Case–Control Studies and Monte Carlo Methods

It is well known that the presence of untractable normalizing constants in a probability density function complicates the calculation of maximum likelihood estimators. Usually numerical or Monte Carlo methods are employed in order to obtain an approximation. We propose a new statistical method for carrying out the calculations regarding maximum likelihood estimation by avoiding calculation of the normalizing constant. We formulate the problem as a semiparametric maximum likelihood estimation problem for a two samples model, where the ratio of two densities is known up to some parameters, but the forms of two densities themselves 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. Specifically, the semiparametric likelihood ratio statistic converges to a chi square random variable. Comparisons between the proposed method and the conventional estimated pseudo-likelihood method are studied. Monte Carlo simulations are implemented.