Comparing Weibull Stress – Strength Reliability Bayesian Estimators for Singly Type II Censored Data under Different loss Functions

The stress(Y) – strength(X) model reliability Bayesian estimation which defines life of a component with strength X and stress Y (the component fails if and only if at any time the applied stress is greater than its strength) has been studied, then the reliability; R=P(Y<X), can be considered as a measure of the component performance. In this paper, a Bayesian analysis has been considered for R when the two variables X and Y are independent Weibull random variables with common parameter α in order to study the effect of each of the two different scale parameters β and λ; respectively, using three different [weighted, quadratic and entropy] loss functions under two different prior functions [Gamma and extension of Jeffery] and also an empirical Bayes estimator Using Gamma Prior, for singly type II censored sample. An empirical study has been used to make a comparison between the three estimators of the reliability for stress – strength Weibull model, by mean squared error MSE criteria, taking different sample sizes (small, moderate and large) for the two random variables in eight experiments of different values of their parameters. It has been found that the weighted loss function was the best for small sample size, and the entropy and Quadratic were the best for moderate and large sample sizes under the two prior distributions and for empirical Bayes estimation.


Introduction:
Weibull models are used to describe various types of observed failures of components and phenomena. They are widely used in reliability and survival analysis (1). A considerable attention for the problem of making inference about the stress-strength reliability (one component or system) model has been received. If X be the strength of a component and Y be the stress applied to the component, then reliability; R=P(Y<X), can be considered as a measure of the component performance.
Various different lifetime distributions are considered to estimate R. (2)(3)(4)(5). As an example; Seuba et al (6) analyzed the applicability of the Weibull analysis to unidirectional microporous yttrium-stabilizedzirconia (YSZ) prepared by ice-tempting, performed crush tests on samples with controlled microstructural features with the loading direction parallel to the porosity. The compressive strength data were fitted using two different fitting techniques, ordinary least squares and Bayesian Markov Chain Monte Carlo, to evaluate whether Weibull statistics are an adequate descriptor of the strength distribution. They assess the effect of different microstructural features (volume, size, densification of the walls, and morphology) on Weibull modulus and strength and found that the key microstructural parameter controlling reliability is wall thickness. In contrast, pore volume is the main parameter controlling the strength.
In this paper the reliability Bayesian analysis when stress X and strength Y are two independent Weibull random variables with parameters(α,β) and (α,λ) respectively is done under two prior functions with three loss distributions. A simulation study has been used to compare by (MSE) the performance of the six different obtained estimators. The results are recorded in Tables (1 to   5 (4) Q for all sample sizes, except for Jeffrey with n=15 (5) E and Q for n=30and n=90, while W is the best for Jeffrey with n=15 (6) W for n=15 and for Jeffrey with n=30, for the other cases E and Q are the best (7) and (8) by E and Q for all sample sizes, except W is the best for Exp. (8) for Jeffrey and gamma when n=15   Table 1 β and λ (as a special case in our research where the other cases can be done as a future work), then the probability distribution function for two independent Weibull r.v.'s are (2) : Let u = (β + λ)y α and du = α(β +λ)y α-1 dy, so by transformation, it will be:

Singly Type II Censored Sample
Let x 1 ,x 2 ,…,x n and y 1 ,y 2 ,….,y m be two random samples, and r < n and r < m , such that : x r ,…., x n and y r ,…., y m . The likelihood function for this type of data is (7) : Where:

Under Gamma Prior
The Gama distribution is used as a prior distribution because of its wide importance in Bayesian analysis. Let β, λ be two independent Gamma random variables with common parameter (a), (one can consider the case of uncommon parameter in other papers for recommendation), the pdf is given by (8)

…… (8) The Bayes Estimators
In this section the Bayes estimators for stressstrength Weibull reliability under three loss functions are derived as following (3) Using these two estimators in (15) and (16) in the reliability estimators obtained above under Gamma prior using three loss functions.

For Extension of Jeffery Prior
The Bayesian estimators for R will be derived in this section for extension of Jeffery prior as in equations (6 and 7) under the three loss functions.

(i) Weighted Loss Function
From equations (8), the Bayesian estimators of R for extension of Jeffery prior under the weighted loss functions; ̂, from equation (9), will be:

…. (20) Empirical Study
To compare between estimators for which is the best to estimate the reliability of stressstrength Weibull model; (Since it is not possible to apply real data in our research, recommending doing so in future researches), an empirical study made by simulation procedure using MATLAB program to compare among them by MSE criteria, under different sample sizes (n = m =15) representing the smallest sample size, (n=m=30) for moderate and (n=m=90) for large (which is known to have a range greater than 75) sample sizes, in eight experiments of different parameters values and when α=0.8. The replication done for (q= 5000).
Equation (20) is used to generate different values of the two random variables X and Y by F(x) and F(y) respectively, where U is uniform random variable on interval (0,1), then by the inverse of distribution function technique got from:

Conclusions:
The results of the simulation study are recorded in Tables (2 to 5) below, where there is a fluctuation in the behavior of the estimated reliability of this system when the sample sizes change using the loss functions. While in Table (1), the best performance of the estimators is recorded as a summary of the experiment conclusions.
As a final result, it is found that for small sample size the best performance was for weighted loss function, and the entropy and Quadratic are the best for moderate and large sample sizes.