Moshe Shaked

STOCHASTIC CONVEXITY AND ITS APPLICATIONS

Abstract : Several notions of stochastic convexity and concavity and their properties are studied in this paper. Efficient sample path approaches are developed in order to verify the occurrence of these notions in various applications. Numerous examples are given. The use of these notions in several areas of probability and statistics is demonstrated. In queueing theory, the convexity (as a function of c) of the steady state mean waiting time in a GI/D/c queue, and as a function of the arrival and service rates in a GI/G/1 queue, is established. Also the convexity of the queue length in the M/M/c case as a function of the arrival rate is shown, thus strengthening previous results while simplifying their derivation. In reliability theory, the convexity of the payoff on the success rate of an imperfect repair is obtained and used to find an optimal repair probability. Also the convexity of the damage as a function of time in a cumulative damage shock model is shown. In branching processes, the convexity of the population size as a function of a parameter of the offspring distribution is proved. In nonparametric statistics, the stochastic concavity (convexity) of the empirical distribution function is established. And, for applications in the theory of probability inequalities, we identify several families of distributions which are convexly parametrized.

Cores of Inventory Centralization Games

Abstract Consider a set of n stores with single-item and single-period demands. Assume an option of centralized ordering and inventory with holding and penalty costs only. In this case, a cooperative inventory “centralization” game “defines” allocations of the cost. We examine the conditions under which such an inventory centralization game has a nonempty core. We prove the existence of nonempty core for demands with symmetric distributions and the existence of nonempty core for joint multivariate normal demand distribution. We establish the equivalency of four different nonempty core conditions for the Newsboy Problem and demonstrate their efficacy for discrete independent and identically distributed (iid) demands. Journal of Economic Literature Classification Numbers: C44, C62, C71.

Two Variability Orders

In this paper we study a new variability order that is denoted by ≤ st:icx . This order has important advantages over previous variability orders that have been introduced and studied in the literature. In particular, X ≤ st:icx Y implies that Var[ h ( X )] ≤ Var[ h ( Y )] for all increasing convex functions h . The new order is also closed under formations of increasing directionally convex functions; thus it follows that it is closed, in particular, under convolutions. These properties make this order useful in applications. Some sufficient conditions for X ≤ st:icx Y are described. For this purpose, a new order, called the excess wealth order, is introduced and studied. This new order is based on the excess wealth transform which, in turn, is related to the Lorenz curve and to the TTT (total time on test) transform. The relationships to these transforms are also studied in this paper. The main closure properties of the order ≤ st:icx are derived, and some typical applications in queueing theory are described.

A Family of Concepts of Dependence for Bivariate Distributions

Abstract A family of concepts of stochastic dependence for bivariate distribution functions is introduced. Each concept gives rise to a family of bivariate distribution functions. We show the equivalence of some of these families with families of positively dependent distribution functions, which are known in the literature, and characterize some of them by notions from reliability theory. Interrelations among the various families are studied, and some moment inequalities are derived. Some examples and applications are discussed.

Multivariate Imperfect Repair

In this paper, we consider models of systems whose components have dependent lifelengths and are imperfectly repaired upon failure until they are scrapped. First, assuming that no more than one component can fail at a time, we study two models that describe imperfect repairs, and derive the resulting density and other probabilistic quantities of interest. We then generalize the models to cover applications in which more than one component can fail at the same time, and obtain various properties of the resulting distributions. Finally, we illustrate the theory through some examples.

Parametric stochastic convexity and concavity of stochastic processes

A collection of random variables {X(θ), θ∈Θ} is said to be parametrically stochastically increasing and convex (concave) in θ∈Θ if X(θ) is stochastically increasing in θ, and if for any increasing convex (concave) function ϕ, Eϕ(X(θ)) is increasing and convex (concave) in θ∈Θ whenever these expectations exist. In this paper a notion of directional convexity (concavity) is introduced and its stochastic analog is studied. Using the notion of stochastic directional convexity (concavity), a sufficient condition, on the transition matrix of a discrete time Markov process {Xn(θ), n=0,1,2,...}, which implies the stochastic monotonicity and convexity of {Xn(θ), θ∈Θ}, for any n, is found. Through uniformization these kinds of results extend to the continuous time case. Some illustrative applications in queueing theory, reliability theory and branching processes are given.

Multivariate Phase-Type Distributions

A univariate random variable is said to be of phase type if it can be represented as the time until absorption in a finite state absorbing Markov chain. Univariate phase type random variables are useful because they arise from processes that are often encountered in applications, they have densities that can be written in a closed form, they possess some useful closure properties, and they can approximate any nonnegative random variable. This paper introduces and discusses several extensions to the multivariate case. It shows that the multivariate random variables possess many of the properties of univariate phase type distributions and derives explicit formulas for various probabilistic quantities of interest. Some examples are included.

The Total Time on Test Transform and the Excess Wealth Stochastic Orders of Distributions

For nonnegative random variables X and Y we write X ≤TTT Y if ∫0 F -1(p)(1-F(x))dx ≤ ∫0 G -1(p)(1-G(x))dx all p ∈ (0,1), where F and G denote the distribution functions of X and Y respectively. The purpose of this article is to study some properties of this new stochastic order. New properties of the excess wealth (or right-spread) order, and of other related stochastic orders, are also obtained. Applications in the statistical theory of reliability and in economics are included.

INFERENCE FOR STEP-STRESS ACCELERATED LIFE TESTS

Abstract In this paper we consider the more realistic aspect of accelerated life testing wherein the stress on an unfailed item is allowed to increase at a preassigned test time. Such tests are known as step-stress tests. Our approach is nonparametric in that we do not make any assumptions about the underlying distribution of life lengths. We introduce a model for step-stress testing which is based on the ideas of shock models and of wear processes. This model unifies and generalizes two previously proposed models for step-stress testing. We propose an estimator for the life distribution under use conditions stress and show that this estimator is strongly consistent.

A general theory of some positive dependence notions

A general theory of concepts of positive dependence, which are weaker than association but stronger than orthant dependence, is developed. A random vector X is associated if and only if P(X [set membership, variant] A [down curve] B) >= P(X [set membership, variant] A) P(X [set membership, variant] B) for all open upper sets A and B. By requiring the above inequality to hold only for some open upper sets A and B various notions of positive dependence which are weaker than association are obtained. First a general theory is given and then the results are specialized to some concepts of a particular interest. Various properties and interrelationships are derived and some applications are discussed.