Maximum Likelihood Estimation and Confidence Intervals of System Reliability for Gompertz Distribution in Stress-Strength Models
A stress-strength model defines life of a component which has strength X and is subjected to stress Y. In this paper, we consider the estimation problem of R=P(Y<X) when X ~ Gompertz(c,??) and Y ~ Gompertz(c,??) are independent with c known. R can be considered to be the reliability of a system and is known to be stress-strength reliability. The maximum likelihood estimate of R is derived and various distributional properties of this estimator is discussed. Exact and asymptotic confidence intervals for R are constructed. Also a simulation study is performed to investigate the coverage probabilities of these intervals.
This article shows how to transform residuals from regression on an arbitrary set of k regressors to a set of values having the same joint distribution as the residuals from regression on a different set L of k regressors. Let d′ denote the value of the statistic $\Sigma (z_{t}-z_{t-1})^{2}/\Sigma z_{t}^{2}$ calculated from these values. It is shown that for a suitable choice of L the distribution of d′ is the same as that of $d_{U}$ , the significance values of which are tabulated in [1].