Ordering results for the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables

ABSTRACT

In this paper, we discuss stochastic comparisons of the smallest and largest order statistics from independent heterogeneous exponential–Weibull random variables. Let $X_1,\dots ,X_n$ be independent random variables with $X_i\sim EW({\alpha }_i,{\beta }_i,{\gamma }_i),i=1,\dots ,n.$ Further, let $X_1^{\ast },\dots ,X_n^{\ast }$ be another set of independent random variables with $X_i^{\ast }\sim EW({\alpha }_i^{\ast },{\beta }_i^{\ast },{\gamma }_i^{\ast }),i=1,\dots ,n.$ First, when ${\gamma }_1=\cdots ={\gamma }_n={\gamma }_1^{\ast }=\cdots ={\gamma }_n^{\ast }$ and a matrix with different parameters ${\alpha }_i,{\beta }_i$ changes to another matrix in the sense of multivariate chain majorization and row majorization, we investigate the usual stochastic order of the largest order statistics. Next, when ${\alpha }_1=\cdots ={\alpha }_n={\alpha }_1^{\ast }=\cdots ={\alpha }_n^{\ast },{\beta }_1=\cdots ={\beta }_n={\beta }_1^{\ast }=\cdots ={\beta }_n^{\ast }$ and $({\gamma }_1,\dots ,{\gamma }_n){⪰}^m({\gamma }_1^{\ast },\dots ,{\gamma }_n^{\ast })$, we establish the usual stochastic order of the largest and smallest order statistics. Finally, we provide sufficient conditions for the hazard rate order of the smallest order statistics.

KEYWORDS:

• exponential–Weibull distribution
• usual stochastic order
• hazard rate order
• multivariate chain majorization

AMS SUBJECT CLASSIFICATION:

• 60E15
• 62D05
• 62G30
• 62N05

Acknowledgments

The authors thank the Editor and anonymous reviewers for their useful suggestions on an earlier version of this manuscript which led to this improved one.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

This research was supported by the Provincial Natural Science Research Project of Anhui Colleges, the National Natural Science Foundation of Anhui Province [No. 1408085MA07], the Distinguished Young Scholars Foundation of Anhui Province [No. 1608085J06], and the Ph.D. research startup foundation of Anhui Normal University [No. 2014bsqdjj34] which facilitated the research visit of the first author to McMaster University, Canada. The research work of the second author was supported by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.