## The Annals of Statistics

### Semiparametric estimation of a two-component mixture model

#### Abstract

Suppose that univariate data are drawn from a mixture of two distributions that are equal up to a shift parameter. Such a model is known to be nonidentifiable from a nonparametric viewpoint. However, if we assume that the unknown mixed distribution is symmetric, we obtain the identifiability of this model, which is then defined by four unknown parameters: the mixing proportion, two location parameters and the cumulative distribution function of the symmetric mixed distribution. We propose estimators for these four parameters when no training data is available. Our estimators are shown to be strongly consistent under mild regularity assumptions and their convergence rates are studied. Their finite-sample properties are illustrated by a Monte Carlo study and our method is applied to real data.

#### Article information

Source
Ann. Statist., Volume 34, Number 3 (2006), 1204-1232.

Dates
First available in Project Euclid: 10 July 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1152540747

Digital Object Identifier
doi:10.1214/009053606000000353

Mathematical Reviews number (MathSciNet)
MR2278356

Zentralblatt MATH identifier
1112.62029

Subjects
Primary: 62G05: Estimation 62G20: Asymptotic properties
Secondary: 62E10: Characterization and structure theory

#### Citation

Bordes, Laurent; Mottelet, Stéphane; Vandekerkhove, Pierre. Semiparametric estimation of a two-component mixture model. Ann. Statist. 34 (2006), no. 3, 1204--1232. doi:10.1214/009053606000000353. https://projecteuclid.org/euclid.aos/1152540747

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