## The Annals of Mathematical Statistics

- Ann. Math. Statist.
- Volume 36, Number 2 (1965), 369-401.

### Asymptotically Optimal Tests for Multinomial Distributions

#### Abstract

Tests of simple and composite hypothesis for multinomial distributions are considered. It is assumed that the size $\alpha_N$ of a test tends to 0 as the sample size $N$ increases. The main concern of this paper is to substantiate the following proposition: If a given test of size $\alpha_N$ is "sufficiently different" from a likelihood ratio test then there is a likelihood ratio test of size $\leqq\alpha_N$ which is considerably more powerful than the given test at "most" points in the set of alternatives when $N$ is large enough, provided that $\alpha_N \rightarrow 0$ at a suitable rate. In particular, it is shown that chi-square tests of simple and of some composite hypotheses are inferior, in the sense described, to the corresponding likelihood ratio tests. Certain Bayes tests are shown to share the above-mentioned property of a likelihood ratio test.

#### Article information

**Source**

Ann. Math. Statist., Volume 36, Number 2 (1965), 369-401.

**Dates**

First available in Project Euclid: 27 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aoms/1177700150

**Digital Object Identifier**

doi:10.1214/aoms/1177700150

**Mathematical Reviews number (MathSciNet)**

MR173322

**Zentralblatt MATH identifier**

0135.19706

**JSTOR**

links.jstor.org

#### Citation

Hoeffding, Wassily. Asymptotically Optimal Tests for Multinomial Distributions. Ann. Math. Statist. 36 (1965), no. 2, 369--401. doi:10.1214/aoms/1177700150. https://projecteuclid.org/euclid.aoms/1177700150