Estimation of reliability P(X > Y) for distributions with power hazard function based on upper record values

For $$X$$X and $$Y$$Y two independent random variables, upper values from the family of distributions with power hazard function are used to obtain the maximum likelihood and the Bayes estimators of $$P(X \gt Y)$$P(X>Y). The Bayes estimator relies on the squared-error loss function given informative and non-informative prior distributions. It is obtained by either Lindley’s approximation, Tierney and Kadane’s method, or Monte Carlo simulation. The Monte Carlo simulation and Tierney and Kadane’s method have smaller mean squared errors than both Lindley’s approximation and the maximum likelihood estimator. The application for lung cancer data shows that the mortality risk by lung cancer is 40% lower for men than for women. The application for lifetimes of steels shows that steel specimen are 40% more likely to break up under 35.0 stress amplitude than under 35.5.