## The Annals of Statistics

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

### A General Classification Rule for Probability Measures

Sanjeev R. Kulkarni and Ofer Zeitouni

#### Abstract

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

**Source**

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

**Dates**

First available in Project Euclid: 11 April 2007

**Permanent link to this document**

https://projecteuclid.org/euclid.aos/1176324714

**Digital Object Identifier**

doi:10.1214/aos/1176324714

**Mathematical Reviews number (MathSciNet)**

MR1353511

**Zentralblatt MATH identifier**

0841.62011

**JSTOR**

links.jstor.org

**Subjects**

Primary: 62F03: Hypothesis testing

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

**Keywords**

Hypothesis testing empirical measure large deviations

#### Citation

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. https://projecteuclid.org/euclid.aos/1176324714