Gerald J. Lieberman was born on December 31, 1925, in Brooklyn, New York, after a hectic New Year's Eve trip to the hospital. His father, Joseph, spelled his last name Liberman, but his mother, Ida, preferred Lieberman, the spelling that she and some of Joseph's siblings used. Joseph and Ida had come to this country from Lithuania. Joseph worked for the Metropolitan Life Insurance Company, and they lived in an “historic” section of Flatbush. The much wanted baby boy was the center of the family, which included two doting older sisters, Shirley and Rosalind. He grew fast — one of the tallest boys in nearby Public School 197 — and achieved his adult height at about the age of 13. As a boy, he was described as towheaded and gawky. Jerry did not realize that he had a middle initial until he was 15 and needed a birth certificate to get a work permit. Jerry asked his parents if they had given him a middle initial, but they did not remember. In any case, since the J does not stand for anything, Jerry likes to quip that his middle name is Jinitial.

In the fall of 1948 in my course on Least Squares in the Department of Mathematical Statistics at Columbia University (and in the spring in Correlation and Chi-Square), I was particularly impressed by one of the students— Gerald J. Lieberman. I was disappointed that this promising student left Columbia after one year, but it was not long until our paths met again. It is a pleasure to dedicate this paper to my colleague and close friend!

We present a new exponential inequality for degenerate U-statistics. The bound on the tail probability is quadratic for small to medium values of the deviation and linear for larger values. We apply this bound to a family of test statistics and provide the key step in an optimality result for adaptive tests [1].

Let x = (x1, x2, …) be an infinite sequence of 0's and 1's, initially unknown to you. On day n = 1,2,…, you observe hn = (x1, …, xn–1), the first n – 1 terms of the sequence, and must predict xn. What is a good prediction method, and how well can you do?

A general type of model that results in components having a negative type of dependence is proposed. Specifically, a system of dependent binary components, in which component i is said to function if at least one of a given number of multinomial trials results in that outcome, is considered. The multinomial probabilities that maximize the probability of a k-of-n system are determined. Some approximations for the system reliability are proposed first for a system that functions unless k consecutive components are failed and then for a general system. A simulation method that uses a variance reduction technique to estimate the reliability is presented.

A discounted-cost, continuous-time, infinite-horizon version of a flexible manufacturing and operator scheduling model is solved. The solution procedure is to convexify the discrete operator-assignment constraints to obtain a linear program and then to regain the discreteness and obtain an approximate manufacturing schedule by deconvexification of the solution of the linear program over time. The strong features of the model are the accommodation of linear inequality relations among the manufacturing activities and the discrete manufacturing scheduling, whereas the weak features are intra-period relaxation of inventory availability constraints and the absence of inventory costs, setup times, and setup charges.

Finite-state-and-action Markov branching decision chains are studied with bounded endogenous expected population sizes and interest-rate-dependent one-period rewards that are analytic in the interest rate at zero. The existence of a stationary strong-maximum-present-value policy is established. Miller and Veinott's [1969] strong policy-improvement method is generalized to find in finite time a stationary n-present-value optimal policy and, when the one-period rewards are rational in the interest rate, a stationary strong-maximum-present-value policy. This extends previous studies of Blackwell [1962], Miller and Veinott [1969], Veinott [1974], and Rothblum [1974, 1975], in which the one-period rewards are independent of the interest rate, and Denardo [1971] in which semi-Markov decision chains with small interest rates are studied. The problem of finding a stationary n-present-value optimal policy is also formulated as a staircase linear program in which the objective function and right-hand sides, but not the constraint matrix, depend on the interest rate, and solutions for all small enough positive interest rates are sought. The optimal solutions of the primal and dual are polynomials in the reciprocal of the interest rate. A constructive rule is given for finding a stationary n-present-value optimal policy from an optimal solution of the asymptotic linear program. This generalizes the linear programming approaches for finding maximum-reward-rate and maximum-present-value policies for Markov decision chains studied by Manne [1960], d'Epenoux [1960, 1963], Balinski [1961], Derman [1962], Denardo and Fox [1968], Denardo [1970], Derman and Veinott [1972], Veinott [1973], and Hordijk and Kallenberg [1979, 1984].

The standard regenerative method for estimating steady-state parameters is extended to permit cycles that begin and end in different states. This result is established using the Dynkin martingale and a related solution to Poisson's equation. We compare the variance constant that appears in the associated central limit theorem with that arising from cycles that begin and end in the same state. The standard regenerative method has a smaller variance constant than does the alternative.

It has long been observed that independent laboratories differ in reporting the results of repeated experiments. The problem is to detect those laboratories that might be considered aberrant. Previous analyses have been based on an analysis of variance framework or on subset selection for the detection of aberrant laboratories. The present procedure is also based on an ANOVA model but uses a Bayesian estimate of the number of aberrant laboratories. Subsequent to the determination of the aberrant laboratories, a linear model is used to separate sampling and laboratory effects.

In a job-shop manufacturing environment, consider a model in which the “job” is processed at several stages with random yields. Nondefective items move through the stages to supply a possibly random demand. At each stage there are linear costs of processing and work-in-process inventories and there are convex costs of end-item inventory. Decisions specify the lot sizes that are processed at the various stages. Easy computations specify a simply structured decision rule that minimizes the expected total cost.