On estimating of the number of constituents of a finite mixture of continuous distributions

Abstract

SummarySuppose that H is a mixture of distributions for a given family ~. A necessary and sufficient condition is obtained under which H is, in fact, a finite mixture. An estimator of the number of distributions constituting the mixture is proposed assuming that the mixture is finite and its asymptotic properties are investigated.

An Estimator of the Number of Components of a Finite Mixture of Multivariate Distributions

“…Then, for each n, the probability P{ m * n = m * } using the method given in this paper is larger than that given by the method in Henna (1985) with a common criterion.…”

“…The importance to estimate the number m is described in McLachlan and Basford (1988), Titterington (1990) and others. Henna (1985), Feng and McCulloch (1994), Chen and Kalbfleisch (1996) and Richardson and Green (1997) have treated one-dimensional finite mixtures. Roeder (1994) has investigated a graphical technique to determine the number of components in a case of normal mixture, and Keribin (2000) has given a method which can be applied to a special type of multivariate normal mixture under the assumption that a superior value Q of m is known.…”

Segmenting magnetic resonance images via hierarchical mixture modelling

“…An enormous body of literature exists regarding the application, computational issues, and theoretical aspects of mixture models when the number of components is known (see, e.g., McLachlan and Peel, 2000;Everitt and Hand, 1981;McLachlan and Basford, 1988;Titterington et al, 1990), but estimating the unknown number of components remains an area of intense research effort. See, for instance, Chen and Kalbfleisch (1996); Gassiat (1997, 1999); Escobar and West (1995);Henna (1985); James et al (2001); Keribin (2000); McLachlan (1987); Priebe and Marchette (2000); Roeder and Wasserman (1997) for recent progress in this area.…”

Consistent estimation of mixture complexity

“…Results are given in Table 1. For comparison, we provide results obtained via the bootstrapping procedure of McLachlan (1987) (Bootstrap), the CDF method of Henna (1985) (Henna), and the Bayesian methodology proposed by Roeder and Wasserman (1997) …”

“…However, the literature regarding consistent estimation of the mixture complexity m is sparse but burgeoning: see Henna (1985); Chen and Kalbfleisch (1996); Gassiat (1997, 1999); Keribin (2000); and Priebe and Marchette (2000). Efforts to address the related problem of testing hypotheses about m have met with mixed results; the limiting distribution for the likelihood ratio test statistic was until recently unavailable [Dacunha-Castelle and Gassiat (1999)], and so bootstrap testing methodologies have been developed [see, e.g., McLachlan (1987)].…”