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.


62F03 Parametric hypothesis testing
62G10 Nonparametric hypothesis testing
62G20 Asymptotic properties of nonparametric inference
Full Text: DOI