Comparing two samples by penalized logistic regression

TL;DR

This paper introduces a penalized density ratio model for comparing multiple samples, demonstrating improved estimator accuracy and reliable testing of distribution equality, especially in small samples, supported by theoretical analysis and simulations.

Contribution

It extends the density ratio model with penalization, reducing mean square error and resolving estimator existence issues, enhancing distribution comparison methods.

Findings

Penalized estimators have lower mean square error than maximum likelihood estimators.

Penalization resolves estimator existence problems.

Simulation studies support theoretical advantages.

Abstract

Inference based on the penalized density ratio model is proposed and studied. The model under consideration is specified by assuming that the log--likelihood function of two unknown densities is of some parametric form. The model has been extended to cover multiple samples problems while its theoretical properties have been investigated using large sample theory. A main application of the density ratio model is testing whether two, or more, distributions are equal. We extend these results by arguing that the penalized maximum empirical likelihood estimator has less mean square error than that of the ordinary maximum likelihood estimator, especially for small samples. In fact, penalization resolves any existence problems of estimators and a modified Wald type test statistic can be employed for testing equality of the two distributions. A limited simulation study supports further the theory.