Asymptotics of multivariate contingency tables with fixed marginals

TL;DR

This paper analyzes the asymptotic behavior of individual cells in multivariate contingency tables with fixed marginals, identifying conditions for Poisson or Gaussian limits as marginals grow large.

Contribution

It derives the asymptotic distribution of table cells, classifies types of Poisson convergence, and generalizes results to tables of arbitrary sizes.

Findings

Cell variance order is established.

Diagnostic for Poisson vs. Gaussian limit is provided.

Growth rates of marginals determine the limit type.

Abstract

We consider the asymptotic distribution of a cell in a 2 x ... x 2 contingency table as the fixed marginal totals tend to infinity. The asymptotic order of the cell variance is derived and a useful diagnostic is given for determining whether the cell has a Poisson limit or a Gaussian limit. There are three forms of Poisson convergence. The exact form is shown to be determined by the growth rates of the two smallest marginal totals. The results are generalized to contingency tables with arbitrary sizes and are further complemented with concrete examples.