Efficiency of Monte Carlo EM and Simulated Maximum Likelihood in Two-Stage Hierarchical Models

Abstract

Likelihood estimation in hierarchical models is often complicated by the fact that the likelihood function involves an analytically intractable integral. Numerical approximation to this integral is an option but it is generally not recommended when the integral dimension is high. An alternative approach is based on the ideas of Monte Carlo integration, which approximates the intractable integral by an empirical average based on simulations. This article investigates the efficiency of two Monte Carlo estimation methods, the Monte Carlo EM (MCEM) algorithm and simulated maximum likelihood (SML). We derive the asymptotic Monte Carlo errors of both methods and show that, even under the optimal SML importance sampling distribution, the efficiency of SML decreases rapidly (relative to that of MCEM) as the missing information about the unknown parameter increases. We illustrate our results in a simple mixed model example and perform a simulation study which shows that, compared to MCEM, SML can be extremely inefficient in practical applications.