## Abstract

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 X_{1}, . . ., X_{n} be independent random variables with X_{i}∼BS(α_{i}, β_{i}), i=1, . . ., n, and X_{1}^{*},. . .,X_{n}^{*} be another set of independent random variables with X_{i}^{*}∼BS(α_{i}^{*},β,_{i}^{*}),i = 1,. . .,n. Then, we first show that when α_{1}=⋯=α_{n}=α_{1}^{*}=⋯=α_{n}^{*}, (β_{1},. . .,β_{n})≽m(β_{1}^{*},. . .,β_{n}^{*}) implies X_{n:n}≥_{st}X_{n:n}^{*} and (1/β_{1},. . .,1/β_{n})≽_{m}(1/β_{1}^{*},. . .,1/β_{n}^{*}) implies X_{1:n}^{*}≥_{st}X_{1:n}. We subsequently generalize these results to a wider range of the scale parameters. Next, we show that when β_{1}=⋯=β_{n}=β_{1}^{*}=⋯=β_{n}^{*}, (1/α_{1},. . .,1/α_{n})≽_{m}(1/α_{1}^{*},. . .,1/α_{n}^{*}) implies X_{n:n}≥_{st}X_{n:n}^{*} and X_{1:n}^{*}≥_{st}X_{1:n}. Finally, we establish similar results for the log Birnbaum-Saunders distribution.

Original language | English |
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Pages (from-to) | 131-136 |

Number of pages | 6 |

Journal | Statistics and Probability Letters |

Volume | 112 |

DOIs | |

Publication status | Published - 1 May 2016 |

Externally published | Yes |

## Keywords

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