Poisson approximation for some statistics based on exchangeable trials

Stein's (1970) method of proving limit theorems for sums of dependent random variables is used to derive Poisson approximations for a class of statistics, constructed from finitely exchangeable random variables.

Let be exchangeable random elements of a space and, for I a k-subset of , let XI be a 0–1 function. The statistics studied here are of the form where N is some collection of k -subsets of .

An estimate of the total variation distance between the distributions of W and an appropriate Poisson random variable is derived and is used to give conditions sufficient for W to be asymptotically Poisson. Two applications of these results are presented.

Poisson approximations for the distribution and moments of ordered m-spacings

This article investigates the accuracy of approximations for the distribution of ordered m-spacings for i.i.d. uniform observations in the interval (0, 1). Several Poisson approximations and a compound Poisson approximation are studied. The result of a simulation study is included to assess the accuracy of these approximations. A numerical procedure for evaluating the moments of the ordered m-spacings is developed and evaluated for the most accurate approximation.

Approximations for the tail probabilities and moments of the scan statistic

The scan statistic is used to test the hypothesis that the observed events occur at random (uniformly distributed) in time or space versus the hypothesis that they cluster within a moving window of size w. To implement the testing procedure based on the scan statistic its tail probabilities have to be effectively evaluated. In this article a survey of results on the approximations of the distribution of the scan statistic and its moments is presented. Numerical results comparing these approximations are also given. Numerous references with applications in epidemiological studies using the scan statistics are mentioned. Related scan statistics that have been used in many other interesting applications are listed in the reference section. The article concludes with the presentation of unsolved problems related to the scan statistic.

An approximation for the distribution of the scan statistic

The scan statistic evaluates whether an apparent cluster of disease in time is due to chance. The statistic employs a 'moving window' of length w and finds the maximum number of cases revealed through the window as it scans or slides over the entire time period T. Computation of the probability of observing a certain size cluster, under the hypothesis of a uniform distribution, is infeasible when N, the total number of events, is large, and w is of moderate or small size relative to T. We give an approximation that is an asymptotic upper bound, easy to compute, and, for the purposes of hypothesis testing, more accurate than other approximations presented in the literature. The approximation applies both when N is fixed, and when N has a Poisson distribution. We illustrate the procedure on a data set of trisomic spontaneous abortions observed in a two year period in New York City.