Planning two or more level constant-stress accelerated life tests with competing risks

Highlights

• ALT with two- or more-level constant-stress under competing risks is studied.

• The optimal stress levels under D/A/variance-optimality criteria are derived.

• The optimal sample allocations at each stress level under D/A/variance-optimality are obtained.

• Progressive type II censoring is used to collect life data from the exponential distribution.

Abstract

In this article, we investigate the optimization problem when the competing risks data come from a progressive type II censoring in an accelerated life test with multiple levels of constant stress. The failure times of the individual causes are assumed to be independent and exponentially distributed with different parameters. We propose three criteria related to the Fisher's information matrix to determine the optimal stress level as well as the optimal sample allocation at each stress level. A real data set is studied to illustrate the application of the proposed criteria.

Introduction

In reliability analysis, censoring is usually applied to collect lifetime data. Type I censoring and type II censoring are the most two common schemes in the life tests. However, in these two schemes, no test units are allowed to be removed from the tests at the points other than the final termination point. Because of this reason, progressive censoring is arisen to achieve this objective. In addition, it is easy to see that complete sample and type II censoring are two special cases of progressive type II censoring. Therefore, the study on progressive censoring has grown very fast in the last decade. An account on progressive censoring can be found in the monograph by Balakrishnan and Aggarwala [2] and Balakrishnan and Cramer [3], or in the review article by Balakrishnan [1].

With the development of technology, the products with high reliability usually can work for a long time. For the highly reliable products, there are very few test units can be observed their lifetimes at use condition. Therefore, some techniques were developed to yield failures quickly. The usual way is to test the products at high stress conditions. This is known as accelerated life test (ALT). The purpose of such test is to quickly obtain lifetime data which, properly modeled and analyzed, yield desired information on product performance at use condition.

The stress loading in an ALT can be applied in various ways. They include constant stress, step stress, progressive stress, and random stress. Nelson [22, Chapter 1] discussed their advantages and disadvantages. In this paper, we only consider the case of constant stress because constant stress is the most common and simply used in the life test. In a multiple-level constant-stress ALT, each unit is run at a constant stress level. Suppose that we have a total of N units available for the life test at L stress levels. We assign n_l units for testing at the l-th stress level with ${\sum }_{l=1}^L{n}_l=N$. The test is run till the termination time point.

Now, let us consider an important issue which is called competing risks or multiple modes of failure. A product usually consists of a variety of the components, and it may fail due to one of several causes. In many applications, product lifetime is defined to be the earliest occurrence among all these risks. Nelson [22, Chapter 7] enumerated engineering situations when a product fails because of two or more risks. Ignoring the information on causes of failure may result in improper inference in reliability analysis. Thus, the competing risks data consist of the failure time and an indicator variable denoting the specific cause of failure of the product. Cox [5] proposed the latent failure model to analyze the data with competing risks. It is usually assumed that these competing risks are independent. Although the assumption of dependence may be more realistic, there are some identifiability issues in the underlying model. Kalbfleisch and Prentice [15] and Crowder [6] pointed out that, without the information of the covariates, it is not possible using data to test the assumption of independence of failure times. To avoid the problem of model identifiability, we adopt the latent failure time model and assume the competing risks are independent.

In the literature, a significant amount of work has been done on planning the ALT for single cause of failure, such as Ng et al. [23], Wu et al. [29], Watkins and John [28], Liu and Tang [17], Ka et al. [14], Hu et al. [13], Fan and Yu [7], Haghighi [10] Zhang and Liao [31], Guan et al. [8], Xu and Tang [30], and Hamada [11]. Most of them are to determine the best choice of the proportion of the units assigned to each stress level based on different criteria. For the ALT with multiple failure modes, Pascual [25] studied ALT planning under independent Weibull competing risks. Suzuki et al. [27] investigated the optimum specimen sizes and sample allocation by achieving the most precise estimation of the Weibull shape parameter for two competing risks. Liu and Qiu [19] presented a method for planning multiple step-stress ALT with independent competing risks. Zhu and Elsayed [32] studied the problem of planning the optimal ALT by minimizing the asymptotic variance of the mean time of the first failure under progressive censoring. Liu and Tang [18] proposed a Bayesian approach to planning an accelerated life test for repairable systems with independent failure modes. Liu [16] obtained the c-optimal test plans by minimizing the large-sample approximate variance of the maximum likelihood estimator of a certain life quantile at use condition when an ALT with multiple dependent failure modes. Balakrishnan et al. [4] provided an EM algorithm for the estimation of the model parameters when data come from a one-shot device testing analysis under an accelerated life test with competing risks. Han and Kundu [12] studied statistical inference for a step-stress model with competing risks under type-I censoring when the different risk factors have independent generalized exponential failure time distribution. Luo et al. [20] developed a testing methodology based on the reliability target allocation for reliability demonstration under competing failure modes at accelerated conditions.

In this paper, we investigate the issue of determining the optimal stress levels for two or more level constant-stress ALTs under progressive type II censoring with competing risks. Sample allocations and number of observed failures under the determined constant-stress levels are also decided. The rest of this paper is organized as follows. Model and assumptions are provided in Section 2. The inference of the maximum likelihood method is derived in Section 3. Three optimization criteria are proposed and the approaches to get the optimal stress levels are investigated in Section 4. The optimal sample allocations are also discussed. An illustrated example is studied to demonstrate the proposed methods in Section 5. Some conclusions and discussions are made in Section 6.