Multivariate Liouville distributions, III

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.