Multivariate Liouville distributions, III

Rameshwar D. Gupta, Donald Richards

Research output: Contribution to journalArticle

21 Citations (Scopus)

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 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.

Original languageEnglish (US)
Pages (from-to)29-57
Number of pages29
JournalJournal of Multivariate Analysis
Volume43
Issue number1
DOIs
StatePublished - Jan 1 1992

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Partial ordering
Distribution functions
Lifetime
K-out-of-n System
Reliability theory
Mixing Distribution
Reliability Theory
Reliability Function
Stochastic Ordering
Sufficient Conditions
Cumulative distribution function
Laplace transforms
Number of Components
Coefficient
Order Statistics
Random variables
Identically distributed
Laplace transform
Spacing
Linear Combination

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Numerical Analysis
  • Statistics, Probability and Uncertainty

Cite this

Gupta, Rameshwar D. ; Richards, Donald. / Multivariate Liouville distributions, III. In: Journal of Multivariate Analysis. 1992 ; Vol. 43, No. 1. pp. 29-57.
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Multivariate Liouville distributions, III. / Gupta, Rameshwar D.; Richards, Donald.

In: Journal of Multivariate Analysis, Vol. 43, No. 1, 01.01.1992, p. 29-57.

Research output: Contribution to journalArticle

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