Estimation of the number of components of finite mixtures of multivariate distributions

Abstract

An estimator of the number of components of a finite mixture of k-dimensional distributions is given on the basis of a one-dimensional independent random sample obtained by a transformation of a k-dimensional independent random sample. Some properties of the estimator are given. Some simulation results also are given for the case of finite mixtures of two-dimensional normal distributions.Key words and phrases: k-dimensional finite mixture, normal pdf, number of components, one-dimensional finite mixture.

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

“…. ), are needed to be orthogonal in Henna (2005). However, these conditions are not necessary in this paper, so that the construction of the sequence becomes easy.…”

“…However, these conditions are not necessary in this paper, so that the construction of the sequence becomes easy. For m n (s), ks and s vectors are needed in Henna (2005) and in this paper, respectively. So, it can be expected that the number of vectors to get the value of m n in this paper is smaller than that in Henna (2005).…”

“…For m n (s), ks and s vectors are needed in Henna (2005) and in this paper, respectively. So, it can be expected that the number of vectors to get the value of m n in this paper is smaller than that in Henna (2005). Furthermore, there is the case where a computation to obtain S n ( c ,n | a γ , ρ γ ) becomes easy with an adjustment of length a γ allowed by Lemma 4.1.…”

“…An estimator which can be considered an improvement of that of Henna (2005) is given. To attain our objectives, we find a column vector a and a scalar ρ which satisfy the following condition, that is, putting Y j = a X j + ρ (j = 1, 2, .…”

The likelihood ratio test for general mixture models with or without structural parameter

“…The main problem is to calibrate the penalization factor. In case the possible number of populations is a priori bounded, one obtains easily a consistent estimator as soon as it is known that the likelihood statistic remains stochastically bounded, see [19,26] Section 2, see also [5,24,25] without prior bounds.…”