Inference on Reliability of Stress-Strength Models for Poisson Data

Researchers in reliability engineering regularly encounter variables that are discrete in nature, such as the number of events (e.g., failures) occurring in a certain spatial or temporal interval. The methods for analyzing and interpreting such data are often based on asymptotic theory, so that when the sample size is not large, their accuracy is suspect. This paper discusses statistical inference for the reliability of stress-strength models when stress and strength are independent Poisson random variables. The maximum likelihood estimator and the uniformly minimum variance unbiased estimator are here presented and empirically compared in terms of their mean square error; recalling the delta method, confidence intervals based on these point estimators are proposed, and their reliance is investigated through a simulation study, which assesses their performance in terms of coverage rate and average length under several scenarios and for various sample sizes. The study indicates that the two estimators possess similar properties, and the accuracy of these estimators is still satisfactory even when the sample size is small. An application to an engineering experiment is also provided to elucidate the use of the proposed methods. 1. Introduction A stress-strength model, in the simplest terms, considers a unit/system that is subjected to an external stress, modeled by r.v. , against which the unit sets its own strength, modeled by r.v. , in order to properly operate. The probability that the unit withstands the stress is then given by , which is usually called reliability. A great deal of work has been done about this topic: most of it deals with the computation of reliability, if the distributions of stress and strength are known, or its estimation under various parametric assumptions on and , when samples from and are available. A complete review is available in [1]. Many applications of the stress-strength model, for its own nature, are related to engineering or military problems, where it is also referred to as a load-strength model [2]. However, there are also natural applications in medicine or psychology, which involve the comparison of two r.v., representing, for example, the effect of a specific drug or treatment administered to two groups (control and test); here, reliability assumes a wider meaning. Almost all of these papers consider continuous distributions for and , since many practical applications of the stress-strength model in engineering fields presuppose continuous quantitative data. A relatively small amount of work is devoted to