Volume 25 - Issue 1 - June 2000

Stirling numbers and Bell numbers: their role in combinatorics and probability

David Branson


Properties of Stirling numbers of the first and second kinds, and of Bell numbers, are derived from combinatorial definitions using combinatorial or probabilistic arguments, and a number of applications in these fields are reviewed. Extensions of Stirling numbers to the case where one or both of the arguments are negative integers are discussed, together with other developments.

A simple immigration–catastrophe process

Randall J. Swift


In this paper, the transient probabilities and moments of a simple immigration–catastrophe process are obtained. Catastrophes occur at a constant rate, and when they occur reduce the population size to zero.

Asymmetric Laplace distributions

Tomasz J. Kozubowski and Krzysztof Podgórski


We study asymmetric Laplace (AL) distributions which arise as the limits of sums of i.i.d. random variables with finite second moment, where the number of terms summed is geometrically distributed, independently of the terms themselves. Ordinary symmetric Laplace laws are a subclass of the AL distributions. We present the main properties of AL distributions, derive their representations, and obtain explicit forms for their densities and distribution functions. We derive explicit formulas and asymptotic properties for moment and maximum likelihood estimators of AL parameters. Finally, we discuss an application of AL laws in finance.

The switching problem and conditionally specified distributions

F. Thomas Bruss and Ludger Rüscgendorf


This article re-examines switching in the two-envelopes problem with a different focus. The question is presented as a problem of the existence of a joint probability model with conditionally specified distributions. We prove the non-existence of a solution for the classical two-envelopes specification in terms of conditional distributions. Then we introduce a generalized version of the problem and, within this framework, characterize those distributions which support the switching paradox. Finally we discuss conditionally specified distributions, in connection with the two-envelopes problem of Cover, and consider possible misinterpretations.

The estimation of the hazard ratio in clinical trials and in meta-analysis

W. F. Scott


We consider the analysis of the results of controlled trials of (say) a certain new drug treatment against a standard treatment. In many cases, the only information which is conveniently available is (a) the initial numbers of patients, and (b) the numbers of deaths (or relapses, adverse reactions, etc.). A statistic, rather similar to that of Peto, is shown to be a suitable estimator of the logarithm of the hazard ratio. The loss of efficiency relative to that obtained by the use of the log-rank statistic is shown to be very small when the total number of observed deaths is less than about half the total number of patients.