Generalized Bayes Estimation Based on a Joint Type-II Censored Sample from K-Exponential Populations

: Generalized Bayes is a Bayesian study based on a learning rate parameter. This paper considers a generalized Bayes estimation to study the effect of the learning rate parameter on the estimation results based on a joint censored sample of type-II exponential populations. Squared error, Linex, and general entropy loss functions are used in the Bayesian approach. Monte Carlo simulations were performed to assess how well the different approaches perform. The simulation study compares the Bayesian estimators for different values of the learning rate parameter and different losses


Introduction
Generalized Bayes is a Bayesian study based on a learning rate parameter (0 < η < 1). As a fractional power on the likelihood function L ≡ L(θ; data) for the parameter θ ∈ Θ, the traditional Bayesian framework is obtained for η = 1. In this paper, we will show the effect of the learning rate parameter on the estimation results. That is, if the prior distribution of the parameter θ is π(θ), then the generalized Bayes posterior distribution for θ is: π * (θ|data) ∝ L η π(θ), θ ∈ Θ, 0 < η < 1. ( For more details on the generalized Bayes method and the choice of the value of the rate parameter, see, for example [1][2][3][4][5][6][7][8][9][10]. In addition, we refer readers to [11,12] for recent work on Bayesian inversion. An exact inference method based on maximum likelihood estimates (MLEs), and compared its performance with approximate, Bayesian, and bootstrap methods developed by [13]. A joint progressive censoring of type-II and the expected values of the number of failures for two populations under joint progressive censoring of type-II introduced and studied by [14]. Exact likelihood inference for two exponential populations under joint progressive censoring of type II was studied by [15]. A precise result based on maximum likelihood estimates developed by [16]. A study of Bayesian estimation and prediction based on a joint type-II censored sample from two exponential populations was presented by [17]. Exact likelihood inference for two populations of two-parameter exponential distributions under joint censoring of type II was studied by [18].
Suppose that products from k different lines are manufactured in the same factory and that k independent samples of size n j , 1 ≤ j ≤ k are selected from these k lines and simultaneously subjected to a lifetime test. To reduce the cost of the experiment and shorten the duration of the experiment, the experimenter can terminate the lifetime test experiment once a certain number (say r) of failures occur. In this situation, one is interested in either a point or interval estimate of the mean lifetime of the units produced by these k lines.
Suppose {X n j j , j = 1, . . . , k} are k-samples where, X n j j = {X j1 , X j2 , . . . , X jn j } are the lifetimes of n j copies of product line A j and assumed to be independent and identically distributed (iid) random variables from a population with cumulative distribution function (cdf) F j (x) and probability density function (pdf) f j (x).
Furthermore, let N = ∑ k j=1 n j be the total sample size and r be the total number of observed failures. Let W 1 ≤ . . . ≤ W N denote the order statistics of the N random variables {X n j j , j = 1, . . . , k}. Under the joint Type-II censoring scheme for the k-samples, the observable data consist of (δ, W), where W = (W 1 , . . . , W r ), W i ∈ {X n j i j i , j i = 1, . . . , k}, and r is a predefined integerand δ = δ 1j , . . . , δ rj associated with (j 1 , . . . , j r ) is defined by: If r j = ∑ r i=1 δ ij denotes the number of X j -failures in W and r = ∑ k j=1 r j , then, the joint density function of (δ, W) is given by: The main goal of this paper is to consider the Bayesian estimation of the parameter based on the learning rate parameter under a joint censoring scheme of type-II for exponential populations when censoring is applied to the samples in a combined manner. Section 2 presents the maximum likelihood and generalized Bayes estimators, using squared error, Linex, and general entropy loss functions in the Bayesian approach to estimate the population parameters. A numerical investigation of the results from Section 2 is presented in Section 3. Finally, we conclude the paper in Section 4.

Estimation of the Parameters
Suppose that for 1 ≤ j ≤ k, the k populations are exponential with the following pdf and cdf: Then, the likelihood function in (3) becomes: where Θ = (θ 1 , . . . , θ k ) and u j = ∑ r i=1 w i δ ij + w r n j − r j .

Maximum Likelihood Estimation
From (5), the MLE of θ j , for 1 ≤ j ≤ k, is given by: Remark 1. MLEs of θ j exist if we have at least k failures (r ≥ k), which means at least one failure from each sample, i.e., 1 ≤ r j ≤ r − k + 1 and r j ≤ n j .
We determined the MLEs to compare their results with those of Bayesian estimation, which uses the three types of loss functions for different values of the rate parameters, as described in Section 3.

Generalized Bayes Estimation
Since the parameters Θ are assumed to be unknown, we can consider the conjugate prior distributions of Θ as independent gamma prior distributions, i.e., θ j ∼ Gam a j , b j . Therefore, the joint prior distribution of Θ is given by: where and Γ(·) denotes the complete gamma function. Combining (5) and (7), after raising (5) to the fractional power η, the posterior joint density function of Θ is then: Since π j is a conjugate prior, we see that if θ j ∼ Gam a j , b j , then it has the posterior density function as θ j data ∼ Gam r j η + a j , u j η + b j .
In generalized Bayes estimation, we consider three types of loss functions: (i) The squared error loss function (SE), which is classified as a symmetric function and gives equal importance to losses for overestimates and underestimates of the same magnitude; (ii) The Linex loss function, which is asymmetric; (iii) The generalization of the entropy (GE) loss function.
Using (9), the Bayesian estimators of θ j under the squared error (SE) loss function are: Under the Linex loss function, the Bayesian estimators of θ j are given by: and under the GE loss function, the Bayesian estimators of θ j are given bŷ Remark 2. Obviously,θ j for 1 ≤ j ≤ k in the above three cases are the unique Bayes estimators of θ j and thus admissible. The estimatorsθ jJ , are Bayes estimators of θ j using the noninformative Jeffreys priors π J ∝ ∏ k j=1 1 θ j , obtained directly by substituting a j = b j = 0 into (9), so that (10) leads to MLEsθ jM .

Remark 3.
For c = 1, −1, −2, the Bayes estimatesθ jE agree with the Bayes estimates under the following losses: weighted squared error loss function, squared error loss function, and precautionary loss function.

Numerical Study
This section examines the results of a Monte Carlo simulation study to evaluate the performance of the inference procedures derived in the previous section. An example is then presented to illustrate the inference methods discussed here.

Simulation Study
We have considered the different choices for the three populations sample sizes (n 1 , n 2 , n 3 ) and also for r. We choose the exponential parameters (θ 1 , θ 2 , θ 3 ) as (1, 2, 3) and for the Monte Carlo simulations, we use 10,000 replicates. Using (6), we obtain the MLEs of θ 1 , θ 2 , θ 3 and their estimated risk, which are shown in Table 1. Table 1. Average value of (r 1 , r 2 , r 3 ) and the average value and estimated risk (ER) of the MLEŝ θ 1M ,θ 2M ,θ 3M for different choices of n 1 , n 2 , n 3 , r. In the simulation study, it should be noted that some of the simulated samples do not meet the condition in Remark 1 and, therefore, must be discarded. Thus, the average values of the observed failures (r 1 , r 2 , r 3 ) are calculated and reported in Table 1. For the Bayesian study, the hyperparameters are represented by ∆ = (a 1 , b 1 , a 2 , b 2 , a 3 , b 3 ), where ∆ = ∆ 1 = (1, 1, 2, 1, 3, 1).

Conclusions
In this work, we considered a joint type-II censoring scheme when the lifetimes of three populations have exponential distributions. We obtained the MLEs and the Bayesian estimates of the parameters using different values for the learning rate parameter η and the loss functions SE, GE, and Linex in a simulation study and an illustrative example. In both methods, the MLEs and the generalized Bayes estimatesθ 1 ,θ 2 ,θ 3 become better with more significant n i ; i = 1, 2, 3 for different values of r; of course, the estimates become better, but the Bayes estimators are better than the MLEs. From Table 2 to Table 7, it can be seen that the results improve as c increases. Generally, the best result is obtained for generalized Bayes estimators for η = 0.1, i.e., the result improve better when η becomes small. Studying this work with a different type of censoring might be interesting.