Bayesian and Non-Bayesian Inference for Weibull Inverted Exponential Model under Progressive First-Failure Censoring Data

: In this article, the estimation of the parameters and the reliability and hazard functions for Weibull inverted exponential (WIE) distribution is considered based on progressive ﬁrst-failure censoring (PFFC) data. For non-Bayesian inference, maximum likelihood (ML) estimators are acquired; meanwhile, their existence is veriﬁed. Via asymptotic normality of ML estimators and delta method, the corresponding conﬁdence intervals (CIs) of the parameters and the reliability and hazard functions are constructed. For Bayesian inference, Lindley’s approximation and Markov chain Monte Carlo (MCMC) techniques are proposed to obain the Bayes estimators and the corresponding credible intervals (CRIs). To this end, both symmetric and asymmetric loss functions are used. A large number of Monte Carlo simulations are implemented to evaluate the efﬁciency of the developed methods. Eventually, a numerical example is analyzed for illustrative purposes.


Introduction
In life-testing experiments, it is impossible for experimenters to obtain complete information on failure items. For the sake of reducing costs and saving time, the sampling process is truncated in accordance with the pre-specified censoring plan. Hence, the final sample we obtain is named the censored sample. A relatively good censoring scheme design can achieve efficient statistical inference and cost savings, simultaneously. To improve the search efficiency on incomplete data, statisticians have already proposed a variety of censored data. The most common schemes are considered type I and type II censoring. These types have been studied by several statisticians; see, for instance [1][2][3]. In terms of the procedure, in type I censoring, all items n are put in the test for a pre-specified time and at the end of the specified time, the test ends. In type II censoring, all units n are put in the test, and the test is terminated at the failure of the pre-specified m-th unit (1 ≤ m ≤ n). The disadvantages of these types are represented in that the units cannot be removed during the test. Thus, a progressive type-II censoring (Pro-II-C) was proposed, which has more flexibility in allowing units to be withdrawn within the duration of the test. In this censoring, the n items are put to a life test, then at the first failure occurrence X 1 , the surviving units R 1 are withdrawn at random, then at the second failure occurrence X 2 , the surviving units R 2 are withdrawn at random, and so on until the occurrence of the m-th are called a Pro-II-C sample with a censoring scheme R = (R 1 , . . . , R m ). For more details about progressive censoring, see [4][5][6][7][8][9]. Although the experimental efficiency under Pro-II-C can be significantly improved, the duration of the test is still too long. So, ref. [10] proposed another life test in which the test units are divided into several groups of the same size in which all units are run simultaneously until, in each group, the first failure occurs. This type is called first-failure censoring; for more details, see, for example [11,12]. Unfortunately, in the first-failure censoring, the groups during the test cannot be removed. So, to overcome this obstacle and to better improve the test efficiency, a new life test that combines progressive censoring with first-failure censoring was proposed by [13], which is called a PFFC scheme. It is an extension and improvement of progressive censoring, and it is widely used in experimental design due to its flexibility. In this scheme, assume that n independent groups with k units within each group are put on a life test. As soon as the first failure X R 1 1:m:n:k appears, R 1 random groups as well as the group which includes the first failure are withdrawn; as soon as the second failure X R 2 2:m:n:k appears, R 2 random groups as well as the group which includes the second failure are withdrawn, and so on. Finally, when the m-th failure X R m m:m:n:k appears, the remaining groups R m are withdrawn, and the test is terminated. Hence, the observed failure times X R 1 1:m:n:k < X R 2 2:m:n:k < · · · < X R m m:m:n:k are called a PFFC order statistics of size m with a censoring plan R = (R 1 , . . . , R m ).
If the failure times of the N = n × k units are from a continuous population with the cumulative distribution function (CDF) F(x) and the probability density function (PDF) f (x), then the joint PDF for the observed PFFC: X R 1 1:m:n:k < X R 2 2:m:n:k < · · · < X R m m:m:n:k is given by [13] as follows: . (1) where 0 < x R 1 1:m:n:k < x R 2 2:m:n:k < · · · < x R m m:m:n:k < ∞ and c is defined as From Equation (1), some sampling schemes can be obtained from the PFFC as special cases, such as the following: Under the PFFC scheme, studies of many distribution models have been performed by many authors. Wu and Huang [14] developed a reliability sampling plans for the Weibull distribution in presence of PFFC scheme. In [15], point and interval estimations for the parameters of the exponentiated exponential (EE) distribution were studied based on PFFC data. Ahmadi and Doostparast [16] considered the problem of estimating the lifetime performance index on the basis of PFFC samples under the Pareto distribution; moreover, the problem of testing hypotheses on the lifetime performance index was studied based on both Bayesian and non-Bayesian approaches. Kayal et al. [17] discussed one-and twosample prediction problems for the two-parameter Chen distribution on the basis of PFFC. Zhang and Gui [18] used Bayesian and non-Bayesian approaches to estimate the unknown parameters and the reliability and failure rate functions of the inverted exponentiated half-logistic distribution under PFFC plan. Moreover, ref. [19] used the Tierney-Kadane approximation method for Bayesian computation and constructed the highest posterior credible intervals of the parameters of the inverse power Lomax distribution under the PFFC scheme.
Chandrakant et al. [20] proposed a three-parameter WIE distribution, which is considered an extension of the inverted exponential distribution. The WIE distribution is flexible in nature and can take several shapes, such as J-reversed, positively skewed, and symmetric as well. Additionally, the shape of the WIE distribution could either be unimodal or decreasing. The shape of the hazard function can be decreasing, increasing and an inverted bathtub (depending upon the values of the parameters). According to the previous features, the WIE distribution can be used to fit different data in several vital fields, such as engineering, industry, biomedical studies, and medicine, to contribute to solving many obstacles.
In our study, the failure times are assumed to be from the WIE (α, β, λ) distribution. Thus, the PDF and CDF of WIE (α, β, λ) distribution with random variable X are, respectively, as follows: and where λ is the scale parameter, and α and β are the shape parameters whose role is to vary the tail weight. The reliability function S(t) and hazard function H(t) are as follows: and For some statistical properties as well as the properties of order statistics of the WIE distribution, one can refer to [20]. To our best knowledge, statistical inference for unknown parameters and reliability characteristics of a three-parameter WIE distribution has not yet been studied under PFFC data. The objective of this paper is to make an inference about the WIE distribution parameters under the PFFC scheme. Hence, ML inference is implemented. Additionally, the approximate confidence intervals (ACIs) of the parameters are constructed. By using the Lindley approximation form as well as the Metropolis-Hasting (M-H) algorithm within the Gibbs sampler, the Bayes estimates are obtained. Additionally, the corresponding CRIs are constructed under the MCMC technique.
The rest of the paper is as follows: In Section 2, we highlight the ML estimators as well as the proof of their existence, the observed Fisher information matrix (FIM), and asymptotic interval inference. Bayes estimates using Lindley's approximation and MCMC technique are provided in Section 3. In Section 4, a simulation study is provided to inspect the performance of the proposed methods. Section 5, discusses a numerical example for illustrative purposes. Eventually, a brief conclusion is given in Section 6.

Maximum Likelihood Inference
In this section, the ML estimates of the parameters α, β, λ, S(t), and H(t) are discussed under the PFFC scheme. In addition, the ACIs are constructed. By substituting (3) and (4) into (1), the likelihood function is as follows: where x i is used instead of x R i i:m:n:k . The log-likelihood function = ln L(α, β, λ|x) is given as After equating each differentiating with respect to α, β, and λ to zero, the likelihood equations are given as and From (9), we obtain the ML estimator of α as follows: It is noticeable that Equations (9)-(12), analytically, cannot be solved, so a numerical method, such as Newton-Raphson iteration, is necessary to construct the estimates. The ML estimators of S(t) and H(t) can be obtained according to the invariance property of ML estimators by puttingα,β, andλ instead of α, β, and λ in (5) and (6) as follows:

Existence of the Maximum Likelihood Estimators
In this part, the necessary and sufficient conditions for the existence of the ML estimators for arbitrary PFFC data are discussed. To this end, the behavior of the Equations (9)- (11) is examined on the positive real line (0, ∞).

Bayesian Inference
Bayesian approach is a widely used and more effective alternative to the frequentist approach. In this section, two Bayesian inference procedures (Lindley's approximation and MCMC) are proposed to estimate the parameters α, β, λ, S(t), and H(t). Due to the influential role that the loss function plays, both squared error (SE) and linear exponential (LINEX) loss functions are considered to deduce the Bayes estimates. Additionally, the corresponding CRIs are constructed under the MCMC technique.
Assume that the parameters α, β, and λ follow the following gamma prior distributions where µ i and η i , i = 1, 2, 3 reflect the prior knowledge about α, β, and λ, and they are assumed to be known and nonnegative hyperparameters. In addition, the parameters α, β, and λ as well as the corresponding priors are assumed here to be independent. Note that, when (µ 1 , µ 2 , µ 3 , η 1 , η 2 , η 3 ) equal or approach to zero, the prior distributions are said to be non-informative priors, but otherwise, the prior distributions are said to be informative priors of α, β, and λ.
The joint prior of the parameters α, β, and λ can be written as follows: Via the Bayes theorem across combining (7) with (21), the posterior distribution of α, β, and λ can be written as follows: where h 1 is the normalizing constant. Therefore, for any function of α, β, and λ, say ω(α, β, λ), the Bayes estimator under SE and LINEX loss functions can be obtained as follows: It is noticeable that Equations (23) and (24), analytically, can not be solved. So, we will apply Lindley's approximation and MCMC technique.

Lindley's Approximation
It is clear from Equations (23) and (24) that the Bayes estimators under SE and LINEX loss functions include integrals, so an explicit expression for each parameter cannot be acquired. Therefore, Lindley's approximation, which was introduced by [24] to approximate two integral ratios, can be utilized to obtain the Bayes estimators. This approximation formula has been used by several authors; see, for example [25]. Now, we consider the ratio of integral I(x), where where ψ(α, β, λ) is the function of the parameters. (α, β, λ) is the log-likelihood function. ρ(α, β, λ) is the log of the joint prior. According to [24], I(x) in (25) can be calculated as All terms in (26) are evaluated at the ML estimators, wherê a i =ρ 1σi1 +ρ 2σi2 +ρ 3σi3 for i = 1, 2, 3. a 4 =ψ 12σ12 +ψ 13σ13 +ψ 23σ23 , From (21), ρ(α, β, λ) can be written as follows: (27) and then we obtain Now for various parameters, the Bayes estimates can be obtained under symmetric and asymmetric loss functions as follows: -Under the SE loss function, the approximate Bayes estimators for Φ = (α, β, λ) can be obtained as follows: -Under the LINEX loss function, the approximate Bayes estimators for Φ = (α, β, λ) can be obtained as follows: The approximate Bayes estimators of S(t), and H(t) can be obtained similarly. These are omitted for the sake of conciseness. Unfortunately, the CRIs of the unknown parameters cannot be constructed since Lindley's approximation makes the point estimation only. So, we will highlight overcoming this obstacle in the MCMC technique.

MCMC Technique
The MCMC technique is one of the most general techniques for an estimation, which is provided here to compute the Bayes estimates and the corresponding CRIs for the parameters α, β, λ, S(t), and H(t) of the WIE distribution. MCMC procedures, such as Gibbs sampler and the M-H algorithm are widely applied in several statistical analyses. For the Gibbs method proposed by [26], the conditional distributions of each parameter are required for sampling. As for the M-H algorithm, which is proposed by [27,28], random samples can be generated from any complex distribution. Here, a more general procedure of the MCMC algorithm that will be used is the M-H algorithm within Gibbs sampling.
From Equation (22), the conditional density functions of α, β, and λ are as follows: and It is noticeable that Equation (31) represents a gamma density with shape parameter (m + µ 1 ) and scale parameter Hence, by implementing any gamma generating routine, samples of α can be simply generated. It is noticeable that Equations (32) and (33) cannot be reduced to a common distributions, but their plots (see Figures 1 and 2) show that they are semi-normal. So, the Gibbs sampler is not an accurate option; alternatively, the M-H algorithm is implemented to conduct the MCMC methodology. To run the procedure of the M-H algorithm within the Gibbs sampler, we start with the ML estimatorsα,β, andλ as initial values. Now, the procedure is illustrated in the following steps: (1): Start with (α (0) , β (0) , λ (0) ) as a guess of initial, and put J = 1.

Simulation Study
For the sake of evaluating the effectiveness of the proposed estimates (point and interval), Monte Carlo simulations were implemented in this section. Hence, a comparison of different estimation methods was conducted in terms of the average mean (AM) and mean square error (MSE) for point estimates and the average width (AW) and coverage probability (CP) for interval estimates. Distinct combinations of group size k = (2, 4), various values of n (number of groups), and m (observed data) with different censoring schemes R are taken into consideration. We consider the following censoring schemes (CSs): CS III: A large number of PFFC samples were generated from WIE distribution with the parameters α = 0.3, β = 0.2, and λ = 0.1 by implementing the proposed algorithm by [29] with distribution function 1 − (1 − F(x)) k . For informative priors, the associated hyperparameters are selected as µ 1 = 9.0052, η 1 = 27.8632, µ 2 = 42.1144, η 2 = 207.2652, µ 3 = 4.4958, and η 3 = 36.7804; this guarantees that the expectation of the prior distribution corresponding to each parameter is equivalent to the true value. This approach was also used by some authors, such as [30]. The true values of S(t) and H(t) at t = 0.24 are S(t = 0.24) = 0.7101 and H(t = 0.24) = 0.3488. Additionally, the Bayes estimates were derived depending on 12,000 and getting rid of the first 2000 iterations. This procedure was repeated 1000 times. The results of the simulation study are displayed in Tables 1-5. Note that the MSEs of α, β, λ, S, and H are multiplied by 10 −2 . It is known that model identification is a common problem for distribution with two or more shape parameters. In this article, we overcame this problem using (NMaximize package in MATHEMATICA ver. 12), and we obtained the estimates for the three parameters of the model from different estimation methods. In accordance with the simulation results, we note the following:

1.
As the effective sample sizes m increase for fixed n, the MSEs and the average widths of α, β, λ, S(t), and H(t) decrease.

2.
When n and m are fixed but k increases, the MSEs have no obvious trend on the whole.

3.
Lindley's approximation performs better than the ML estimation on the basis of the smallest MSEs.

4.
Bayes estimates with respect to the informative priors perform better and more accurately than the ML estimators according to the MSE. When the prior information is lacking in practice, ML estimation can be taken into account as a good procedure to perform point estimates. 5. Table 5 shows that the CRIs of the MCMC technique are better and more accurate than the ACIs obtained via normal approximation to the ML estimation, due to the existence of smaller widths. 6.
Finally, for large sample sizes, scheme I performs mostly better than schemes II and III due to the smaller MSEs.

Numerical Example
For illustrative purposes, real-life data are presented in this section. These data represent the survival times (in years) of a group of patients given a combination of chemotherapy and radiation therapy in the treatment of nonresectable gastric carcinoma. These data are a subset of data reported by [31] and used by [32]. The data include 45 survival times for 45 patients as follows: For the goodness of fit test, we compute the Kolmogorov-Smirnov (K-S) distance between the fitted and the empirical distribution functions. The K-S is 0.0855 and the associated p-value is 0.8691. So, in accordance with the p-value, the WIE distribution fits quite well with the above data. Figure 3 shows the empirical and fitted survival functions. In our study, a PFFC sample is obtained as follows: From the original data, the items are divided at random into n = 15 groups with k = 3 (units within each group). The PFFC sample of size 10 is acquired as (0.047, 0.132, 0.458, 0.54, 0.644, 0.863, 1.271, 1.589, 2.416, 3.743) with R = (3, 0, 0, 0, 1, 0, 0, 1, 0, 0), where m = 10, and 5 groups are censored.
By implementing the method of the Newton-Raphson iteration, the ML estimators of α, β, λ, S(t), and H(t) are obtained. Additionally, the corresponding CIs are constructed in accordance with the asymptotic normality of the ML estimators and delta method. Now, to compute the Bayes estimators, the informative gamma priors is considered and the hyperparameters are selected as µ 1 = 3.3739, η 1 = 80.0577, µ 2 = 62.5384, η 2 = 69.5859, µ 3 = 0.1023, and η 3 = 1.0285. Firstly, under Lindley's approximation, the Bayes estimators are obtained as described in Section 3.1. Secondly, under MCMC method, the posterior analysis is implemented by combining the M-H algorithm within Gibbs sampling. Hence, the initial values for the parameters α, β, and λ are taken to be their ML estimators; additionally, 12,000 MCMC iterations are generated. In order to erase the effect of the initial values, we neglect the first 2000 iterations as 'burn-in', which is considered enough to achieve that purpose. Further, Lindley's approximation and MCMC techniques are conducted under both SE loss function and LINEX loss function with the parameter c having different values. Tables 6 and 7 show the point and interval estimates of the proposed methods. We can observe the convergence in the MCMC technique through the trace plots of the parameters, which are shown in Figure 4. Figure 5 shows the histograms of the parameters obtained from MCMC technique.

Conclusions
Throughout the article, we developed different methods to estimate the parameters as well as the reliability and hazard functions for the WIE distribution using the PFFC scheme. The ML estimators were obtained and the observed FIM and the delta method were used to construct the CIs. Proof of the existence of the ML estimators was also provided. On the other hand, Bayes estimates were proposed under both symmetric and asymmetric loss functions, which fit some real-life situations well. Since it is hard to produce Bayes estimates in closed form, two approximation procedures were used: Lindley's approximation and MCMC techniques. One advantage of the MCMC technique is that it can be used to derive the CRIs. In order to check and compare the performance and efficacy of the proposed methods, a simulation study was conducted with different sample sizes (n, m) and different CSs. Eventually, an application to real-life data was used to illustrate how the proposed methods work.