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

is called a finite mixture of fθ1(x ), fθ2(x ), . . . , fθ◦m(x ) (Titterington et al. (1985)), where Am = (p1, p2, . . . , pm;θ1,θ2, . . . ,θm) with ∑m i=1 p ◦ i = 1, 0 < p ◦ i ≤ 1 and θi ∈ Θ (i = 1, 2, . . . ,m). So a single fθ(x ) in F is also considered a finite mixture for m = 1 as a special case. Each fθi (x ) is called a component of f(x | A ◦ m) and each pi a mixing ratio of fθi (x ). The purpose of this paper is to give an estimator m̂n of the number m of components on the basis of an independent random sample (X 1,X 2, . . . ,X n) from the distribution (1.1). 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. Methods to determine the number of components are described in McLachlan and Peel (2000). Chen et al. (2001) and Garel (2001) have given a test for m in a univariate case.