The Annals of Mathematical Statistics

Asymptotically Optimal Tests for Multinomial Distributions

Wassily Hoeffding

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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.

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

First available in Project Euclid: 27 April 2007

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

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