Asymptotic results for a stationary random process generated by the stochastic EM algorithm in mixture analyses

Abstract

Implementing the stochastic EM (or SEM) algorithm for parameter estimation in finite mixture models yields a random process which is normally distributed AR(1) as the sample size tends to infinity. The normal distribution underlying this process follows from a regularity assumption pertaining to theory concerning SEM. In the present paper we consider a centred version {Xi} of this process and focus on the particular case where this is generated from the stationary state of SEM, denoted as process {X(i)}. We study distribution properties of as n tends to infinity without assuming a normal distribution for {X(i)}. To this end a multivariate central limit theorem is shown to apply and relationship between the variance of as n tends to infinity and that of for r sufficiently large is established.