The Annals of Statistics

On Moderate and Large Deviations in Multinomial Distributions

Wilbert C. M. Kallenberg

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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.

Article information

Ann. Statist., Volume 13, Number 4 (1985), 1554-1580.

First available in Project Euclid: 12 April 2007

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 60F10: Large deviations
Secondary: 62E20: Asymptotic distribution theory 62E15: Exact distribution theory 62F20

Moderate and large deviations in multinomial distributions likelihood ratio statistic for the multinomial distribution multinomial distribution Pearson's chi squared statistic contingency table


Kallenberg, Wilbert C. M. On Moderate and Large Deviations in Multinomial Distributions. Ann. Statist. 13 (1985), no. 4, 1554--1580. doi:10.1214/aos/1176349755.

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