Inference for stress – strength problem

The basic impulsion to these developments can perhaps be recognized to the specific practical problem of applied statistics summarized by the term “Stress-Strength”. In the simplest terms this can be described as an assessment of “Reliability” of a “Component” in terms of random variables X representing “Stress” experienced by the component and Y representing the “Strength” of the component available to overcome the stress.

Here I have considered the inference for Stress-Strength Model and different distributions like Normal, Geometric, Bivariate Exponential. we will derive confidence limits on R involving a noncentral t distribution, obtained by approximating the distribution of a weighted sum of independent X² by the distribution of a scaled X² variable by equating the first two moments. A familiar approximation to the noncentral t distribution is then used to derive a different confidence interval.we summarize a Bayesian approach to derive a posterior probability interval for R and then these confidence intervals will be compared in Simulation Study.

Here three different estimators of R have been found out for independent set-up. Using ML Estimator of parameters, an estimator of R has been given and procedure for finding out lower bound to R has been mentioned. UMVUE of R has been derived and through simulation study we will compare these two estimators. A bayes Estimator of R and a lower bound to R have been derived under Beta prior for parameters θ₁ and θ₂. An n-standby system which is working under cyclically impinging stresses is discussed. We will be obtained reliability of the system at the r^th cycle, R(r) for n ≤ 4 only, as the complexly of expression increases rapidly with n. In particular, when stress-strength follow exponential or gamma or normal distribution we have obtained the expression for R(r), where n ≤ 4. Here we considered an n-cold standby system with imperfect switch in stress-strength model. Only one switch is considered through the system. Both stress-strength are assumed to be random variables. We have taken identical strength of the components following particular distribution. Some numerical values of reliability are also tabulated.