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

Longxiang Fang, Xiaojun Zhu*, N. Balakrishnan

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (Scopus)


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, . . ., Xn be independent random variables with Xi∼BS(αi, βi), i=1, . . ., n, and X1*,. . .,Xn* be another set of independent random variables with Xi*∼BS(αi*,β,i*),i = 1,. . .,n. Then, we first show that when α1=⋯=αn1*=⋯=αn*, (β1,. . .,βn)≽m(β1*,. . .,βn*) implies Xn:nstXn:n* and (1/β1,. . .,1/βn)≽m(1/β1*,. . .,1/βn*) implies X1:n*stX1:n. We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β1=⋯=βn1*=⋯=βn*, (1/α1,. . .,1/αn)≽m(1/α1*,. . .,1/αn*) implies Xn:nstXn:n* and X1:n*stX1:n. Finally, we establish similar results for the log Birnbaum-Saunders distribution.

Original languageEnglish
Pages (from-to)131-136
Number of pages6
JournalStatistics and Probability Letters
Publication statusPublished - 1 May 2016
Externally publishedYes


  • Birnbaum-Saunders distribution
  • Log Birnbaum-Saunders distribution
  • Majorization
  • Parallel systems
  • Series systems
  • Usual stochastic order

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