Testing Reliability in a Stress-Strength Model When X and Y are Normally Distributed

Abstract

We consider the stress-strength problem in which a unit of strength X is subjected to environmental stress Y. An important problem in stress-strength reliability concerns testing hypotheses about the reliability parameter R = P[X > yl. In this article, we consider situations in which X and Y are independent and have normal distributions or can be transformed to normality. We do not require the two population variances to be equal. Our approach leads to test statistics which are exact p values that are represented as one-dimensional integrals. On the basis of the p value, one can also construct approximate confidence intervals for the parameter of interest. We also present an extension of the testing procedure to the case in which both strength and stress depend on covariates. For comparative purposes, the Bayesian solution to the problem is also presented. We use data from a rocket-motor experiment to illustrate the procedure.

KEY WORDS:

• Confidence interval
• Covariates
• Generalized p value
• Lognormal distribution