A Note on Monte Carlo Maximization by the Density Ratio Model

Abstract

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.

Key Words:

• Biased sampling
• empirical likelihood
• density ratio model
• likelihood ratio
• normalizing constant