Inference on P(Y<X) in Bivariate Rayleigh Distribution
  This paper deals with the estimation of reliability $R=P(Y<X)$ when $X$ is a random strength of a component subjected to a random stress $Y$ and $(X,Y)$ follows a bivariate Rayleigh distribution. The maximum likelihood estimator of $R$ and its asymptotic distribution are obtained. An asymptotic confidence interval of $R$ is constructed using the asymptotic distribution. Also, two confidence intervals are proposed based on Bootstrap method and a computational approach. Testing of the reliability based on asymptotic distribution of $R$ is discussed. Simulation study to investigate performance of the confidence intervals and tests has been carried out. Also, a numerical example is given to illustrate the proposed approaches. 
Abbas Pak and Nayereh Bagheri Khoolenjani and Ali Akbar Jafari
math.ST stat.ME stat.TH