The Annals of Statistics

A General Classification Rule for Probability Measures

Sanjeev R. Kulkarni and Ofer Zeitouni

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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 $\mathscr{M}_1(\Sigma)$ 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 $A_i \subset \mathscr{M}_1(\Sigma), i = 1,2,\ldots$, 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.

Article information

Ann. Statist., Volume 23, Number 4 (1995), 1393-1407.

First available in Project Euclid: 11 April 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Primary: 62F03: Hypothesis testing
Secondary: 62G10: Hypothesis testing 62G20: Asymptotic properties

Hypothesis testing empirical measure large deviations


Kulkarni, Sanjeev R.; Zeitouni, Ofer. A General Classification Rule for Probability Measures. Ann. Statist. 23 (1995), no. 4, 1393--1407. doi:10.1214/aos/1176324714.

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