Usual stochastic and reversed hazard orders of parallel systems with independent heterogeneous components

Abstract

In this paper, we present some new ordering properties between two parallel systems comprising general independent heterogeneous components. More precisely, let $X_1,\cdots ,X_n$ and $Y_1,\cdots ,Y_n$ be independent non-negative random variables with $X_i\sim F(x;{\alpha }_i,{\beta }_i)$ and $Y_i\sim F(x;{\theta }_i,{\lambda }_i),i=1,\cdots ,n,$ where $F(.)$ is an absolutely continuous distribution function with reversed hazard rate function $\tilde{r}(·).$ In this paper, under certain conditions, by using the concept of vector majorization, unordered order, p-majorization and related orders, we discuss stochastic comparisons between parallel systems in the sense of usual stochastic and reversed hazard rate orders. The results developed in this paper generalize some known results in the literature.

Keywords:

• Unordered majorization order
• multivariate chain majorization order
• parallel systems
• reversed hazard rate order
• usual stochastic order

Acknowledgements

Thanks are due to the Editor and anonymous Reviewers for their useful comments and suggestions on an earlier version of this manuscript which led to this improved version.

Additional information

Funding

This work has been conducted by University of Zabol (Grant Number UOZ-GR-9618-14).