Abstract
Surles and Padgett [Inference for reliability and stress-strength for a scaled Burr type X distribution. Lifetime Data Anal. 2001;7:187-200] introduced a two-parameter Burr-type X distribution, which can be described as a generalized Rayleigh distribution. In this paper, we consider the estimation of the stress-strength parameter R=P[Y<X], when X and Y are both three-parameter generalized Rayleigh distributions with the same scale and locations parameters but different shape parameters. It is assumed that they are independently distributed. It is observed that the maximum-likelihood estimators (MLEs) do not exist, and we propose a modified MLE of R. We obtain the asymptotic distribution of the modified MLE of R, and it can be used to construct the asymptotic confidence interval of R. We also propose the Bayes estimate of R and the construction of the associated credible interval based on importance sampling technique. Analysis of two real data sets, (i) simulated and (ii) real, have been performed for illustrative purposes.