Computational Statistics & Data Analysis 29 (1999) 429–444 Monte Carlo EM with importance reweighting and its applications in random eects models  Fernando A. Quintana a;∗ , Jun S. Liu b , Guido E. del Pino a a Departamento de Probabilidad y Estad stica, Facultad de Matem atica, Ponticia Universidad Cat olica de Chile, Casilla 306, Correo 22, Santiago, Chile b Department of Statistics, Stanford University, Stanford, CA 94305-4065, USA Received 1 September 1997; received in revised form 1 August 1998 Abstract In this paper we propose a new Monte Carlo EM algorithm to compute maximum likelihood estimates in the context of random eects models. The algorithm involves the construction of ecient sampling distributions for the Monte Carlo implementation of the E-step, together with a reweighting procedure that allows repeatedly using a same sample of random eects. In addition, we explore the use of stochastic approximations to speed up convergence once stability has been reached. Our algorithm is compared with that of McCulloch (1997). Extensions to more general problems are discussed. c  1999 Elsevier Science B.V. All rights reserved. Keywords: Importance sampling; Metropolis–Hastings algorithm; Stochastic approximations 1. Introduction 1.1. Random eects models The last decade has witnessed the modication of many standard (xed eects) statistical models through the incorporation of the so-called random eects. The mo- tivation behind these modications has been the quest for more realistic models that take into account heterogeneity in population parameters and correlations between  This work was partially supported by Project FONDECYT 1960915 and by a visiting professorship grant from DIPUC. * Corresponding author. E-mail: quintana@mat.puc.cl. 0167-9473/99/$ – see front matter c  1999 Elsevier Science B.V. All rights reserved. PII: S0167-9473(98)00075-9