Concave Extended Linear Modeling: a Theoretical Synthesis

Extended linear modeling provides a flexible framework for functional estimation problems with multiple covariates. Such problems include ordinary and generalized regression, density and conditional density estimation, hazard regression, spectral density estimation and polychotomous regression. In this paper, we develop a general theory on the rate of convergence of maximum likelihood estimation in extended linear modeling. The role of concavity of the log-likelihood function is highlighted. Both correctly specified and misspecified models are treated in a unified manner. Applications are made to a variety of structural models: saturated models, partly linear models, and functional ANOVA models. Two specific contexts, counting process regression and conditional density estimation, are used to illustrate the general theory.