Frank Proschan’s research while affiliated with Florida State University and other places

Let λ=(λ1,…,λn), λ1 ⩽…⩽λn, and . A function is said to be arrangement increasing (AI) if (i) ƒ is permutation invariant in both arguments λ and , and (ii) ) ⩾ whenever x and x′ differ in two coordinates only, say i and j, (xi-xj)(i–j)⩾0, and xi′=xj, xj′=xi. This paper reviews concepts and many of the basic properties of AI functions, their preservation properties under mixtures, compositions and integral transformations. The AI class of functions includes as special cases other well-known classes of functions such as Schur functions, totally positive functions of order two and positive set functions. We present a number of applications of AI functions to problems in probability, statistics, reliability theory and mathematics. A multivariate extension of the arrangement ordering is also reviewed.

A composition theorem for functions obeying certain positive ordering is proved. The novelty of the present version is that unlike earlier results which assume both components of the composition to be distributions or survival functions, one of the components is allowed to be negative and unbounded. The theorem is applied to yield very simple proof of characterizations for failure rate orderings of distributions given recently by Capéraà (1988). We also use this composition theorem to give a characterization of two distributions with ordered mean residual life functions.

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 (MTP 2 ). Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

The concept of max-infinite divisibility is viewed as a positive dependence concept. It is shown that every max-infinitely divisible distribution function is a multivariate totally positive function of order 2 MTP2. Inequalities are derived, with emphasis on exchangeable distributions. Applications and examples are given throughout the paper.

The hazard rate ordering is an ordering for random variables which compares lifetimes with respect to their hazard rate functions. It is stronger than the usual stochastic order for random variables, yet is weaker than the likelihood ratio ordering. The hazard rate ordering is particularly useful in reliability theory and survival analysis, owing to the importance of the hazard rate function in these areas. In this paper earlier work on the hazard rate ordering is reviewed, and extensive new results related to coherent systems are derived. Initially we fix the form of a coherent structure and investigate the effect on the hazard rate function of the system when we switch the lifetimes of its components from the vector ( T 1 , · ··, T n ) to the vector ( T′ 1 , · ··, T′ n ), where the hazard rate functions of the two vectors are assumed to be comparable in some sense. Although the hazard rate ordering is closed under the formation of series systems, we see that this is not the case for parallel systems even when the system is a two-component parallel system with exponentially distributed lifetimes. A positive result shows that for two-component parallel systems with proportional hazards ( λ 1 r ( t ), λ 2 r ( t ))), the more diverse ( λ 1 , λ 2 ) is in the sense of majorization the stronger is the system in the hazard rate ordering. Unfortunately even this result does not extend to parallel systems with more than two components, demonstrating again the delicate nature of the hazard rate ordering. The principal result of the paper concerns the hazard rate ordering for the lifetime of a k -out-of- n system. It is shown that if τ k|n is the lifetime of a k -out-of- n system, then τ k|n is greater than τ k+ 1 |n in the hazard rate ordering for any k. This has an interesting interpretation in the language of order statistics. For independent (not necessarily identically distributed) lifetimes T 1 , · ··, T n , we let T k:n represent the k th order statistic (in increasing order). Then it is shown that T k + 1 :n is greater than T k:n in the hazard rate ordering for all k = 1, ···, n − 1. The result does not, however, extend to the stronger likelihood ratio order.

Convolutions of random variables which are either exponential or geometric are studied with respect to majorization of parameter vectors and the likelihood ratio ordering ([greater-or-equal, slanted]lr) of random variables. Let X[lambda], ..., X[lambda]n be independent exponential random variables with respective hazards [lambda]i (means 1/[lambda]i), i = 1 ..., n. Then if [lambda] = ([lambda]1, ..., [lambda]n) [greater-or-equal, slanted]m ([lambda]1', ..., [lambda]n') = [lambda]', it follows that [Sigma]i = 1n X[lambda] [greater-or-equal, slanted]lr [Sigma]i = 1n X[lambda]'1. Similarly if Xp1, ..., Xpn are independent geometric random variables with respective parameters p1, ..., pn, then p = (p1, ..., pn) [greater-or-equal, slanted]m(p'1, ..., p'n) = p' or log p = (log p1, ..., log pn) [greater-or-equal, slanted] m (log p1, ..., log pn) = log p' implies [Sigma]i = 1n Xpl [greater-or-equal, slanted] lr [Sigma]i = 1n XP'1. Applications of these results are given yielding convenient upper bounds for the hazard rate function of convolutions of exponential (geometric) random variables in terms of those of gamma (negative binomial) distributions. Other applications are also given for a server model, the range of a sample of i.i.d. exponential random variables, and the duration of a multistate component performing in excess of a given level.

In this paper we first prove an arrangement-decreasing property of partial sums of independent random variables when they are partially ordered through the likelihood ratio ordering. We then apply a similar argument to obtain a stochastic ordering of random processes via a comparison of their parameter functions, with special applications to Poisson and Wiener processes. Finally, in Section 4 we present some applications in reliability theory, queueing, and first-passage problems.

We give short proofs by total positivity arguments that: (1) the reliability function h(k\n)(p) of a k-out-of-n system crosses an arbitrary coherent system reliability function h(p) at most once, and if a crossing does occur, h(k\n)(p) crosses h(p) from below for 0 less-than-or-equal-to p less-than-or-equal-to 1, and (2) h(p) crosses the reliability function p(m) (alternatively, 1 - q(m)) of a system of m independent components in series (alternatively, parallel) at most once, and if it does occur, it crosses from below, 1 less-than-or-equal-to m less-than-or-equal-to n-1.