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December, 1985 On Moderate and Large Deviations in Multinomial Distributions
Wilbert C. M. Kallenberg
Ann. Statist. 13(4): 1554-1580 (December, 1985). DOI: 10.1214/aos/1176349755


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.


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Wilbert C. M. Kallenberg. "On Moderate and Large Deviations in Multinomial Distributions." Ann. Statist. 13 (4) 1554 - 1580, December, 1985.


Published: December, 1985
First available in Project Euclid: 12 April 2007

zbMATH: 0581.60023
MathSciNet: MR811510
Digital Object Identifier: 10.1214/aos/1176349755

Primary: 60F10
Secondary: 62E15, 62E20, 62F20

Rights: Copyright © 1985 Institute of Mathematical Statistics


Vol.13 • No. 4 • December, 1985
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