## The Annals of Mathematical Statistics

### Asymptotically Optimal Tests for Multinomial Distributions

Wassily Hoeffding

#### 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

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