Estimation of Failure Probability and Its Applications in Lifetime Data Analysis

Since Lindley and Smith introduced the idea of hierarchical prior distribution, some results have been obtained on hierarchical Bayesian method to deal with lifetime data. But all those results obtained by means of hierarchical Bayesian methods involve complicated integration compute. Though some computing methods such as Markov Chain Monte Carlo (MCMC) are available, doing integration is still very inconvenient for practical problems. This paper introduces a new method, named E-Bayesian estimation method, to estimate failure probability. In the case of one hyperparameter, the definition of E-Bayesian estimation of the failure probability is provided; moreover, the formulas of E-Bayesian estimation and hierarchical Bayesian estimation and the property of E-Bayesian estimation of the failure probability are also provided. Finally, calculation on practical problems shows that the provided method is feasible and easy to perform. 1. Introduction In the area related to the reliability of industrial products, engineers often deal with the truncated data in life testing of products, where the data sometimes have small sample size or have been censored, and the products of interest have high reliability. In the literature, Lindley and Smith [1] first introduced the idea of hierarchical prior distribution. Han [2] developed some methods to construct hierarchical prior distribution. Recently, hierarchical Bayesian methods have been applied to data analysis [3]. However, complicated integration compute is a hard work by using hierarchical Bayesian methods in practical problems, though some computing methods such as Markov Chain Monte Carlo (MCMC) methods are available [4, 5]. Han [6] introduced a new method—E-Bayesian estimation method—to estimate failure probability in the case of two hyperparameters, proposed the definition of E-Bayesian of failure probability, and provided formulas for E-Bayesian estimation of the failure probability under the cases of three different prior distributions of hyperparameters. But we did not provide formulas for hierarchical Bayesian estimation of the failure probability nor discuss the relations between the two estimations. In this paper, we will introduce the definition for E-Bayesian estimation of the failure probability, provide formulas both for E-Bayesian estimation and hierarchical Bayesian estimation, and also discuss the relations between the two estimations in the case of only one hyperparameter. We will see that the E-Bayesian estimation method is really simple. Conduct type I censored life testing time, denote the