Stochastic comparisons of parallel and series systems with heterogeneous Birnbaum-Saunders components

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• 作者：Longxiang Fang^a ; ^b ; ^{lxfang@fudan.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Xiaojun Zhu^b ; ^{xiaojunzhu8899@hotmail.com" class="auth_mail" title="E-mail the corresponding author} ; N. Balakrishnan^b ; ^{bala@mcmaster.ca" class="auth_mail" title="E-mail the corresponding author}
• 关键词：60E15 ; 62G30 ; 62D05 ; 62N05
• 刊名：Statistics & Probability Letters
• 出版年：2016
• 出版时间：May 2016
• 年：2016
• 卷：112
• 期：Complete
• 页码：131-136
• 全文大小：370 K

文摘

In this paper, we discuss stochastic comparisons of lifetimes of parallel and series systems with independent heterogeneous Birnbaum–Saunders components with respect to the usual stochastic order based on vector majorization of parameters. Specifically, let X₁,…,X_n$X_1,\dots ,X_n$ be independent random variables with 620be5e9f4a043f" title="Click to view the MathML source">X_i∼BS(α_i,β_i),i=1,…,n$X_i\sim BS({\alpha }_i,{\beta }_i),i=1,\dots ,n$, and $X_1^{\ast },\dots ,X_n^{\ast }$ be another set of independent random variables with $X_i^{\ast }\sim BS({\alpha }_i^{\ast },{\beta }_i^{\ast }),i=1,\dots ,n$. Then, we first show that when ${\alpha }_1=\cdots ={\alpha }_n={\alpha }_1^{\ast }=\cdots ={\alpha }_n^{\ast }$, 62d6805c3addbc49714">$({\beta }_1,\dots ,{\beta }_n){⪰}_m({\beta }_1^{\ast },\dots ,{\beta }_n^{\ast })$ implies $X_{n:n}{\ge }_{st}{X}_{n:n}^{\ast }$ and $(\frac{1}{{\beta }_1},\dots ,\frac{1}{{\beta }_n}){⪰}_m(\frac{1}{{\beta }_1^{\ast }},\dots ,\frac{1}{{\beta }_n^{\ast }})$ implies $X_{1:n}^{\ast }{\ge }_{st}{X}_{1:n}$. We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when ${\beta }_1=\cdots ={\beta }_n={\beta }_1^{\ast }=\cdots ={\beta }_n^{\ast }$, $(\frac{1}{{\alpha }_1},\dots ,\frac{1}{{\alpha }_n}){⪰}_m(\frac{1}{{\alpha }_1^{\ast }},\dots ,\frac{1}{{\alpha }_n^{\ast }})$ implies $X_{n:n}{\ge }_{st}{X}_{n:n}^{\ast }$ and $X_{1:n}^{\ast }{\ge }_{st}{X}_{1:n}$. Finally, we establish similar results for the log Birnbaum–Saunders distribution.