On the Convergence of Monte Carlo Maximum Likelihood Calculations

by Charles J. Geyer
Technical Report No. 571
School of Statistics
University of Minnesota
February, 1992
Revised September 24, 1992

Research supported in part by grant DMS-9007833 from the National Science Foundation.

Abstract

Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family [ h(sub-theta) : (theta) in (THETA) ] of nonnegative integrable functions, maximum likelihood estimates in the family obtained by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the evaluation maps [ (theta) maps to h(sub-theta)(x) ] remains continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of approximate likelihood (support regions) converge to the exact sets (in Painleve-Kuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 1991) if a Wald-type integrability condition is satisfied. Asymptotic normality of the Monte Carlo error and convergence of the Monte Carlo approximation to the observed Fisher information are also shown.

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