Abstract: Density-ratio models are receiving increasing attention, particularly because of their relationship with generalized linear models and their applications in missing-data analyses. The density-ratio assumption, however, may not be true in some applications, and an important limitation is that the standard density-ratio model does not accommodate heterogeneity within the underlying population. To address these issues, we propose a new density-ratio model that incorporates a stratification procedure and dispersion parameters. The resulting stratified density-ratio model 1) retains attractive properties of the standard density-ratio model, while allowing the density-ratio assumption to be violated for some covariate, and 2) provides a validation tool, using a Kolmogorov-Smirnov-type statistic, to check the modeling assumption. We estimate the finite-dimensional and infinite-dimensional parameters simultaneously using an efficient nonparametric maximum likelihood approach. The resulting estimators are shown to be consistent and asymptotically normal. The asymptotic covariance matrix of the estimators for the finite-dimensional parameters attains the semiparametric efficiency bound.

Key words and phrases: Bootstrap test, density ratio models, generalized linear models, Kolmogorov-Smirnov test, nonparametric maximum likelihood estimation, semiparametric efficiency.