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

Longxiang Fang; Xiaojun Zhu; N. Balakrishnan

doi:10.1080/02331888.2017.1322086

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

Longxiang Fang^*
, Xiaojun Zhu
, N. Balakrishnan

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

6 Citations (Scopus)

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 X₁, X₂ and X^*₁, X^*₂ follow BVBS(α₁,β₁,(α₂,β₂,ρ) and BVBS(α^*₁,β^*₁,(α^*₂,β^*₂,ρ) distributions, respectively. Suppose 0< υ≤2. We then prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.) implies (Formula presented.). These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.). Analogous results are then deduced for bivariate log BS distributions as well.

Original language	English
Pages (from-to)	147-155
Number of pages	9
Journal	Statistics
Volume	52
Issue number	1
DOIs	https://doi.org/10.1080/02331888.2017.1322086
Publication status	Published - 2 Jan 2018
Externally published	Yes

Keywords

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

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10.1080/02331888.2017.1322086

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title = "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 X1, X2 and X*1, X*2 follow BVBS(α1,β1,(α2,β2,ρ) and BVBS(α*1,β*1,(α*2,β*2,ρ) distributions, respectively. Suppose 0< υ≤2. We then prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.) implies (Formula presented.). These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.). Analogous results are then deduced for bivariate log BS distributions as well.",

keywords = "Bivariate Birnbaum–Saunders distribution, bivariate log Birnbaum–Saunders distribution, majorization, maxima, minima, usual stochastic order",

author = "Longxiang Fang and Xiaojun Zhu and N. Balakrishnan",

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year = "2018",

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TY - JOUR

T1 - Stochastic ordering of minima and maxima from heterogeneous bivariate Birnbaum–Saunders random vectors

AU - Fang, Longxiang

AU - Zhu, Xiaojun

AU - Balakrishnan, N.

PY - 2018/1/2

Y1 - 2018/1/2

N2 - 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 X1, X2 and X*1, X*2 follow BVBS(α1,β1,(α2,β2,ρ) and BVBS(α*1,β*1,(α*2,β*2,ρ) distributions, respectively. Suppose 0< υ≤2. We then prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.) implies (Formula presented.). These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.). Analogous results are then deduced for bivariate log BS distributions as well.

AB - 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 X1, X2 and X*1, X*2 follow BVBS(α1,β1,(α2,β2,ρ) and BVBS(α*1,β*1,(α*2,β*2,ρ) distributions, respectively. Suppose 0< υ≤2. We then prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.) implies (Formula presented.). These results are subsequently generalized to a wider range of scale parameters. Next, we prove that when (Formula presented.), (Formula presented.) implies (Formula presented.) and (Formula presented.). Analogous results are then deduced for bivariate log BS distributions as well.

KW - Bivariate Birnbaum–Saunders distribution

KW - bivariate log Birnbaum–Saunders distribution

KW - majorization

KW - maxima

KW - minima

KW - usual stochastic order

UR - https://www.scopus.com/pages/publications/85019238100

U2 - 10.1080/02331888.2017.1322086

DO - 10.1080/02331888.2017.1322086

M3 - Article

AN - SCOPUS:85019238100

SN - 0233-1888

VL - 52

SP - 147

EP - 155

JO - Statistics

JF - Statistics

IS - 1

ER -

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

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Keywords

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