E-Bayesian Estimation of Reliability Characteristics of a Weibull Distribution with Applications

: Given a progressively type-II censored sample, the E-Bayesian estimates, which are the expected Bayesian estimates over the joint prior distributions of the hyper-parameters in the gamma prior distribution of the unknown Weibull rate parameter, are developed for any given function of unknown rate parameter under the square error loss function. In order to study the impact from the selection of hyper-parameters for the prior, three different joint priors of the hyper-parameters are utilized to establish the theoretical properties of the E-Bayesian estimators for four functions of the rate parameter, which include an identity function (that is, a rate parameter) as well as survival, hazard rate and quantile functions. A simulation study is also conducted to compare the three E-Bayesian and a Bayesian estimate as well as the maximum likelihood estimate for each of the four functions considered. Moreover, two real data sets from a medical study and industry life test, respectively, are used for illustration. Finally, concluding remarks are addressed.


Introduction
The random variable X has the two-parameter Weibull distribution, WEI(δ, γ), if its probability density function (pdf), f (x), cumulative distribution function (CDF), F(x), and hazard rate functions (HAR), h(x), are respectively given as, h(x; δ, γ) = δγx γ−1 , where γ > 0 is the shape parameter and δ > 0 is the rate parameter. Figure 1 shows the different representative plots of f (x), F(x), R(x) = 1 − F(x) and h(x) for WEI(δ, γ) given δ = 1. The shape of the WEI(δ, γ) failure rate function depends upon the value of the shape parameter, γ. When γ > 1, the failure rate is increasing and concave up. When γ < 1, the failure rate is decreasing and concave up. When γ = 1, it reduces to a horizontal line that is the failure rate shape of the well-known conventional exponential distribution. Due to the flexibility of the failure rate function, WEI(δ, γ) has been proposed in statistics literature to analyze real applications for industrial and engineering reliability inference and medical survival analysis. This distribution was introduced by Weibull [1]. Since then, it has been one of most important probability models for lifetime distributions. Abernethy [2] produced a Handbook for WEI(δ, γ). Zhang et al. [3] studied the weighted least square estimation for the parameters. Al Omari and Ibrahim [4] investigated Bayesian survival estimation utilizing censored data from WEI(δ, γ). Technology advancement prolongs the process of collecting a random sample of lifetimes. To shorten the process of collecting a sample, many censoring schemes have been cooperated to develop the parameter inference of WEI(δ, γ). Wu [5] developed the point and interval estimations for WEI(δ, γ) parameters by using the maximum likelihood procedure under a progressive censoring. Zhang and Xie [6] analyzed the reliability and maintainability based on the upper truncated sample from WEI(δ, γ). Ahmed [7] investigated Bayes estimation by means of the balanced square error loss function under the progressive type-II censoring. Murthy et al. [8] provided many useful modelings of WEI(δ, γ) until 2003. In this paper, the estimation methods for any function of the unknown rate parameter, δ, are considered based on a progressive type-II censored sample that can be obtained via the progressive type-II censoring scheme during the life test. The progressive type-II censoring scheme can be implemented as follows. At the beginning of the experiment, all n subjects are under the treatment simultaneously at the time labeled by X 0:s:n = 0. Once the ith failure time, X i:s:n , is observed, R i items will be randomly withdrawn from the remaining survival items, for i = 1, 2, 3, · · · , s where 1 ≤ s ≤ n and R i , i = 1, 2, 3, · · · , s are determined before the experiment based on administrative concern. Then the observed failure data set, which contains X i:s:n , i = 1, 2, 3, · · · , s, is called the progressively type-II censored sample under the progressive type-II censoring scheme, R i , i = 1, 2, 3, · · · , s.
Recently, Han [9] studied the structure of the hierarchical prior distribution and a new Bayesian estimation, which is called the Expected Bayesian or E-Bayesian method. Han [10] compared the E-Bayesian estimation method and hierarchical Bayesian estima-tion for the system failure rate by utilizing the quadratic loss function and established the E-Bayesian estimation properties under three different priors for hyper-parameters. Jaheen and Okasha [11] obtained the E-Bayesian inference of the outer power parameter and reliability from a Burr type-XII distribution with type-II censoring based on squared error loss (SEL) and LINEX loss functions. Okasha [12] computed the E-Bayesian estimates for the unknown rate parameter, reliability (parallel and series systems) and failure rate functions using the SEL function based on type-II censored samples from WEI(δ, γ). Okasha [13] suggested using a balanced SEL function for the Bayesian estimation of the unknown power parameter and reliability of a Lomax distribution with type-II censored data. Okasha and Wang [14] provided the geometric model to E-Bayesian estimation for the unknown parameters based on record statistics using different balance loss functions. Okasha et al. [15] investigated E-Bayesian point and interval predictions when only outer power parameter unknown based on type-II censored with two samples from the Burr XII distribution. The aforementioned references have the common conclusion that indicates that the E-Bayes estimate method provides better estimation than the Bayes estimate method does. Other work on E-Bayesian estimation of parameters under different loss functions can be seen in [16][17][18][19].
A scenario throughout this paper is discussing the E-Bayesian estimates of any function of unknown rate parameter, which includes rate parameter, reliability, failure rate and quantile functions as special cases, based on the progressively type-II censored sample. In Section 2, the Bayesian and non-Bayesian estimators will be presented for any given function of rate parameter. Section 3 presents the expected Bayesian or E-Bayesian estimation methods for any given function of rate parameter under three priors for the hyper-parameters. The properties of the E-Bayesian estimators under the SEL function are developed for four special functions in Section 4. In Section 5, a simulation study will be conduced to compare the performance of the different estimators. In the following, two real data sets, one is regarding the remission times from 128 bladder cancer patients reported by Lee and Wang [20] and the other is about the breaking stresses of carbon fibers used in the fibrous composite materials reported by Padgett and Spurrier [21], will be used for the application illustration. Finally, Section 7 provides some concluding remarks.
Bayesian estimation based on the type-II censored sample from WEI(δ, γ) has been studied by Canavos and Tsokos [22], Panahi and Asadi [23] and Singh et al. [24]. In these three referenced papers, gamma or inverted gamma distributions have been used as the prior of the rate parameter or scale parameter, and uniform or gamma distributions have been used as the prior for the shape parameter. Other possible joint priors used were Jeffery priors without hyper-parameters. Whenever a prior is used for the shape parameter in Bayesian estimation for WEI(δ, γ), it raises the level of computational complexity and it is intractable for getting a closed form of the Bayesian estimate. Moreover, the selection of the hyper-parameters for the Bayesian estimation is an important issue. In the current study, we will utilize the E-Bayesian method to investigate the impact of the hyper-parameters to the Bayesian estimate method given a progressively type-II censored sample from WEI(δ, γ). However, there is no closed form for the Bayesian estimate of the shape parameter. Hence, the shape parameter, γ, is treated as a known constant in WEI(δ, γ) and the estimation procedure for any given function, η(δ), of the rate parameter, δ, will be focused on in this study. The given function, η(δ), includes the identity function, δ, the reliability, R(t) = e −δt γ , hazard rate function, h(t; δ, γ) = δγt γ−1 , and quantile function, ξ(p; δ, γ) = (− ln(1 − p)) 1/γ δ −1/γ , where 0 < p < 1, as special cases. First, the maximum likelihood estimator (MLE) of δ can be obtained asδ and the MLE of η(δ) is obtained by the plugging in method and can be denoted bŷ η = η(δ ML ). For getting the Bayesian estimation of η(δ) under the SEL function, the gamma conjugate prior of the unknown rate parameter δ, where β > 0 and ρ > 0 are hyper-parameters, will be used in this study. Combining Equation (4) with Equation (6), the posterior density of δ can be presented as where Under the SEL function, the Bayesian estimate of η(δ) can be given as E(η(δ) | Φ) where E(· | Φ) is taking the posterior expectation. When η(δ) = δ, the Bayesian estimate, that is, the posterior mean of (7). When η(δ) = e −δt γ , the Bayes estimate, E(η(δ) | Φ), can be obtained asR where D * = t γ . When η(δ) = δγt −γ−1 , the Bayes estimate, E(η(δ) | Φ), can be shown aŝ When η(δ) = (− ln(1 − p)) 1/γ δ −1/γ , the Bayes estimate, E(η(δ) | Φ), can be derived as, Hence, the Bayesian estimate of η(δ) is a function of hyper-parameters, β and ρ, and is denoted byη BS (β, ρ). To deal with the selection of these two hyper-parameters, the following E-Bayesian estimation method will be discussed.
The E-Bayesian estimate of η(δ) considered in the current research is defined as the expectation of the Bayesian estimate,η BS (β, ρ), with respect to the joint prior distribution of the hyper-parameters and can be expressed aŝ where is the domain of β and ρ such that the prior density is decreasing in δ andη BS is the Bayes estimate of η(δ) evaluated respectively via (8), (9), (10) or (11) For more details about E-Bayesian, readers may refer to References [10][11][12][13][14][15][16][17].
The properties of the E-Bayesian estimate of η(δ) rely on different joint priors of the hyper-parameters β and ρ. In order to investigate the E-Bayesian estimation of η(δ), we use the following three joint priors, where r > 0, v > 0 and B(r, v) is the beta function. Therefore, the E-Bayesian estimators of the η(δ) under the SEL function can be evaluated as follows, Again, the closed form ofη BS (β, ρ) is needed to establish the properties of the E-Bayesian estimate of η(δ). Therefore, the aforementioned four different functions of η(δ) will be used for discussions Therefore, the E-Bayesian estimators of η(δ) = δ under the SEL function can be obtained by plugging (8) into (14), (15) and (16) to respectively produce the following,

Proof of Proposition 2. From
. Thus, the proof is complete.
Remark 4. The first conclusion of Proposition 4 provides the comparison among E-Bayesian estimationsξ EBSj for j = 1, 2, 3 and the second conclusion of Proposition 4 implies that allξ EBSj for j = 1, 2, 3 are asymptotically equivalent when the number of items under the life test approaches to infinity.

Simulation Study
Given n subjects for the life test experiment, the number of observed failure times, s(≤ n), a progressive type-II censoring scheme mentioned above, a pair of Weibull distri-bution parameters, (δ, γ), a pair of (β, ρ), a value of a and a pair of (r, v), the simulation is conducted according to the following steps:

1.
A progressive type-II sample with n test items and s observed failure times is generated from WEI(δ, γ) by implementing the progressive type-II censoring R i , i = 1, 2, 3, · · · , s using a MATLAB program that was developed by the authors.

2.
The MLEδ ML can be obtained by Equation (5), and the MLEs of R(t), h(t) and ξ(p) can be obtained by pluggingδ ML into R(t), h(t) and ξ(p), respectively.
The above simulation was carried out by using a MATLAB program. Let Θ = (θ 1 , θ 2 , θ 3 , θ 4 ) = (δ, R(t), h(t), ξ(p)) andΘ j = (θ 1,j ,θ 2,j ,θ 3,j ,θ 4.j ) for j = 1, 2, · · · , 10,000 be the corresponding 10,000 estimates for any estimator mentioned in the simulation study. Then, we calculated the average value estimates (AVEs) asθ i = 1 10,000 ∑ 10,000 j=1θ i,j and mean square errors (MSEs) to be 1 10,000 ∑ 10,000 Therefore, we recommend to use the E-Bayesian procedure to estimate the parameters δ, R, h and ξ when type-II censoring scheme is used for life test because of the better performance over the other estimates in terms of minimum MSE.

Applications to Real Data
In this section, two real data sets will be used for the E-Bayesian estimation application. The first data set, that was originally reported by Lee and Wang [20], contains the remission times (in months) of 128 bladder cancer patients. The second data set, that was originally reported by Padgett and Spurrier [21], is regarding breaking stresses (in GPa) of carbon fibers of a certain type used in fibrous composite materials. For easy reference, both complete data sets are also reported in Tables 3 and 4, respectively. Before the E-Bayesian estimation procedure is applied to these two data sets, a Kolmogorov-Smirnov (K-S) test and the scaled total time on test (TTT) plot mentioned by Aarset [26] will be applied to examine these two data sets. Since the K-S test can be conducted through current existing software, the scaled TTT transform will be addressed briefly in this section. The scaled TTT transform is defined as t 0 R(u)du and 0 ≤ t ≤ 1. The corresponding empirical scaled TTT transform will be addressed by g(r/n) = H −1 . . , n denotes the ordered statistic of the lifetime sample {X 1 < X 2 < . . . < X n } and r = 1, 2, 3, · · · , n. Then the empirical scaled TTT plot will be {(x, g(x))|0 ≤ x ≤ 1}. Aarset [26] mentioned that the scaled TTT transform is convex (concave) if the hazard function decreases (increases) and the hazard function is a bathtub (unimodal) if the scaled TTT transform changes from convex (concave) to concave (convex).  First, we use the data set from Table 3 to fit with WEI(δ, γ). The MLEs of the unknown δ and γ of WEI(δ, γ) are given byδ = 0.0939 andγ = 1.0478. By using the K-S test with a significance level of 0.05, the test statistic value is 0.069449 with p-value = 0.5675. Figure 2 shows the empirical scaled TTT plot of the 128 remission times of bladder cancer patients and indicates concave on the left lower part and slightly curved up on the right upper part. The MLE of γ is larger than 1 but very closed to 1, indicating the hazard function is an increasing or constant function. Therefore, based on the K-S test result, it is still reasonable to assume that the Weibull distribution is accepted as a good fit. The true parameters of WEI(δ, γ) will be assumed as δ = 0.0939 and γ = 1.0478 in this section for the practical application purpose.  Table 3.
From Table 3, the first 15 ordered statistics or the first 30 ordered statistics will be used as two type-II censored samples with s = 15 and 30, respectively, to obtain the aforementioned estimates of δ, R(t), h(t) and ξ(p) functions assuming that the shape parameter γ is known as 1.0478. All different estimates of δ, R(t), h(t), ξ(p) are calculated and displayed in Table 5, where R(t) and h(t) functions are estimated at t = 0.9, ξ(p) is estimated at p = 0.5 and Bayes estimates are obtained by using hyper-parameters, β = 0.5 and ρ = 0.5, for the prior because no other information about the unknown rate parameter of the Weibull distribution is available.

Example 2
The data set from Table 4 is used to fit WEI(δ, γ) and the MLEs of the unknown δ and γ of WEI(δ, γ) are obtained asδ = 0.0491 andγ = 2.792. The K-S test with a significance level of 0.05 provides the test statistic value 0.061136 and p-value = 0.8489. Figure 3 shows the empirical scaled TTT plot of the 100 breaking stresses of carbon fibers and indicates a concave shape. It means that the hazard function is an increasing function and shape parameter γ must be greater than 1. The MLE of γ presents a consistent conclusion. Hence, the Weibull distribution is accepted as a good fit probability model for the carbon fiber breaking stresses. In this application, WEI(δ, γ), with δ = 0.0491 and γ = 2.792, is assumed as the true distribution for the carbon fiber breaking stresses.  Table 4.
The first 15 ordered statistics or the first 30 ordered statistics in Table 4 are used as two type-II censored samples with s = 15 and 30, respectively. To derive the aforementioned estimates of δ, R(t), h(t) and ξ(p) functions, we assumed that the shape parameter γ is known as 1.0478. All estimation results for δ, R(t), h(t) and ξ(p) are calculated and displayed in Table 6, where R(t) and h(t) functions are estimated at t = 0.9, ξ(p) is estimated at p = 0.5 and Bayes estimates are calculated by using hyper-parameters, β = 0.5 and ρ = 0.5, for the prior because no other information about the unknown rate parameter of the Weibull distribution is available.

Concluding Remarks
Based on the progressively type-II censored sample, the expected Bayesian estimate method, called the E-Bayesian method, for any function of rate parameter has been established by using the square error loss function. The aforementioned function includes the identity function, i.e., rate parameter, reliability, failure rate and quantile functions of the two-parameter Weibull distribution as special cases. Three different priors of the hyper-parameters are introduced to investigate the impact of hyper-parameters on the E-Bayesian estimators. Some theoretical properties among three E-Bayesian estimators for each of these four functions have been derived. The performance comparison among three E-Bayesian, Bayesian and maximum likelihood estimators of the rate parameter, reliability, failure rate and quantile functions is also carried out through a simulation study. Based on the minimum mean square error criterion, the simulation results showed that the E-Bayesian method performs quite well in estimating the rate parameter as well as reliability, failure rate and quantile functions. Finally, two real data sets regarding the remission times from 128 bladder cancer patients and breaking stresses (in GPa) of carbon fibers used in fibrous composite materials were used to demonstrate two-parameter Weibull modeling and to obtain the MLE, Bayesian and E-Bayes estimates for the rate parameter, reliability, failure rate and quantile functions under the SEL function.
The E-Bayesian estimation using different loss functions for reliability characteristics of the two-parameter Weibull distribution under other different censoring schemes as well as theoretical properties of the E-Bayesian estimate for many different families of η(δ) functions is interesting and difficult work that needs more time. We are currently working on the corresponding problems.