### Abstract

We present a panoply of results on partial orderings for the Liouville distributions, including sufficient conditions for two Liouville vectors to be comparable under the stochastic, convex, concave, and Laplace transform orderings. Further, we derive partial orderings for the order statistics and spacings from certain exchangeable Liouville distributions. As applications to reliability theory, we obtain stochastic orderings for N(t) and bounds for R_{k}(t), the number of components working at time t ≥ 0 and the reliability function, respectively, for a "k-out-of-n" system consisting of components whose lifetimes have a joint Liouville distribution. When the component lifetimes are distributed as a mixture of independent, identically distributed exponential random variables, we derive some results for a conjecture of Lefevre and Malice (J. Appl. Prob. 26 (1989), 202-208) on variation comparisons for R_{k}(t) as the mixing distribution is varied. Following a suggestion and using the methods of Diaconis and Perlman (in Topics in Statistical Dependence, IMS Lecture Notes, 1991), we compare the cumulative distribution functions of two linear combinations of an exchangeable Liouville vector when the first vector of coefficients majorizes the second vector of coefficients. We derive sufficient conditions under which the two distribution functions cross exactly once, and obtain bounds for the location of the unique crossing point.

Original language | English (US) |
---|---|

Pages (from-to) | 29-57 |

Number of pages | 29 |

Journal | Journal of Multivariate Analysis |

Volume | 43 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1992 |

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### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Numerical Analysis
- Statistics, Probability and Uncertainty

### Cite this

*Journal of Multivariate Analysis*,

*43*(1), 29-57. https://doi.org/10.1016/0047-259X(92)90109-S

}

*Journal of Multivariate Analysis*, vol. 43, no. 1, pp. 29-57. https://doi.org/10.1016/0047-259X(92)90109-S

**Multivariate Liouville distributions, III.** / Gupta, Rameshwar D.; Richards, Donald.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Multivariate Liouville distributions, III

AU - Gupta, Rameshwar D.

AU - Richards, Donald

PY - 1992/1/1

Y1 - 1992/1/1

N2 - We present a panoply of results on partial orderings for the Liouville distributions, including sufficient conditions for two Liouville vectors to be comparable under the stochastic, convex, concave, and Laplace transform orderings. Further, we derive partial orderings for the order statistics and spacings from certain exchangeable Liouville distributions. As applications to reliability theory, we obtain stochastic orderings for N(t) and bounds for Rk(t), the number of components working at time t ≥ 0 and the reliability function, respectively, for a "k-out-of-n" system consisting of components whose lifetimes have a joint Liouville distribution. When the component lifetimes are distributed as a mixture of independent, identically distributed exponential random variables, we derive some results for a conjecture of Lefevre and Malice (J. Appl. Prob. 26 (1989), 202-208) on variation comparisons for Rk(t) as the mixing distribution is varied. Following a suggestion and using the methods of Diaconis and Perlman (in Topics in Statistical Dependence, IMS Lecture Notes, 1991), we compare the cumulative distribution functions of two linear combinations of an exchangeable Liouville vector when the first vector of coefficients majorizes the second vector of coefficients. We derive sufficient conditions under which the two distribution functions cross exactly once, and obtain bounds for the location of the unique crossing point.

AB - We present a panoply of results on partial orderings for the Liouville distributions, including sufficient conditions for two Liouville vectors to be comparable under the stochastic, convex, concave, and Laplace transform orderings. Further, we derive partial orderings for the order statistics and spacings from certain exchangeable Liouville distributions. As applications to reliability theory, we obtain stochastic orderings for N(t) and bounds for Rk(t), the number of components working at time t ≥ 0 and the reliability function, respectively, for a "k-out-of-n" system consisting of components whose lifetimes have a joint Liouville distribution. When the component lifetimes are distributed as a mixture of independent, identically distributed exponential random variables, we derive some results for a conjecture of Lefevre and Malice (J. Appl. Prob. 26 (1989), 202-208) on variation comparisons for Rk(t) as the mixing distribution is varied. Following a suggestion and using the methods of Diaconis and Perlman (in Topics in Statistical Dependence, IMS Lecture Notes, 1991), we compare the cumulative distribution functions of two linear combinations of an exchangeable Liouville vector when the first vector of coefficients majorizes the second vector of coefficients. We derive sufficient conditions under which the two distribution functions cross exactly once, and obtain bounds for the location of the unique crossing point.

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U2 - 10.1016/0047-259X(92)90109-S

DO - 10.1016/0047-259X(92)90109-S

M3 - Article

VL - 43

SP - 29

EP - 57

JO - Journal of Multivariate Analysis

JF - Journal of Multivariate Analysis

SN - 0047-259X

IS - 1

ER -