Stochastic comparisons on extremes of burr type XII samples associated with Archimedean copula and heterogeneous shape parameters

Abstract

This study deals with random variables from Burr Type XII samples having heterogeneous shape parameters and homogeneous frailty parameters. For samples whose survival copula (whose copula) is Archimedean copula, sufficient conditions based on the heterogeneity of shape parameter vector for the stochastic tail ordering and usual stochastic ordering between sample minimums (sample maximums) are provided. As for independent samples, the convex transform order and tail hazard rate order (tail reversed hazard rate order) between sample minimums (maximums) are discussed. Several numerical examples are also presented to illustrate the theoretical findings.

Keywords:

• Archimedean copula
• convex transform order
• majorization
• tail hazard rate order
• tail reversed hazard rate order

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60E15
• Secondary 62G30

Acknowledgments

The authors would like to express their sincere thank to two referees for their detailed and constructive comments, which led to an significant improvement of the original manuscript.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 The transformation $x=-\hspace{0.17em}\text{ln}\hspace{0.17em}(1-s)$ for $s\in (0,1)$ is performed to show the complete curves on $x\in (0,\infty ).$

Additional information

Funding

This study has been funded Guangdong Natural Science Foundation (2020A1515010436).