The Annals of Mathematical Statistics

Domains of Optimality of Tests in Simple Random Sampling

David K. Hildebrand

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Abstract

This paper deals with the structure of sets $\Omega$ of distributions for which a particular test is the most powerful for testing a simple hypothesis $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$, that is, with the domain of optimality of a test. The context is restricted to these $\Omega$ consisting of probabilities having continuous positive densities, and to one-sample tests. The important concept is that of a family of tests, one for each significance level. This concept allows us to use the full power of the Neyman-Pearson Lemma. The main results are: (1) The domain of optimality of a test family $\Phi$ is essentially a multiplicatively-convex (convex in the logarithms) cone; hence there are distributions both "near to" and "far from" the null distribution for which $\Phi$ is optimal. (Theorems 1, 2, and 3). (2) If $\Phi$ is uniformly most powerful for testing $H:f = f_0 \operatorname{vs.} K:f \varepsilon\Omega$ with $n \geqq 2$ then the class of distributions has a monotone likelihood ratio. (Theorem 4).

Article information

Source
Ann. Math. Statist., Volume 40, Number 1 (1969), 308-312.

Dates
First available in Project Euclid: 27 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.aoms/1177697827

Digital Object Identifier
doi:10.1214/aoms/1177697827

Mathematical Reviews number (MathSciNet)
MR242308

Zentralblatt MATH identifier
0177.22804

JSTOR
links.jstor.org

Citation

Hildebrand, David K. Domains of Optimality of Tests in Simple Random Sampling. Ann. Math. Statist. 40 (1969), no. 1, 308--312. doi:10.1214/aoms/1177697827. https://projecteuclid.org/euclid.aoms/1177697827


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