E-Bayesian Estimation Based on Burr-X Generalized Type-II Hybrid Censored Data

In this article, we are concerned with the E-Bayesian (the expectation of Bayesian estimate) method, the maximum likelihood and the Bayesian estimation methods of the shape parameter, and the reliability function of one-parameter Burr-X distribution. A hybrid generalized Type-II censored sample from one-parameter Burr-X distribution is considered. The Bayesian and E-Bayesian approaches are studied under squared error and LINEX loss functions by using the Markov chain Monte Carlo method. Confidence intervals for maximum likelihood estimates, as well as credible intervals for the E-Bayesian and Bayesian estimates, are constructed. Furthermore, an example of real-life data is presented for the sake of the illustration. Finally, the performance of the E-Bayesian estimation method is studied then compared with the performance of the Bayesian and maximum likelihood methods.


Introduction
Hybrid censoring scheme (HCS) is introduced by [1] as a mixture of Type-I and Type-II censoring schemes.In such schemes, experiments are stopped as soon as a pre-specified number r of items from n items fails or when a pre-fixed time T runs out.Thus, there have been two types of censored schemes, namely: Type-I HCS: in which we terminate the test at T * = min{Y k:n , T}, where Y k:n , is the time of k-th item failure, T is the maximum time point for the test and k ∈ {1, 2, • • • , n} and T ∈ (0, ∞) are determined in advance, i.e., the experiment is terminated when a number k failures out of n items or a pre-fixed time T is reached.Type-II HCS: in this type, the experiment is terminated at T * = max{Y k:n , T}, i.e., we end the experiment when the latter of the stopping rules is reached.Hence, we guarantee that at least k failures are obtained.
There are advantages and disadvantages in these schemes.HCS Type-I possesses the advantage of having a fixed time.However, there are a small number of failures obtained before the stopping time of the life-testing.These disadvantages may impact the accuracy of estimators.However, a drawback of Type-I HCS is overcome by Type-II HCS, and we guarantee The obtaining of a specified number of failures.It also has a disadvantage in that the experimenter cannot know when the required specified failures are observed.Ref. [2] proposed two generalized HCSs to overcome these drawbacks, described as follows.Generalized Type-I HCS: Let r, k ∈ {1, 2, • • • , n}, k < r < n, and time T ∈ (0, ∞).When k-th failure is obtained before time T, the experiment is terminated at min{Y r:n , T}.If the k-th failure is obtained after time T, the experiment is ended at Y k:n .Generalized Type-II HCS: Fix r ∈ {1, 2, • • • , n} and T 1 , T 2 ∈ (0, ∞) where T 1 < T 2 .The experiment is terminated at T 1 , if r-th failure occurs before T 1 .When the r-th failure occurs between T 1 and T 2 , we terminate at Y r:n .In the third case, if r-th failure is observed after time T 2 , the experiment is terminated at T 2 .This scheme has been studied by many authors, such as [3], who presented details on censoring scheme developments in addition to generalized and unified HCS.Ref. [4] discussed Bayesian analysis and prediction based on generalized Type-II HCS for exponential and Pareto models.Ref. [5] studied maximum likelihood, Bayes and percentile bootstrap methods for unknown parameters, failure rate function, the survival function and the coefficient of variation of the exponential Rayleigh distribution with generalized Type-II HCS.
Here, generalized Type-II HCS is considered.We observe one of these types of the censored data.Case 1: In this case, the r-th failure is obtained before T 1 , so the experiment is stopped at T 1 and d 1 number of failures is obtained at time T 1 .
Case 2: In this case, the r-th failure occurs after T 1 , so the experiment is terminated at Y r:n , and r number of failures is obtained.
Case 3: In this case, the r-th failure occurs after T 2 , so the experiment is ended at T 2 and d 2 number of failures is obtained at time T 2 .where T 1 and T 2 are time points determined by the experimenter according to how the experiment should continue based on the information about the product.

Burr-X as a Lifetime Model
Burr-X model is part of Burr distribution family suggested by [6].This distribution is important in many fields such as operations research and statistics.It is widely used in health, agriculture, and biology.For more details on the applications of this model, one can refer to [7], as they discussed the cumulative distribution function (CDF) of Burr-X distribution with the cumulative damage process and the shock model.Also, they assumed a mathematical model for the expected lifetime of AIDS patients, then fitted the observed data of infected persons for Burr-X distribution.The probability density function (PDF) of one-parameter Burr-X distribution is given by and the CDF takes the form where θ is the shape parameter.The reliability function R(t) and the hazard rate function h(t) for one-parameter Burr-X distribution are, respectively, expressed as follows: Many publications discuss Burr-X distribution, such as [8], who discussed maximum likelihood estimate (MLE) and Bayesian estimates for parameters of Burr-X model under double Type-II censored sample of dual generalized order statistics.Ref. [9] considered the Bayesian and non-Bayesian estimators of the parameter of the Burr-X model based on grouped data.Ref. [10] studied inferences and predictions for the Burr Type-X model based on records.Ref. [11] considered the statistical analysis of the parameter through different priors and loss functions.Ref. [12] discussed E-Bayesian method and ML and Bayesian methods for the parameter and the reliability function of the Burr-X model under Type-I HCS.
We assume Y 1:n , Y 2:n , • • • , Y n:n are n lifetimes of failure observations under a generalized Type-II hybrid censored sample following one-parameter Burr-X distribution.Let D 1 and D 2 denote several items that fail before T 1 and T 2 , respectively; the likelihood function for three cases is given by ( where By taking the natural logarithm of Equation ( 5), ignoring the constant term, as it does not depend on θ, we get the log-likelihood function as follows: The MLE of the unknown parameter θ is obtained by differentiating Equation (6) regarding θ then equating the derivative to zero (see Appendix A) then solving numerically the following equation: we get θ, the MLE of the parameter θ.Also, we can obtain R (t) the MLE of R(t) by replacing θ with θ in Equation (3), as follows: The rest of the article is presented as follows.The Bayesian estimation under SEL and LINEX functions is studied in Section 2. Section 3 discusses the E-Bayesian method under SEL and LINEX loss functions.The Markov chain Monte Carlo (MCMC) algorithm is presented in Section 4. Section 5 provides an example of real-life data sets.The results are compared and concluded in Section 6.

Bayesian Estimation Method
Here, the Bayesian estimation of the parameter θ and the reliability function R(t) of one-parameter Burr-X distribution based on generalized Type-II HCS is considered.The Bayesian approach needs to specify the prior PDF of the parameter θ.The gamma PDFs are widely used in the Bayesian analysis, so we suppose the gamma conjugates prior PDF for θ (as suggested by [8,9]) with the PDF of the form.
the posterior PDF of θ can be given from ( 5) and ( 9), as follows: where K is the normalizing constant given as

Estimates under Squared Error Loss Function
The Bayesian estimate of θ under SEL function is given by The Bayesian estimate for the reliability function R(t) by using the SEL function is given as follows:

Estimates under LINEX Loss Function
The LINEX loss function with parameters k, h is given by where θ is the estimate of θ and h is the shape parameter of loss function.For more details, see [13].
The Bayesian estimate θBL of the parameter θ using the LINEX loss function is θ, which minimizes Equation (13) given by By using the LINEX loss, the Bayesian estimate of R(t) is given by

E-Bayesian Estimation Method
The possibility of the rate of failure for high-reliability products is less than the possibility of the failure rate of low-reliability products.For this, according to [14], we select the hyper-parameters a and b to prove that π(θ) is a decreasing function of θ.The first derivative of π(θ) regarding θ is as follows: Thus, for 0 < a < 1 and b > 0, the prior PDF π(θ) is a decreasing function of θ.The thinnertailed prior PDF often decreases the robustness of the Bayesian estimate; therefore, b should be smaller than an upper bound c, where c > 0 is a constant.Accordingly, hyper-parameters a and b should be selected with the restriction of 0 < a < 1 and 0 < b < c.We assume that a and b are independent with bi-variate PDF given by p(a, b) = p 1 (a)p 2 (b), the E-Bayesian estimates of θ and R(t) are, respectively, obtained as follows: where D stands for the set of possible values of a and b, θB and R (t) B are the Bayesian estimates of θ and R(t), respectively, under the SEL and the LINEX loss functions.See, for more details, [15][16][17][18].

The E-Bayesian Estimate of θ
To obtain the E-Bayesian estimate of θ, we use three prior PDFs of a and b to study the influence of these prior PDFs on the E-Bayesian estimation.We suggest the following prior PDFs The E-Bayesian estimate of θ, under the SEL are computed from ( 11), ( 16) and ( 18) as follows: also, under the LINEX loss function, the E-Bayesian estimate of θ, can be derived from ( 14), ( 16) and ( 18) as

The E-Bayesian Estimate of R(t)
The E-Bayesian estimate can be computed by adopting (18) based on the SEL and LINEX loss functions as follows.
The E-Bayesian estimate of R(t) under SEL function is written from ( 12), ( 17) and (18) as also, the E-Bayesian estimate of R(t) by using the LINEX loss function can be computed from ( 15), ( 17) and ( 18) as follows: R (t It is noted that E-Bayesian and Bayesian estimates cannot be obtained analytically; therefore, we use the MCMC method for deriving the Bayesian and E-Bayesian estimates of θ and R(t).

The MCMC Algorithm
In this work, we consider the MCMC method for generating a sample of θ from the posterior PDF; after that, we derive Bayesian and E-Bayesian estimates of θ and R(t) under LINEX loss and SEL functions.A Metropolisâ€"Hastings sampler is used to obtain the posterior sample of the parameter θ from the full conditional posterior PDF.From (10), discarding all terms do not depend on θ; we get the full conditional posterior PDF as follows. where For more details, see, for example, [19,20].The following steps indicate the Metropolisâ€ "Hastings algorithm for simulating the posterior samples, then derive the Bayesian estimates and the credible related intervals (CRIs) as follows.
(3) By applying a Metropolisâ€"Hastings sampler, θ (j) is simulated from g * (θ (j−1) |y) • Generate a proposal θ ( * ) from a normal distribution as a proposal distribution ignoring negative draws, as they lead to a high rejection rate.• A sample u is generated from the Uni f orm(0, 1) distribution.
(6) Steps (3-6) are repeated N times to get a sequence of the parameter θ with optional burn-in period.
(7) The Bayesian estimates of θ and R(t) under SEL function are, respectively, given as where M represents a burn-in period of Markov chain discarded to get to the rapid convergence and cancel the effect of choosing the starting point of the parameter.(8) The Bayesian estimates for θ and R(t) by the LINEX loss function are, respectively, given by,  Where REBS1 , REBS3 and REBS2 are the E-Bayesian estimates of R(t) relative to SEL based on p 1 (a, b), p 2 (a, b) and p 3 (a, b), respectively, REBL1 , REBL2 and REBL3 are the E-Bayesian estimates of R(t) relative to LINEX based on p 1 (a, b), p 2 (a, b) and p 3 (a, b), respectively.In addition, RBS and RBL denote the Bayesian estimates of R(t) under SEL and LINEX loss, respectively.

Illustrative Example (Real Data Set)
An illustrative example is provided to investigate the performance of proposed methods and clarify how they behave in practice.These data were reported by [21], representing minority electron mobility for Ga 1−x Al x As.Two data sets at the mole fractions of 0.25 and 0.30 were used by [22], who stated that Burr-X distribution presents a good fit for Sets 1 and 2 of data.Each set contains 21 observations, as given below: Data With respect to real data Sets 1 and 2, all estimates are obtained according to the same procedure as before and results are displayed in Tables 3-6.Tables 3 and 5, give average estimates, MSE, and 95% CIs and CRIs of MLEs, Bayesian, and E-Bayesian estimates for θ under LINEX and SEL functions belong to real data Sets 1 and 2, respectively.Tables 4 and 6, display average estimates, MSE and 95% CIs and CRIs of MLEs, Bayesian, and E-Bayesian estimates of R(t = 1.2) based on LINEX and SEL functions belong to real data Sets 1 and 2, respectively.

Concluding Remarks
In this article, we used the Bayesian and E-Bayesian approaches and the ML method for estimating the parameter θ and the reliability function R(t) of one-parameter Burr-X distribution under generalized Type-II HCS.It is noted that Bayesian estimators cannot be derived analytically, so we applied the MCMC method to derive estimates of the parameter θ and the reliability function R(t).Based on SEL and LINEX loss functions, Bayesian and E-Bayesian estimates are derived.Also, CIs of MLEs and CRIs of Bayesian and E-Bayesian estimates are constructed.Furthermore, an example of real data testing is provided for illustration.
From Tables 1 and 2, one can observe that the MSEs decrease when the sample sizes n, T 1 , and T 2 increase.Also, the length of CIs and CRIs decreases when time points T 1 and T 2 increase for constant sample size n.The MSE of E-Bayesian estimator for the parameter θ and the reliability function R(t) is the smallest compared to the MSE of MLEs and Bayesian estimates.Furthermore, the estimator by using LINEX loss function is more efficient than the estimator based on SEL function in terms of MSE.Generally, The LINEX loss function is better than SEL function in most cases.Ref. [23] compared the difference between the LINEX and SEL functions and stated that the LINEX loss is more appropriate than the SEL function.For the shape parameter h of LINEX close to zero, it is approximately SEL and, therefore, almost symmetric.Based on the above results, we can state that a large sample size n gives a better estimate with a smaller MSE.Finally, we can conclude that the E-Bayesian method is more convenient and efficient when compared with ML and Bayesian methods.The prior PDFs affect the E-Bayesian estimation because of selecting the restriction of 0 < a < 1 and 0 < b < c.
From Tables 3-6, we observe that the E-Bayesian method is more efficient than ML, and Bayesian methods in the sense of having smaller MSE.On the other hand, MSEs of all estimators and the length of CIs and CRIs decrease when time points T 1 and T 2 increase, i.e., the proposed methods perform efficiently in the application as well.

Table 1 .
Estimates, MSE, and 95%CIs and CRIs of MLEs, Bayesian, and E-Bayesian methods for θ under LINEX and SEL functions when θ = 1.22, a = 0.8, b = 0.7, h = 1.5, c = 1.Bayesian estimates of θ relative to SEL based on p 1 (a, b), p 2 (a, b) and p 3 (a, b), respectively, θEBL1 , θEBL2 and θEBL3 are the E-Bayesian estimates of θ relative to LINEX based on p 1 (a, b), p 2 (a, b) and p 3 (a, b), respectively.In addition, θBS and θBL denote the Bayesian estimates of θ under SEL and LINEX loss, respectively.