Some Ordering Results for the Marshall and Olkin's Family of Distributions

In this paper, we consider Marshall and Olkin's family of distributions. The parent (baseline) distribution is taken to be a scaled family of distributions. Two models: (i) modified proportional hazard rate scale and (ii) modified proportional reversed hazard rate scale, are considered. Some stochastic comparison results in terms of the usual stochastic, hazard rate and reversed hazard rate orders are studied to compare order statistics formed from two sets of independent observations following these models. Most of the sufficient conditions are obtained depending on various majorization-type partial orderings. Further, the setup with multiple-outlier model is taken. Various stochastic orders between the smallest and largest order statistics are developed. Several numerical examples are provided to illustrate the effectiveness of the established theoretical results.