The estimation of the order of a mixture model

Abstract

We propose a new method to estimate the number of different populations when a large sample of a mixture of these populations is observed. It is possible to define the number of different populations as the number of points in the support of the mixing distribution. For discrete distributions having a finite support, the number of support points can be characterized by Hankel matrices of the first algebraic moments, or Toeplitz matrices of the trigonometric moments. Namely, for one-dimensional distributions, the cardinality of the support may be proved to be the least integer such that the Hankel matrix (or the Toeplitz matrix) degenerates. Our estimator is based on this property. We first prove the convergence of the estimator, and then its exponential convergence under wide assumptions. The number of populations is not a priori bounded. Our method applies to a large number of models such as translation mixtures with known or unknown variance, scale mixtures, exponential families and various multivariate models. The method has an obvious computational advantage since it avoids any computation of estimates of the mixing parameters. Finally we give some numerical examples to illustrate the effectiveness of the method in the most popular cases.