## A general classification rule for probability measures.(English)Zbl 0841.62011

Summary: We consider the composite hypothesis testing problem of classifying an unknown probability distribution based on a sequence of random samples drawn according to this distribution. Specifically, if $$A$$ is a subset of the space of all probability measures $${\mathcal M}_1 (\Sigma)$$ over some compact Polish space $$\Sigma$$, we want to decide whether or not the unknown distribution belongs to $$A$$ or its complement. We propose an algorithm which leads a.s. to a correct decision for any $$A$$ satisfying certain structural assumptions. A refined decision procedure is also presented which, given a countable collection $$A_i \subset {\mathcal M}_1 (\Sigma)$$, $$i=1, 2, \dots$$, each satisfying the structural assumption, will eventually determine a.s. the membership of the distribution in any finite number of the $$A_i$$. Applications to density estimation are discussed.

### MSC:

 62F03 Parametric hypothesis testing 62G10 Nonparametric hypothesis testing 62G20 Asymptotic properties of nonparametric inference
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