝑛−𝑚)−∑𝑚−1𝑖=1(𝑚−𝑖)𝑟𝑖}𝑛−𝑚−∑^𝑖−1_𝑙=1𝑟_𝑙!∏^𝑚−1_𝑖=1𝑟_𝑖!. (10) Now, using (5), (8), and (10), we can write the full likelihood function in the following form: 𝐿𝛼,𝜆,𝑝;𝑥=𝐴𝐿₁(𝛼,𝜆)𝐿₂𝑝, (11) where 𝑐𝐴=^∗(𝑛−𝑚)!𝑛−𝑚−∑^𝑖−1_𝑙=1𝑟_𝑙!∏^𝑚−1_𝑖=1𝑟_𝑖!,𝐿₁(𝛼;𝜆)=^𝑚_𝑖=1𝛼𝜆𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖𝛼−1}1−𝑒^{−𝜆/𝑥𝑖𝛼𝑟𝑖},𝐿₂𝑝=𝑝^{∑𝑚−1𝑖=1𝑟𝑖}1−𝑝^{(𝑚−1)(𝑛−𝑚)−∑𝑚−1𝑖=1(𝑚−𝑖)𝑟𝑖}. (12)(13)

3. Classical and Bayesian Estimation of Parameters

3.1. Maximum Likelihood Estimation

In this section, we have obtained the MLEs of the parameters 𝛼, 𝜆, and 𝑝 based on progressive type-II censored data with binomial removals. We observe from (11), (12), and (13) that likelihood function is multiplication of three terms, namely, 𝐴, 𝐿₁, and 𝐿₂. Out of these, 𝐴 does not depend on the parameters 𝛼, 𝜆, and 𝑝. 𝐿₁, does not involve 𝑝 and it is function of 𝛼 and 𝜆 only, whereas 𝐿₂ involves 𝑝 only. Therefore, the MLEs of 𝛼 and 𝜆 can be derived by maximizing 𝐿₁. Similarly, the MLE of 𝑝 can be obtained by maximizing 𝐿₂.

Taking log of both sides of (12), we have ln𝐿₁(𝛼;𝜆)=𝑚ln(𝜆)+𝑚ln(𝛼)−𝜆^𝑚_𝑖=11𝑥_𝑖+(𝛼−1)^𝑚_𝑖=1ln1−𝑒^{−𝜆/𝑥𝑖}+^𝑚_𝑖=1𝑟_𝑖ln1−𝑒^{−𝜆/𝑥𝑖𝛼}. (14) Thus, MLEs of 𝛼 and 𝜆 can be obtained by simultaneously solving the following nonlinear normal equations which are as follows: 𝜕ln𝐿₁(𝛼)=𝑚𝜕𝛼𝛼+^𝑚_𝑖=1𝑟_𝑖+1ln1−𝑒^{−𝜆/𝑥𝑖}=0,𝜕ln𝐿₁(𝜆)=𝑚𝜕𝜆𝜆−^𝑚_𝑖=11𝑥_𝑖+^𝑚_𝑖=1𝛼𝑟_𝑖𝑒+1−1^{−𝜆/𝑥𝑖}𝑥_𝑖1−𝑒^{−𝜆/𝑥𝑖}=0. (15)(16) From (15), we obtain the MLE of 𝛼 as a function of 𝜆, say 𝛼(𝜆), where 𝛼(𝜆)=−𝑚∑^𝑚_𝑖=1𝑟_𝑖+1ln1−𝑒^{−𝜆/𝑥𝑖}. (17) Putting 𝛼(𝜆) in (14), we obtain ln𝐿₁(𝛼(𝜆);𝜆)=𝑚ln(𝜆)+𝑚ln(𝛼(𝜆))−𝜆^𝑚_𝑖=11𝑥_𝑖+(𝛼(𝜆)−1)^𝑚_𝑖=1ln1−𝑒^{−𝜆/𝑥𝑖}+^𝑚_𝑖=1𝑟_𝑖ln1−𝑒^{−𝜆/𝑥𝑖}𝛼_(𝜆). (18) Therefore, the MLE of 𝜆 can be obtained by maximizing (18) with respect to 𝜆. Once 𝜆 is obtained, 𝛼 can be obtained from (17) as 𝛼(𝜆). Therefore, it reduces the two-dimensional problem to a one-dimensional problem which is relatively easier to solve with fixed point iteration method. For details about the fixed point iteration method, readers may refer Rao [17].

The log of 𝐿₂(𝑝) takes the following form: ln𝐿₂𝑝=ln𝑝^𝑚−1_𝑖=1𝑟_𝑖×()−+ln1−𝑝𝑚−1)(𝑛−𝑚^𝑚−1_𝑖=1𝑟_𝑖(𝑚−𝑖). (19) The first order derivative of ln𝐿₂(𝑝) with respect to 𝑝 is 𝜕ln𝐿₂𝑝=∑𝜕𝑝^{𝑚−1𝑖=1}𝑟_𝑖𝑝−(𝑚−1)(𝑛−𝑚)−∑^𝑚−1_𝑖=1𝑟_𝑖(𝑚−𝑖).1−𝑝 (20) Setting (𝜕ln𝐿₂(𝑝))/𝜕𝑝=0 and solving, we get the MLE of 𝑝 as 𝑝_mle=∑^𝑚−1_𝑖=1𝑟_𝑖(𝑚−1)(𝑛−𝑚)−∑^𝑚−1_𝑖=1𝑟_𝑖(𝑚−𝑖−1). (21)

3.2. Bayes Estimators

In order to obtain the Bayes estimators of the parameters 𝛼 and 𝜆 based on progressively type-II censored data with binomial removals. We must assume that the parameters 𝛼 and 𝜆 are random variables. Following Nassar and Eissa [18] and Kim et al. [19], we assume that these are independently distributed. The random variables 𝛼 and 𝜆 have prior distribution with respective prior pdfs 𝑔₁(𝛼)=𝑎₁^𝑏1𝑒^−𝑎1𝛼𝛼^𝑏1−1Γ𝑏₁;0<𝛼<∞,𝑎₁>0,𝑏₁𝑔>0,₂(𝜆)=𝑎₂^𝑏2𝑒^−𝑎2𝜆𝜆^𝑏2−1Γ𝑏₂;0<𝜆<∞,𝑎₂>0,𝑏₂>0, (22) respectively. It may be noted that the gamma prior 𝑔₁(𝛼) and 𝑔₂(𝜆) are flexible enough to cover wide variety of the prior believes experimenter. Based on the assumptions stated above, the joint prior pdf of 𝛼 and 𝜆 is 𝑔(𝛼,𝜆)=𝑔₁(𝛼)𝑔₂(𝜆);𝛼>0,𝜆>0. (23) Combining the priors given by (22) with likelihood given by (8), we can easily obtain joint posterior pdf of (𝛼,𝜆) as 𝜋()=𝐽𝛼,𝜆∣𝑥,𝑟¹𝐽₀, (24) where 𝐽₁=𝛼^{(𝑚+𝑏1−1)}𝜆^{(𝑚+𝑏2−1)}𝑒^{−(𝑎1𝛼+𝑎2𝜆)}×^𝑚_𝑖=11𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1},𝐽₀=^∞₀𝐽₁𝑑𝛼𝑑𝜆. (25)

Hence, the respective marginal posterior pdfs of 𝛼 and 𝜆 are given by 𝜋₁(𝛼∣𝑥,𝑟)=^∞₀𝐽₁𝐽₀𝜋𝑑𝜆,₂(𝜆∣𝑥,𝑟)=^∞₀𝐽₁𝐽₀𝑑𝛼. (26) Usually the Bayes estimators are obtained under square error loss function (SELF) 𝑙₁𝜙,𝜙=∈₁𝜙−𝜙²;∈₁>0, (27) where 𝜙 is the estimate of the parameter 𝜙 and the Bayes estimator 𝜙_𝑆 of 𝜙 comes out to be 𝐸_𝜙[𝜙], where 𝐸_𝜙 denotes the posterior expectation. However, this loss function is symmetric loss function and can only be justified if over estimation and under estimation of equal magnitude are of equal seriousness. But in practical situations, this may not be true. A number of asymmetric loss functions are available in statistical literature. Let us consider the general entropy loss function (GELF) proposed by Calabria and Pulcini [20] defined as follows: 𝑙₂𝜙,𝜙=∈₂𝜙𝜙^𝛿𝜙−𝛿ln𝜙−1;∈₂>0. (28) The constant 𝛿, involved in (28), is its shape parameter. It reflects departure from symmetry. When 𝛿>0, then over estimation (i.e., positive error) causes more serious consequences than under estimation (i.e., negative error) and converse for 𝛿<0. The Bayes estimator 𝜙_𝐺 of 𝜙 under GELF is given by 𝜙_𝐺=𝐸_𝜙𝜙^{−𝛿(−1/𝛿)} (29) provided that the posterior expectation exits. It may be noted here that, for 𝛿=−1, the Bayes estimator under loss (27) coincides with the Bayes estimator under SELF 𝑙₁. Expressions for the Bayes estimators 𝛼_𝐺 and 𝜆_𝐺 for 𝛼 and 𝜆, respectively, under GELF can be given as 𝛼_𝐺=^∞₀𝛼^−𝛿𝜋₁(𝛼∣𝑥,𝑟)𝑑𝛼^(−1/𝛿),𝜆_𝐺=^∞₀𝜆^−𝛿𝜋₂(𝜆∣𝑥,𝑟)𝑑𝜆^(−1/𝛿). (30)(31) Substituting the posterior pdfs from (26) in (30) and (31), respectively, and then simplifying then, we get the Bayes estimators 𝛼_𝐺 and 𝜆_𝐺 of 𝛼 and 𝜆 as follows: 𝛼_𝐺=^∞₀𝜆^{𝑏2+𝑚−1}𝑒^{−𝑎2𝜆∞}₀𝛼^{𝑏1+𝑚−𝛿−1}𝑒^−𝑎1𝛼×^𝑚_𝑖=11𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1}×𝑑𝛼𝑑𝜆_∞0𝜆^{𝑏2+𝑚−1}𝑒^{−𝑎2𝜆∞}₀𝛼^{𝑏1+𝑚−1}𝑒^−𝑎1𝛼×^𝑚_𝑖=11𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}×1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1}𝑑𝛼𝑑𝜆^{−1−1/𝛿},𝜆_𝐺=^∞₀𝜆^{𝑏2+𝑚−𝛿−1}𝑒^{−𝑎2𝜆∞}₀𝛼^{𝑏1+𝑚−1}𝑒^−𝑎1𝛼×^𝑚_𝑖=11𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1}×𝑑𝛼𝑑𝜆_∞0𝜆^{𝑏2+𝑚−1}𝑒^{−𝑎2𝜆∞}₀𝛼^{𝑏1+𝑚−1}𝑒^−𝑎1𝛼×^𝑚_𝑖=11𝑥²_𝑖𝑒^{−𝜆/𝑥𝑖}×1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1}𝑑𝛼𝑑𝜆^{−1−1/𝛿}. (32) It may be noted here that the integrals involved in the expressions for the Bayes estimators 𝛼_𝐺 and 𝜆_𝐺 cannot be obtained analytically and one needs numerical techniques for computations. We, therefore, have proposed to use Markov Chain Monte Carlo (MCMC) methods. In MCMC techniques, we considered the Metropolis-Hastings algorithms to generate samples from posterior distributions and these samples are used to compute Bayes estimates. The Gibbs is an algorithm for simulating from the full conditional posterior distributions while the metropolis-hastings algorithm generates samples from an arbitrary proposal distribution. The conditional posterior distributions of the parameters 𝛼 and 𝜆 can be written as 𝜋^∗₁(𝛼∣𝜆,𝑥,𝑟)∝𝛼^{(𝑚+𝑏1−1)}𝑒^{−(𝑎1−∑𝑚𝑖=1(𝑟𝑖+1)log(1−𝑒𝑖−𝜆/𝑥))𝛼},𝜋^∗₂(𝜆∣𝛼,𝑥,𝑟)∝𝜆^{(𝑚+𝑏2−1)}𝑒^−𝑎2𝜆×^𝑚_𝑖=1𝑒^{−𝜆/𝑥𝑖}1−𝑒^{−𝜆/𝑥𝑖(𝑟𝑖+1)𝛼−1}, (33) respectively. For the Bayes estimators, the following MCMC procedure is followed.(I)Set the initial guess of 𝛼 and 𝜆 say 𝛼₀ and 𝜆₀.(II)Set 𝑖=1.(III)Generate 𝛼_𝑖 from 𝜋^∗₁(𝛼∣𝜆_𝑖−1,𝑥,𝑟) and 𝜆_𝑖 from 𝜋^∗₂(𝜆∣𝛼_𝑖,𝑥,𝑟).(IV)Repeat steps (II)-(III), 𝑁 times.(V)Obtain the Bayes estimates of 𝛼 and 𝜆 under GELF as 𝛼_𝐺=𝐸(𝛼^−𝛿∣data)^−1/𝛿=1𝑁−𝑁₀^𝑁−𝑁0_𝑖=1𝛼_𝑖^{−𝛿−1/𝛿},𝜆_𝐺=𝐸(𝜆^−𝛿∣data)^−1/𝛿=1𝑁−𝑁₀^𝑁−𝑁0_𝑖=1𝜆_𝑖^{−𝛿−1/𝛿}, (34) where 𝑁₀ is the burn-in-period of Markov Chain. Substituting 𝛿 equal to −1 in step (V), we get Bayes estimates of 𝛼 and 𝜆 under SELF.(VI)To compute the HPD interval of 𝛼 and 𝜆, order the MCMC sample of 𝛼 and 𝜆 (say 𝛼₁,𝛼₂,𝛼₃,…,𝛼_𝑁 as 𝛼_[1],𝛼_[2],𝛼_[3],…,𝛼_[𝑁]) and (𝜆₁,𝜆₂,𝜆₃,…,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.

4. Expected Experiment Times

In practical situations, an experimenter may be interested to know whether the test can be completed within a specified time. This information is important for an experimenter to choose an appropriate sampling plan because the time required to complete a test is directly related to cost. Under progressive censoring with a fixed number of removal, the time is given by 𝑋_𝑚. Following Balakrishnan and Aggarwala [21], the expected value of 𝑋_𝑚 is given by 𝐸𝑋_𝑚∣𝑅=𝐶(𝑟)^𝑟1_𝑙1^𝑟₌₀²_𝑙2=0⋯^𝑟𝑚_𝑙𝑚=0(−1)^{𝐵𝑟1𝑙1}⋯^𝑟𝑚_𝑙𝑚∏^𝑚−1_𝑖=1ℎ𝑙_𝑖×^∞₀𝑥𝑓(𝑥)𝐹^{ℎ(𝑙𝑚)−1}(𝑥)𝑑𝑥, (35) where 𝐵=∑^𝑚_𝑖=1𝑙_𝑖, ℎ(𝑙_𝑖)=𝑙₁+𝑙₂+⋯+𝑙_𝑖+𝑖, 𝐶(𝑟)=𝑛(𝑛−𝑟₁−1)(𝑛−𝑟₁−𝑟₂−2)⋯[𝑛−∑^𝑚−1_𝑖=1(𝑟_𝑖+1)] and 𝑖 is the number of live units removed from experiment (number of failure units). Using the pdf and cdf of GIED, the equation will be 𝐸𝑋_𝑚∣𝑅=𝐶(𝑟)^𝑟1_𝑙1^𝑟₌₀²_𝑙2=0⋯^𝑟𝑚_𝑙𝑚=0(−1)^{𝐵𝑟1𝑙1}⋯^𝑟𝑚_𝑙𝑚∏^𝑚−1_𝑖=1ℎ𝑙_𝑖𝑆₁, (36) where 𝑆₁=^∞₀𝛼𝜆𝑥𝑒^−𝜆/𝑥1−𝑒^{−𝜆/𝑥𝛼−1}×1−1−𝑒^{−𝜆/𝑥𝛼ℎ(𝑙𝑚)−1}𝑑𝑥. (37) After simplification it reduces to 𝑆₁=𝛼𝜆^{ℎ(𝑙𝑚)−1}_𝑘=0(−1)^𝑘ℎ𝑙_𝑚𝑘×−1_∞0𝑒^−𝜆/𝑥𝑥1−𝑒^{−𝜆/𝑥𝛼(𝑘+1)−1}𝑑𝑥. (38) Putting this value in (36), the expected test time is given by 𝐸𝑋_𝑚∣𝑅=𝐶(𝑟)^𝑟1_𝑙1^𝑟₌₀²_𝑙2=0⋯^𝑟𝑚_𝑙𝑚=0(−1)^{𝐵𝑟1𝑙1}⋯^𝑟𝑚_𝑙𝑚∏^𝑚−1_𝑖=1ℎ𝑙_𝑖×^{ℎ(𝑙𝑚)−1}_𝑘=0(−1)^𝑘ℎ𝑙_𝑚𝑘×−1_∞0𝑒^−𝜆/𝑥𝑥1−𝑒^{−𝜆/𝑥𝛼(𝑘+1)−1}𝑑𝑥. (39) The expected test time for progressively type-II censored data with binomial removals is evaluated by taking expectation on both sides (36) with respect to the 𝑅. That is, 𝐸𝑋_𝑚=𝐸_𝑅𝐸𝑋_𝑚=∣𝑅=𝑟_{𝑔(𝑟1)𝑟1=0𝑔(𝑟2)𝑟2=0}⋯^{𝑔(𝑟𝑚−1)}_{𝑟𝑚−1=0}𝑃𝑅=𝑟_𝑖;𝑝𝐸𝑋_𝑚∣𝑅=𝑟, (40) where 𝑔(𝑟_𝑖)=𝑛−𝑚−𝑟₁−⋯−𝑟_𝑖−1 and 𝑃(𝑅=𝑟_𝑖;𝑝) is given in (10). For the expected time of a complete sampling, case with 𝑛 test units is obtained by taking 𝑚=𝑛 and 𝑟_𝑖=0 for all 𝑖=1,2,…,𝑚, in (39). We have 𝐸𝑋^∗_𝑛=𝑛𝛼𝜆^∞₀𝑒^−𝜆/𝑥𝑥1−1−𝑒^{−𝜆/𝑥𝛼𝑛−1}×1−𝑒^{−𝜆/𝑥𝛼−1}𝑑𝑥 (41) and the expected time of a type-II censoring is defined by the expected value of the 𝑚th failure time; then, 𝐸𝑋^∗_𝑚𝑛𝑚=𝑚𝛼𝜆^∞₀𝑒^−𝜆/𝑥𝑥1−1−𝑒^{−𝜆/𝑥𝛼𝑚−1}×1−𝑒^{−𝜆/𝑥𝛼(𝑛−𝑚+1)−1}𝑑𝑥. (42) The ratio of the expected experiment time (REET) denoted as 𝛿_REET is computed between progressive type-II censored data with binomial removals (PT-II CBR) and the complete sampling. We define 𝛿_REET=𝐸𝑋_𝑚underPT-IICBR𝐸𝑋^∗_𝑛.undercompletesampling (43) It can be noted that 𝛿_REET gives an important information in order to determine the shortest experiment time significantly, and if a much larger sample of 𝑛 test units is used, the test is stopped, when 𝑚th failures are observed. But we are interested here, in order to consider various values of 𝑛, 𝑚 and 𝑝 numerically calculated, the expected experiment times under PT-II CBR and complete sample, which are derived in equations (40) and (41), respectively. The results are presented in Table 1. As mentioned earlier, analytical comparison of the expected test times under PT-II CBR and complete sampling test is very difficult. Hence, it is calculated for different values of 𝑛, 𝑚, and 𝑝. Here, we have considered 𝑛=6,10, and 15 and the choices of 𝑚 are listed in Table 1. The various values of removal probability 𝑝 considered are 𝑝=0.05, 0.1, 0.3, 0.5, 0.7, and 0.9. The results thus obtained are summarized in Table 1. It is noted from the results that for a fixed value of effective sample size 𝑚, the values of 𝛿_REET decrease as 𝑛 increases. For fixed 𝑛, the values of 𝛿_REET and expected termination time under PT-II CBR and complete sampling test increase as 𝑚 increases. Moreover, from Table 1, the expected test time is influenced by the value of the removal probability 𝑝 from effected sample size 𝑚. So, 𝑝 is an important factor on the expected test time and when 𝑝 is large, (𝑛−𝑚) units are removed at the earlier stage of life test out of 𝑛 units. Hence, this result gives an idea to the observed lifetimes much closer to tail of the failure time distribution. Thus, the expected test time of PT-II CBR is closed to that of complete sample. Figure 1 represents the ratio of the expected test time under PT-II CBR to the expected test time under complete sample versus 𝑛 for 𝑚=8 and different values of the removal probability 𝑝. Finally, we observed that form Figure 1, for larger values of 𝑝, the ratio is approaching to 1 quickly. Hence, up 𝑝≤0.5 is more significant for the reduction of expected test time. So, the expected termination time for binomial removal with 𝑝=0.5 is to be taken for further calculation.

Table 1 

Expected experiment time 𝐸(𝑋_𝑚) and 𝛿_REET (in the brackets) for (𝛼,𝜆)=(2,2) under PT-II CBR.

Figure 1 

𝛿REET under PT-II CBR to 𝛿REET under complete sample.

5. Algorithm for Sample Simulation under PT-II CBR

We need to simulate PT-II CBR from specified GIED and propose the use of the following algorithm.(I)Specify the value of 𝑛.(II)Specify the value of 𝑚.(III)Specify the value of parameters 𝛼, 𝜆, and 𝑝.(IV)Generate random number 𝑟_𝑖 from 𝐵(𝑛−𝑚−∑^𝑖−1_𝑙=0𝑟_𝑙,𝑝), for 𝑖=1,2,3,…,𝑚−1.(V)Set 𝑟_𝑚 according to the following relation:(VI)𝑟_𝑚=^{𝑛−𝑚−∑𝑚−1𝑙=1𝑟𝑙,if𝑛−𝑚−∑𝑚−1𝑙=1𝑟𝑙>0,}_0,otherwise.(VII)Generate 𝑚 independent 𝑈(0,1) random variables 𝑊₁,𝑊₂,…,𝑊_𝑚.(VIII)For given values of the progressive type-II censoring scheme 𝑟_𝑖(𝑖=1,2,…,𝑚) set 𝑉_𝑖=𝑊^{1/(𝑖+𝑟𝑚+⋅+𝑟𝑚−𝑖+1)}_𝑖  (𝑖=1,2,…,𝑚).(IX)Set 𝑈_𝑖=1−𝑉_𝑚𝑉_𝑚−1,…,𝑉_{𝑚−𝑖+1}  (𝑖=1,2,…,𝑚), then 𝑈₁,𝑈₂,…,𝑈_𝑚 are PT-II CBR samples of size 𝑚 from 𝑈(0,1).(X)Finally, for given values of parameters 𝛼 and 𝜆, set 𝑥_𝑖=𝐹⁻¹(𝑈)  (𝑖=1,2,…,𝑚). Then, (𝑥₁,𝑥₂,…,𝑥_𝑚) is the required PT-II CBR sample of size 𝑚 from the GIED.

6. Simulation Studies

The estimators 𝛼_𝑀 and 𝜆_𝑀 denote the MLEs of the parameters 𝛼 and 𝜆, respectively, while 𝛼_𝑆, 𝜆_𝑆 and 𝛼_𝐺, 𝜆_𝐺 are the corresponding Bayes estimators under SELF and GELF, respectively. We compare the estimators obtained under GELF with corresponding Bayes estimators under SELF and MLEs. The comparisons are based on the simulated risks (average loss over sample space) under GELF. Here, ((𝛼_𝐿^𝑐𝛼_𝑈^𝑐),(𝜆_𝐿^𝑐𝜆_𝑈^𝑐)) and ((𝛼_𝐿^ℎ𝛼_𝑈^ℎ),(𝜆_𝐿^ℎ𝜆_𝑈^ℎ)) represent confidence interval (CI) and HPD of 𝛼 and 𝜆, respectively. Also (avg.𝐿_𝛼𝑐,avg.𝐿_𝜆^𝑐) and (avg.𝐿_𝛼ℎ,avg.𝐿_𝜆^ℎ) represent the average length of CI and HPD of 𝛼 and 𝜆, respectively. It may be mentioned here that the exact expressions for the risks cannot be obtained as estimators are not found in nice closed form. Therefore, the risks of the estimators are estimated on the basis of MonteCarlo simulation study of 5000 samples. It may be noted that the risks of the estimators will depend on values of 𝑛, 𝑚, 𝑝, 𝜆, 𝛼, and 𝛿. The choice of hyperparameters 𝛼 and 𝜆 can be taken in such a way that if we consider any two independent information as prior mean and variance of 𝛼 and 𝜆, then, (𝜇₁=𝑏₁/𝑎₁,𝜎₁=𝑏₁/𝑎²₁) and (𝜇₂=𝑏₂/𝑎₂,𝜎₂=𝑏₂/𝑎²₂), respectively whereas 𝜇₁ and 𝜇₂ are considered as true values of the parameters 𝛼 and 𝜆 for different confidence in terms of smaller, moderate, and larger variances, On the basis of this information, the hyper parameters of 𝛼 and 𝜆 (say (𝑏₁,𝑎₁) and (𝑏₂,𝑎₂)) can be easily evaluated from this relation, (𝑎₁=𝜇₁/𝜎₁,𝑏₁=𝜇²₁/𝜎₁) and (𝑎₂=𝜇₂/𝜎₂,𝑏₁=𝜇²₂/𝜎₂), respectively.

In order to consider the variation in the values of 𝑚,𝛿, and 𝑣, we have obtained the simulated risks for 𝑚=9,12,15, 𝜆=2=𝜇₂ (say prior mean of 𝜆), 𝛼=2=𝜇₁ (say prior mean of 𝛼), 𝜎₁=(0.4,1,10) (say prior variance of 𝛼), 𝜎₂=(0.4,1,10) (say prior variance of 𝜆) and 𝛿=(−0.25,0.5) (say GELF loss parameter). Generating the samples of PT-II CBR as mentioned in Section 5, the simulated risks under SELF and GELF have been obtained for selected values of 𝑛, 𝑚, 𝑝, 𝜆, 𝛼, and 𝛿. The results are summarized in Tables 2–4. It is noted from Table 2 that for almost all the considered values of 𝛿, the risks of the estimators 𝛼_𝐺 and 𝜆_𝐺 have minimum as compared to the considered competitive estimators, respectively, under GELF. To know the effect of variation in the value of other parameters on the risks of the estimators of 𝛼 and 𝜆, we arbitrarily fixed 𝛿=0.5 (for the case when overestimation is more serious than underestimation) and 𝛿=−0.25 in reverse situation. Tables 3 and 4 represent the risks of estimators of 𝛼 and 𝜆 under both losses in under and overestimation situations when the prior mean is same as the true value of the parameters 𝛼=2 and 𝜆=2 for smaller, moderate, and larger values of the prior variance of the parameters in order to consider the hyperparameters (𝑎₁=𝑎₂=5,𝑏₁=𝑏₂=10), (𝑎₁=𝑎₂=2,𝑏₁=𝑏₂=4) and (𝑎₁=𝑎₂=1/5,𝑏₁=𝑏₂=2/5), respectively. When the effective sample size 𝑚 increases, the risks of all the estimators of 𝛼 and 𝜆 under both losses decrease for 𝛿=(−0.25,0.5) and the simulated risks of 𝛼_𝐺 and 𝜆_𝐺 are smaller than those of (𝛼_𝑆, 𝜆_𝑆) and (𝛼_𝑀, 𝜆_𝑀) for all the considered cases, including those where under estimation is considered to be more serious than over estimation or viceversa. Under different prior variances along with variation of effective sample size 𝑚, the 95% HPD and CI intervals are obtained. In Table 5, it is observed that the average length of CI and HPD interval decreases when the effective sample size 𝑚 increases and average length of HPD interval is always less than that of CI, which are represented in Figure 2 also.

Table 2 

Risks of estimators of 𝛼 and 𝜆 under GELF for fixed value of 𝑛=15, 𝑚=9, 𝑝=0.5, 𝑎₁=0.00001, 𝑏₁=0.0001, 𝑎₂=0.00001, and 𝑏₂=0.0001.

Table 3 

Risks of estimators of 𝛼 and 𝜆 under SELF for fixed, 𝑛=15, 𝛼=2, 𝜆=2, and 𝑝=0.5.

Table 4 

Risks of estimators of 𝛼 and 𝜆 under GELF for fixed, 𝑛=15, 𝛼=2, 𝜆=2, and 𝑝=0.5.

Table 5 

95% CI and HPD for different sample 𝑚 and prior variances 𝑣 for fixed 𝑛=15, 𝛼=2, and 𝜆=2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 2 

Average HPD and CI for different prior variances for fixed confidence coefficient.

7. Real Data Analysis

Here, we consider the real data set presented in Lawless [22] which represent the number of revolutions to failure for each of 23 ball bearing in a life test. But for the purpose of illustrating the method discussed in this paper, a PT-II CBR is generated from real data set, and we consider three schemes under PT-II CBR which are given in Table 6. Table 7 shows the MLEs and Bayes estimators of 𝛼 and 𝜆 under SELF and GELF, CI/HPD interval based on complete real data set. For this real data set, Abouammoh and Alshingiti [5] indicated that the GIED provides satisfactory fit. This real data set was originally discussed by Lieblein and Zelen [23]. In Section 3, MCMC algorithm and mathematical treatments are given, for long run, we take noninformative prior for this purpose, and the value of the hyper parameters 𝛼 and 𝜆 are taken as (𝑏₁=0.00001,𝑎₁=0.0001) and (𝑏₂=0.00001,𝑎₂=0.0001), respectively. Hence, on the basis of Table 6, we use noninformative prior under different degrees of censoring; the MLEs and the Bayes estimators and CI/HPD interval of 𝛼 and 𝜆 under SELF and GELF for 𝛿=±0.5 are presented in Tables 8 and 9, respectively. Hence, finally, the study of Tables 8 and 9 observed that the MLEs and Bayes estimators and length of CI/HPD interval of 𝛼 and 𝜆 decrease as degree of censoring decreases, respectively.

Table 6 

PT-II CBR under different censoring scheme (𝑆_𝑛∶𝑚) for fixed 𝑛=23, 𝑝=0.5.

Table 7 

Bayes and ML estimates based on real data set for 𝑛=23, 𝑝=0.05.

Table 8 

Bayes and ML estimates, CI and HPD intervals for 𝛼 with fixed 𝑛=23 and 𝑝=0.5 under PT-II CBR.

Table 9 

Bayes and ML estimates, CI and HPD intervals for 𝜆 with fixed 𝑛=23 and 𝑝=0.5 under PT-II CBR.

8. Conclusion

(1) On the basis of simulation study, we observed that the maximum likelihood and Bayes methods are used for estimating the parameters under GELF and SELF of GIED based on PT-II CBR. Bayes estimators have been obtained on the basis of MCMC method. These methods are applied to real data set based on the number of revolutions to failure for each of 23 ball bearing in a life test.

(2) It has been noticed, under consideration of different prior believes, from tables, that the estimated risks of estimators decrease as effective sample size increases and Bayes estimates have the smallest estimated risks as compared with their corresponding MLEs. Hence, the proposed estimators (𝛼_𝐺,𝜆_𝐺) perform better than (𝛼_𝑆,𝜆_𝑆) and (𝛼_𝑀,𝜆_𝑀) for different degree of censoring, when under estimation is serious than over estimation and vice-versa. The CI/HPD interval is also obtained. We found the Bayes estimates are superior than those of the corresponding MLEs.

(3) We have obtained the expected test times under PT-II CBR and complete sampling to compare it. In Table 1, the numerical results indicate that the expected test time depends very much times on the values of removal probability. When the probability of removal is large, a slight reduction in the expected test time can be achieved only by increasing the total number of test units 𝑛.