Stochastic ordering of minima and maxima from heterogeneous bivariate Birnbaum–Saunders random vectors

ABSTRACT

In this paper, we discuss stochastic comparisons of minima and maxima arising from heterogeneous bivariate Birnbaum–Saunders (BS) random vectors with respect to the usual stochastic order based on vector majorization of parameters. Suppose the bivariate random vectors $(X_1,X_2)$ and $(X_1^{\ast },X_2^{\ast })$ follow $\mathrm{BVBS}({\alpha }_1,{\beta }_1,{\alpha }_2,{\beta }_2,\rho )$ and $\mathrm{BVBS}({\alpha }_1^{\ast },{\beta }_1^{\ast },{\alpha }_2^{\ast },{\beta }_2^{\ast },\rho )$ distributions, respectively. Suppose $0<\nu \le 2.$ We then prove that when ${\alpha }_1={\alpha }_2={\alpha }_1^{\ast }={\alpha }_2^{\ast }$, $({\beta }_1^{-1/\nu },{\beta }_2^{-1/\nu }){⪰}_m({\beta }_1^{\ast -1/\nu },{\beta }_2^{\ast -1/\nu })$ implies $X_{2:2}{\ge }_{st}{X}_{2:2}^{\ast }$ and $({\beta }_1^{1/\nu },{\beta }_2^{1/\nu }){⪰}_m({\beta }_1^{\ast \left(1/\nu \right)},{\beta }_2^{\ast \left(1/\nu \right)})$ implies $X_{1:2}^{\ast }{\ge }_{st}{X}_{1:2}$. These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when ${\beta }_1={\beta }_2={\beta }_1^{\ast }={\beta }_2^{\ast }$, $(1/{\alpha }_1,1/{\alpha }_2){⪰}_m(1/{\alpha }_1^{\ast },1/{\alpha }_2^{\ast })$ implies $X_{2:2}{\ge }_{st}{X}_{2:2}^{\ast }$ and $X_{1:2}^{\ast }{\ge }_{st}{X}_{1:2}$. Analogous results are then deduced for bivariate log BS distributions as well.

KEYWORDS:

• Bivariate Birnbaum–Saunders distribution
• bivariate log Birnbaum–Saunders distribution
• maxima
• minima
• majorization
• usual stochastic order

AMS SUBJECT CLASSIFICATION:

• 60E15
• 62G30
• 62D05
• 62N05

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 [No.KJ2016A263, KJ2017ZD27], the National Natural Science Foundation of Anhui Province [No.1608085J06], and the PhD research startup foundation of Anhui Normal University [No. 2014bsqdjj34]. The research work of the last author was supported by an Individual Discovery Grant from the Natural Sciences and Engineering Research Council of Canada.