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 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.
SummaryIn this note, we will study a consistent estimator of a mixing distribution function (mixing d.f.). The estimator discussed in this note is that of Choi and Bulgren [4]. Since there is some doubt about the way of proving Lemma in [4] which is used for showing the consistency of the estimator in [2], [3] and [4], we will give different lemmas. We will show that their result (which is still true by using our lemmas) holds under a weaker assumption than theirs. The existence of the estimator is not discussed in [4]. So, we will give conditions under which the existence is guaranteed.
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