Stochastic ordering of minima and maxima from heterogeneous bivariate Birnbaum–Saunders random vectors. Issue 1 (2nd January 2018)

• Record Type: Journal Article
• Authors: Fang, Longxiang; Zhu, Xiaojun; Balakrishnan, N.
• Is Part Of: Statistics. Volume 52:Issue 1(2018)
• Page Start: 147
• Page End: 155
• Publication Date: 2018-01-02
• Subjects: Bivariate Birnbaum–Saunders distribution -- bivariate log Birnbaum–Saunders distribution -- maxima -- minima -- majorization -- usual stochastic order; 60E15 -- 62G30 -- 62D05 -- 62N05; Mathematical statistics -- Periodicals; 519.505
• DOI: 10.1080/02331888.2017.1322086
• Languages: English
• ISSNs: 0233-1888

Abstract

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 ∗, X 2 ∗ ) followBVBS ( α 1, β 1, α 2, β 2, ρ ) andBVBS ( α 1 ∗, β 1 ∗, α 2 ∗, β 2 ∗, ρ ) distributions, respectively. Suppose0 < ν ≤ 2. We then prove that whenα 1 = α 2 = α 1 ∗ = α 2 ∗, ( β 1 − 1 / ν, β 2 − 1 / ν ) ⪰ m ( β 1 ∗ − 1 / ν, β 2 ∗ − 1 / ν ) impliesX 2 : 2 ≥ s t X 2 : 2 ∗ and( β 1 1 / ν, β 2 1 / ν ) ⪰ m ( β 1 ∗ ( 1 / ν ), β 2 ∗ ( 1 / ν ) ) impliesX 1 : 2 ∗ ≥ s t X 1 : 2 . These results are subsequently generalized to a wider range of scale parameters. Next, we prove that whenβ 1 = β 2 = β 1 ∗ = β 2 ∗, ( 1 / α 1, 1 / α 2 ) ⪰ m ( 1 / α 1 ∗, 1 / α 2 ∗ ) impliesX 2 : 2 ≥ s t X 2 : 2 ∗ andX 1 : 2 ∗ ≥ s t X 1 : 2 . Analogous results are then deduced for bivariate log BS distributions as well.