Inequalities for linear combinations of gamma random variables

We study the behavior of the tail probabilities of weighted averages of certain independently and identically distributed random variables as the weights are varied. We show that the upper and lower tails are smallest when all the weights are equal. Our results apply to exponential, chi-squared, gamma, and Weibull random variables.     

Wild card

In 1992 Persi Diaconis of Stanford University told the world just how many times a pack of cards should be shuffled. He is one of the more colourful of professors of statistics. He tells Julian Champkin about crooked dice, tossing coins and conjuring tricks—and how statistics makes sense of his world.     

Gray codes for randomization procedures

We introduce a simple combinatorial scheme for systematically running through a complete enumeration of sample reuse procedures such as the bootstrap, Hartigan's subsets, and various permutation tests. The scheme is based on Gray codes which give tours through various spaces, changing only one or two points at a time. We use updating algorithms to avoid recomputing statistics and achieve substantial speedups. Several practical examples and computer codes are given.     

EXCHANGEABLE PAIRS OF BERNOULLI RANDOM VARIABLES,KRAWTCHOUCK POLYNOMIALS,AND EHRENFEST URNS

This paper derives characterizations of bivariate binomial distributions of the Lancaster form with Krawtchouk polynomial eigenfunctions. These have been characterized by Eagleson, and we give two further characterizations with a more probabilistic flavour: the first as sums of correlated Bernoulli variables; and the second as the joint distribution of the number of balls of one colour at consecutive time points in a generalized Ehrenfest urn. We give a self‐contained development of Krawtchouck polynomials and Eagleson’s theorem.     

Approximate exchangeability and de Finetti priors in 2022

This is a review paper, beginning with de Finetti's work on partial exchangeability, continuing with his approach to approximate exchangeability, and then his (surprising) approach to assigning informative priors in nonstandard situations. Recent progress on Markov chain Monte Carlo methods for drawing conclusions is supplemented by a review of work by Gerencsér and Ottolini on getting honest bounds for rates of convergence. The paper concludes with a speculative approach to combining classical asymptotics with Monte Carlo. This promises real speed-ups and makes a nice example of how theory and computation can interact.