Relationship between a Central Limit Theorem and Hotelling's \(T^2\)-Statistic in the context of the Stochastic EM Algorithm used in Mixture Analysis

Abstract

In earlier research the asymptotic distribution of a test statistic, that uses the algebraic representation of Hotelling’s \(T^2\) and is pertaining to a process generated from the Stochastic EM (SEM) algorithm, was established in order to assess the performance of the EM algorithm in the estimation of the number of components in finite mixtures; theory concerning the distribution of \(T^2\) was based on a regularity assumption stating that the vector random process generated from SEM is normally distributed. In the present paper a central limit theorem and some theory concerning second order moments are used in order to investigate corresponding results obtained in case the process is generated from the stationary state of SEM without making any assumption of normality. A comparison between our findings and usual asymptotic theory for independently distributed vector random variables is also provided.