On Moderate and Large Deviations in Multinomial Distributions

Abstract

In this paper moderate and large deviation theorems are presented for the likelihood ratio statistic and Pearson's chi squared statistic in multinomial distributions. Let $k$ be the number of parameters and $n$ the number of observations. Moderate and large deviation theorems are available in the literature only if $k$ is kept fixed when $n \rightarrow \infty$. Although here attention is focussed on $k = k(n) \rightarrow \infty$ as $n \rightarrow \infty$, explicit inequalities are obtained for both $k$ and $n$ fixed. These inequalities imply results for the whole scope of moderate and large deviations both for fixed $k$ and for $k(n) \rightarrow \infty$ as $n \rightarrow \infty$. It turns out that the $\chi^2$ approximation continues to hold in some sense, even if $k \rightarrow \infty$. The results are applied in studying the influence of the choice of the number of classes on the power in goodness-of-fit tests, including a comparison of Pearson's chi squared test and the likelihood ratio test. Also the question of combining cells in a contingency table is discussed.