Shaul K. Bar-Lev

The diagnonal multivariate natural exponential families and their classification

A natural exponential family (NEF)F in ℝn,n>1, is said to be diagonal if there existn functions,a1,...,an, on some intervals of ℝ, such that the covariance matrixVF(m) ofF has diagonal (a1(m1),...,an(mn)), for allm=(m1,...,mn) in the mean domain ofF. The familyF is also said to be irreducible if it is not the product of two independent NEFs in ℝk and ℝn-k, for somek=1,...,n−1. This paper shows that there are only six types of irreducible diagonal NEFs in ℝn, that we call normal, Poisson, multinomial, negative multinomial, gamma, and hybrid. These types, with the exception of the latter two, correspond to distributions well established in the literature. This study is motivated by the following question: IfF is an NEF in ℝn, under what conditions is its projectionp(F) in ℝk, underp(x1,...,xn)∶=(x1,...,xk),k=1,...,n−1, still an NEF in ℝk? The answer turns out to be rather predictable. It is the case if, and only if, the principalk×k submatrix ofVF(m1,...,mn) does not depend on (mk+1,...,mn).

Bayesian inference for the power law process

The power law process has been used to model reliability growth, software reliability and the failure times of repairable systems. This article reviews and further develops Bayesian inference for such a process. The Bayesian approach provides a unified methodology for dealing with both time and failure truncated data. As well as looking at the posterior densities of the parameters of the power law process, inference for the expected number of failures and the probability of no failures in some given time interval is discussed. Aspects of the prediction problem are examined. The results are illustrated with two data examples.

On the EOQ model with inventory-level-dependent demand rate and random yield

This paper deals with an EOQ-type inventory problem where the demand rate is a function of the inventory level. It has been noted by marketing researchers and practitioners that an increase in a products shelf space usually has a positive impact on the sales of the product. In such a case, the demand rate is no longer a constant, but it depends on the amount of on-hand inventory. Our objective is to develop a model that can accommodate the dependency of demand on the inventory level. We also assume that the yield is random, i.e., when Q units are ordered, the amount received is YQ where the yield rate Y is a non-negative random variable. With these assumptions of inventory-dependent demand rate and random yield, we develop an extension of the EOQ model. Expected long-run average cost function for the resulting continuous review stochastic inventory model is obtained using the renewal reward theorem. We choose order quantity as the decision variable of the model. Since the computation of the objective function requires the expected stationary inventory level, we develop the stationary distribution of thos stochastic process by using level crossing theory. We discuss three special cases of the objective function which correspond to the following special models: The standard EOQ model; the EOQ model modified to consider random yield; and the EOQ model modified to consider inventory level dependent demand rate. We then discuss the general case where demand rate is a power function of the inventory level and yield rate Y is distributed as a beta random variable. Numerical examples for all four models are presented.

Distributions of stopping times for compound poisson processes with positive jumps and linear boundaries

Let be a compound Poisson process, i.e. positive random variables and a time homogeneous Poisson jump process. We consider two linear boundaries and and the stopping times and . Laplace Stieltjes transforms are developed for the distributions of TU and T for general distribution of . These transforms are obtained by analyzing the sample path behavior of the process {Yt }. The results are applicable to reliability theory, sequential analysis, queuing theory, dam theory, risk analysis, and more

A Common Conjugate Prior Structure For Several Randomized Response Models

In this paper we present a common Bayesian approach to four randomized response models, including Warners (1965) and other modification for it that appeared thereafter in the literature. Suitable truncated beta distributions are used throughout in a common conjugate prior structure to obtain the Bayes estimates for the proportion of a “sensitive” attribute in the population of interest. The results of this common conjugate prior approach are contrasted with those of Winkler and Franklins (1979), in which non-conjugate priors have been used in the context of Warners model. The results are illustrated numerically in several cases and exemplified further with data reported in Liu and Chow (1976) concerning incidents of induced abortions.

Incomplete identification models for group‐testable items

A set of items is called group-testable if for any subset of these items it is possible to carry out a simultaneous test with two possible outcomes: “success,” indicating that all items in the subset are good, and “failure,” indicating a contaminated subset. In this article we compare two alternatives of purchasing group-testable items in order to meet a demand of d good items. These two alternatives are (i) purchasing d good items from a 100% quality population with a relatively high cost per item, and (ii) purchasing N items, N>d, from a 100q% (0<q<1) quality population with a relatively low cost per item, group-testing groups of fixed size m, and recording good groups until d good items are accumulated, where both N and m are decision variables. We present three models (of which two are probabilistic and one is deterministic) which are related to purchasing items by alternative (ii) and are costwise competitive with alternative (i).

Minimum variance unbiased estimation for families of distributions involving two truncation parameters

Abstract In this paper we treat the problem of minimum variance unbiased estimation for families of distributions involving two truncation parameters. Under mild conditions, an expression for a minimum variance unbiased estimator of a general function of these parameters is derived. The results are applied to obtain a minimum variance unbiased estimator of the tail probability of such distributions, and an explicit expression for the variance of this estimator is found.

Large sample properties of the mle and mcle for the natural parameter of a truncated exponential family

SummaryConsider a truncated exponential family of absolutely continuous distributions with natural parameter θ and truncation parameter γ. Strong consistency and asymptotic normality are shown to hold for the maximum likelihood and maximum conditional likelihood estimates of θ with γ unknown. Moreover, these two estimates are also shown to have the same limiting distribution, coinciding with that of the maximum likelihood estimate for θ when γ is assumed to be known.

Group Testing Models with Processing Times and Incomplete Identification

We consider the group testing problem for a finite population of possibly defective items with the objective of sampling a prespecified demanded number of nondefective items at minimum cost. Group testing means that items can be pooled and tested together; if the group comes out clean, all items in it are nondefective, while a “contaminated” group is scrapped. Every test takes a random amount of time and a given deadline has to be met. If the prescribed number of nondefective items is not reached, the demand has to be satisfied at a higher (penalty) cost. We derive explicit formulas for the distributions underlying the cost functionals of this model. It is shown in numerical examples that these results can be used to determine the optimal group size.

On polynomial variance functions

SummaryLet ℱ be a natural exponential family onℝ and (V, Ω) be its variance function. Here, Ω is the mean domain of ℱ andV, defined on Ω, is the variance of ℱ. A problem of increasing interest in the literature is the following: Given an open interval Ω⊂ℝ and a functionV defined on Ω, is the pair (V, Ω) a variance function of some natural exponential family? Here, we consider the case whereV is a polynomial. We develop a complex-analytic approach to this problem and provide necessary conditions for (V, Ω) to be such a variance function. These conditions are also sufficient for the class of third degree polynomials and certain subclasses of polynomials of higher degree.