Showing papers by "Albert W. Marshall published in 1983"
TL;DR: In this paper, the univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions, and random variables with a multi-dimensional extreme value distribution are associated.
Abstract: The univariate conditions of Gnedenko characterizing domains of attraction for univariate extreme value distributions are generalized to higher dimensions. In addition, it is shown that random variables with a multivariate extreme value distribution are associated. Applications are given to a number of parametric families of joint distributions with given marginal distributions.
103 citations
TL;DR: In this paper, it is shown that many useful processes are NBU random variables, such as Gaver-Miller processes and strong Markov processes, which can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.
Abstract: A stochastic process {Z(t), t - 0}, such that P{Z(0) = 0} = 1, is said to be new better than used (NBu) if, for every x, the first-passage time T, = inf {t :Z(t) > x} satisfies P{T, > s + t} s}P{IT, > t} for every s - 0, t i_ 0. In this paper it is shown that many useful processes are NBU. Examples of such processes include processes with shocks and recovery, processes with random repair-times, various Gaver-Miller processes and some strong Markov processes. Applications in reliability theory, queueing, dams, inventory and electrical activity of neurons are indicated. It is shown that various waiting times for clusters of events and for short and wide gaps in some renewal processes are NBU random variables. The NBU property of processes and random variables can be used to obtain bounds on various probabilistic quantities of interest; this is illustrated numerically.
27 citations
01 Jan 1983
TL;DR: In this article, a brief introduction to the theory of majorization and its use in deriving inequalities is provided. But this work assumes that the order-preserving functions are Schur-convex functions.
Abstract: This paper provides a brief introduction to the theory of majorization and its use in deriving inequalities. Majorization is a preordering of vectors, and inequalities are obtained from the fact that ϕ(x) ≤ ϕ(y) whenever x is majorized by y and ϕ is an order-preserving function. For majorization, the order-preserving functions are called Schur-convex functions. Examples of vectors ordered by majorization, and of Schur-convex functions, are given. To illustrate their usefulness, some inequalities are derived.
10 citations