On Moderate and Large Deviations in Multinomial Distributions

Kallenberg, Wilbert C. M.

Ann. Statist., Tome 13 (1985) no. 1, p. 1554-1580 / Harvested from Project Euclid

Résumé

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.

Détail

Publié le : 1985-12-14
Classification:  Moderate and large deviations in multinomial distributions,  likelihood ratio statistic for the multinomial distribution,  multinomial distribution,  Pearson's chi squared statistic,  contingency table,  60F10,  62E20,  62E15,  62F20