Inference of $P\left(X<Y\right)$ for two-parameter exponential-Rayleigh distribution with applications

Let $X$ and $Y$ follow independent exponential-Rayleigh distributions when the shape and scale parameters are different. In this paper, the maximum likelihood and the Bayes estimates of the stress-strength parameter $\delta =P(X<Y)$ are derived. Based on the sampling technique, we use the asymptotic distribution and the Bayes estimate of $\delta$ to construct the corresponding confidence and credible intervals. Analyses of two data sets, one simulated data and the other real-life data, are given for illustrative purposes. Finally, Monte Carlo simulations are used to compare the different methods discussed here.