A Note on Monte Carlo Maximization by the Density Ratio Model
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It is well known that untractable normalizing constants of probability density functions complicate the calculation of maximum likelihood estimators. Usually numerical or Monte Carlo methods are employed in order to obtain an approximation to the solution of the likelihood equations. We propose a new statistical method for carrying out the calculations regarding maximum likelihood estimation by avoiding the explicit calculation of any normalizing constant. We formulate the problem within the framework of semiparametric maximum likelihood estimation for a two samples model, where the ratio of two densities is known up to some parameters, but the form of the two densities are unknown and one of the sample sizes can be chosen arbitrarily large. The two-sample semiparametric model-which is referred as density ratio model-arises naturally in case-control studies. Statistical inference techniques are developed for this model. Comparisons between the proposed method and the conventional estimated pseudo-likelihood method are studied.
AMS Subject ClassificationPrimary 62G05 Secondary 65C05
KeywordsBiased sampling empirical likelihood density ratio model likelihood ratio normalizing constant
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- Breslow, N.E., Day, N.E., 1980. The Analysis of Case-Control Data, Volume 1 of Statistical Methods in Cancer Research. World Helath Organization.Google Scholar
- Geyer, C., 1999. Likelihood inference for spatial point processes. In Stochastic geometry (Toulouse, 1996), Volume 80 of Monogr. Statist. Appl. Probab., pp. 79–140. Chapman & Hall/CRC, Boca Raton, FL.Google Scholar
- Geyer, C.J., 1994. Estimating normalizing constants and reweighting mixtures in Markov chain Monte Carlo. Technical Report 568, School of Statistics, University of Minnesota.Google Scholar