Abstract
Stress-strength reliability problems arise frequently in applied statistics and related fields. Often they involve two independent and possibly small samples of measurements on strength and breakdown pressures (stress). The goal of the researcher is to use the measurements to obtain inference on reliability, which is the probability that stress will exceed strength. This paper addresses the case where reliability is expressed in terms of an integral which has no closed form solution and where the number of observed values on stress and strength is small. We find that the Lagrange approach to estimating constrained likelihood, necessary for inference, often performs poorly. We introduce a penalized likelihood method and it appears to always work well. We use third order likelihood methods to partially offset the issue of small samples. The proposed method is applied to draw inferences on reliability in stress-strength problems with independent exponentiated exponential distributions. Simulation studies are carried out to assess the accuracy of the proposed method and to compare it with some standard asymptotic methods.
Highlights
We consider a stress-strength reliability problem, R = P (Y < X), where X and Y are independently distributed as exponentiated exponential random variables
We present a penalized likelihood method, which was discussed in Smith et al
We report the proportion of ψ that fall outside the lower bound of the confidence interval, the proportion of ψ that fall outside the upper bound of the confidence interval, the proportion of ψ that fall within the confidence interval, and the average bias (Average Bias), which is defined as Average Bias =
Summary
We consider a stress-strength reliability problem, R = P (Y < X), where X and Y are independently distributed as exponentiated exponential random variables In this case, the parameter of interest, R, is an integral with no closed form solution. Using the Lagrange technique, this problem requires that, in each iteration, one parameter has to be used to guarantee that the integral equality constraint is satisfied up to a given level of numerical accuracy. Smith et al (2014) [3] demonstrates the accuracy of the penalty method in a constrained optimization problem When implementing this approach to the problem considered in this paper, we found it was numerically stable and rapid in the sense that parameter estimates from successively more harshly penalized models quickly converged to optimal values and became effectively insensitive to further penalization.
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