## The Annals of Statistics

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

### 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.

#### Article information

**Source**

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

**Dates**

First available in Project Euclid: 12 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176349755

**Digital Object Identifier**

doi:10.1214/aos/1176349755

**Mathematical Reviews number (MathSciNet)**

MR811510

**Zentralblatt MATH identifier**

0581.60023

**JSTOR**

links.jstor.org

**Subjects**

Primary: 60F10: Large deviations

Secondary: 62E20: Asymptotic distribution theory 62E15: Exact distribution theory 62F20

**Keywords**

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

#### Citation

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. https://projecteuclid.org/euclid.aos/1176349755