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The International Journal of Biostatistics

Ed. by Chambaz, Antoine / Hubbard, Alan E. / van der Laan, Mark J.

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Maximum Likelihood Estimation for Semiparametric Density Ratio Model

Guoqing Diao / Jing Ning / jing qin
Published Online: 2012-06-27 | DOI: https://doi.org/10.1515/1557-4679.1372


In the statistical literature, the conditional density model specification is commonly used to study regression effects. One attractive model is the semiparametric density ratio model, under which the conditional density function is the product of an unknown baseline density function and a known parametric function containing the covariate information. This model has a natural connection with generalized linear models and is closely related to biased sampling problems. Despite the attractive features and importance of this model, most existing methods are too restrictive since they are based on multi-sample data or conditional likelihood functions. The conditional likelihood approach can eliminate the unknown baseline density but cannot estimate it. We propose efficient estimation procedures based on the nonparametric likelihood. The nonparametric likelihood approach allows for general forms of covariates and estimates the regression parameters and the baseline density simultaneously. Therefore, the nonparametric likelihood approach is more versatile than the conditional likelihood approach especially when estimation of the conditional mean or other quantities of the outcome is of interest. We show that the nonparametric maximum likelihood estimators are consistent, asymptotically normal, and asymptotically efficient. Simulation studies demonstrate that the proposed methods perform well in practical settings. A real example is used for illustration.

Keywords: semiparametric density ratio model; conditional likelihood; generalized linear model; nonparametric maximum likelihood estimation; semiparametric efficiency

About the article

Published Online: 2012-06-27

Citation Information: The International Journal of Biostatistics, Volume 8, Issue 1, ISSN (Online) 1557-4679, DOI: https://doi.org/10.1515/1557-4679.1372.

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©2012 Walter de Gruyter GmbH & Co. KG, Berlin/Boston. Copyright Clearance Center

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