Estimation of Failure Probability and Its Applications in Lifetime Data Analysis

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 m time, denote the censored times as t_i  (i = 1,2, …, m), the corresponding sample numbers as n_i, and the corresponding failure sample numbers observed in the testing process as r_i  (r_i = 0,1, 2, …, n_i).

In the situation where no information about the life distribution of tested products is available, Mao and Luo [7] introduce a so-called curves fitting distribution method to give the estimation of the failure probability p_i at time t_i, p_i = P(T < t_i) for i = 1,2, …, m, where T is the life of a product.

This paper introduces a new method, called E-Bayesian estimation method, to estimate failure probability. The definition and formulas of E-Bayesian estimation of the failure probability are described in Sections 2 and 3, respectively. In Section 4, formulas of hierarchical Bayesian estimation of the failure probability are proposed. In Section 5, the properties of E-Bayesian estimation are discussed. In Section 6, an application example introduces given. Section 7 is the conclusions.

2. Definition of E-Bayesian Estimation of p_i

According to Han [2], a and b should be chosen to guarantee that π(p_i∣a, b) is a decreasing function of p_i. Thus 0 < a < 1, b > 1.

Given 0 < a < 1, the larger the value of b, the thinner the tail of the density function. Berger had shown in [8] that the thinner tailed prior distribution often reduces the robustness of the Bayesian estimate. Consequently, the hyperparameter b should be chosen under the restriction 1 < b < c, where c is a given upper bound. How to determine the constant c would be described later in an example.

Since a ∈ (0,1) and 1/2 is an expectation of uniform on (0, 1), that we take a = 1/2. When 1 < b < c and a = 1/2, π(p_i∣a, b) also is a decreasing function of p_i.

The definition of E-Bayesian estimation was originally addressed by Han [6] in the case of two hyperparameters. In the case of one hyperparameter, the E-Bayesian estimation of failure probability is defined as follows.

3. E-Bayesian Estimation of p_i

4. Hierarchical Bayesian Estimation

If the prior density function π(p_i∣b) of p_i is given by (3), how can the value of hyperparameter a be determined? Lindley and Smith [1] addressed an idea of hierarchical prior distribution, which suggested that one prior distribution may be adapted to the hyperparameters while the prior distribution includes hyperparameters.

5. Property of E-Bayesian Estimation of p_i

Now we discuss the relations between $$ and $$ in Theorems 2 and 3.

Theorem 4 shows that $$ and $$ are asymptotically equivalent to each other as s_i tends to infinity. In application, $$ and $$ are close to each other, when s_i is sufficiently large.

6. Application Example

Han [6] provided a testing data of type I censored life testing for a type of engine, which is listed in Table 1 (time unit: hour).

By Theorems 2, 3, and Table 1, we can obtain $$, $$, and $$. Some numerical results are listed in Table 2.

From Table 2, we find that for some c (c = 2,3, 4,5, 6), $$, and $$ are very close to each other and satisfy Theorem 4; for different c (c = 2,3, 4,5, 6), $$ and $$ are all robust (when i = 9, there exists some difference). In application, the author suggests selecting a value of c in the middle point of interval [2, 6], that is, c = 4.

When c = 4, some numerical results are listed in Table 3 and Figure 1.

Range: in Figure 1, * is the results of $$ and o is the results of $$.

From Table 3 and Figure 1, we find that $$ and $$ are very close to each other and consistent with Theorem 4.

From Table 3, we find that the results of $$ and $$ are very close to the corresponding results of Han [6].

From (24) and Table 3, we can obtain $$ and $$. Some numerical results of $$ and $$ are listed in Table 4.

From Table 4, we find that the results of $$ and $$ are very close to the corresponding results of Han [6].

From (25) and Table 4, we can obtain the estimate of the reliability with some numerical results of $$ and $$ listed in Table 5 and Figure 2.

Note that $$ is the estimate of reliability at moment t with regard to $$, $$ is the estimate of reliability at moment t with regard to $$, and $$.

Range: in Figure 2, * is the results of $$ and o is the results of $$.

From Table 5, we find that $$ when t = 200,600,1000,1400, 1800,2000, and the results of $$ and $$ are very close to those of Han [6].

7. Conclusions

This paper introduces a new method, called E-Bayesian estimation, to estimate failure probability. The author would like to put forward the following two questions for any new parameter estimation method: (1) how much dependence is there between the new method and the other already-made ones? (2) In which aspects is the new method superior to the old ones?

For the E-Bayesian estimation method, Theorem 4 has given a good answer to the above question (1) and, in addition, the application example shows that $$ and $$ satisfy Theorem 4.

To question (2), from Theorems 2 and 3, we find that the expression of the E-Bayesian estimation is much simple, whereas the expression of the hierarchical Bayesian estimation relies on beta function and complicated integrals expression, which is often not easy.

Reviewing the application example, we find that the E-Bayesian estimation method is both efficient and easy to operate.