Stochastic comparisons on extreme order statistics from observations associated by FGM copula

Abstract

This study deals with the stochastic comparison of extreme order statistics from random samples with FGM copula or FGM survival copula. For general marginal distributions, several sufficient conditions on the usual stochastic order, the reversed hazard rate order, the hazard rate order, the increasing convex order and the increasing concave order on sample maximums or minimums are derived. Then, we apply these conditions to samples following proportional hazard rate model, proportional reversed hazard rate model, Weibull distributions and negative binomial distributions, and generalize some ordering results about samples having independent observations to the case with FGM copula or FGM survival copula. Two comparing results on the second-order statistics from proportional hazard samples are presented. Some numerical examples are provided to illustrate the theoretical results as well.

Keywords:

• FGM copula
• negative binomial distribution
• proportional hazard rate model
• proportional reversed hazard rate model
• stochastic order

MATHEMATICS SUBJECT CLASSIFICATION:

• Primary 60E15
• Secondary 62G30

Acknowledgments

The authors would like to thank the Associate Editor and three anonymous reviewers for their valuable comments, which help improve the presentation of this paper.

Notes

1 To show the whole range, we perform the transform $x=1/t-1$ and plot the curve over the interval $t\in (0,1).$