Semiparametric Generalized Linear Models

Abstract

We propose a new class of generalized linear models. As with the existing models, these new models are specified via a linear predictor and a link function for the mean of response Y as a function of predictors X. However, here, the reference distribution of Y when the linear predictor is zero, is left unspecified and estimated from the data.? The response distribution when the linear predictor differs from zero is then generated via exponential tilting of the baseline distribution, yielding a response model that is a member of the natural exponential family, with corresponding canonical link and variance functions. The resulting model has a similar level of flexibility as the proportional odds model. Properties are discussed. Maximum likelihood estimators are developed for response distribution with finite support. Recent results complete the theory for two key cases of infinite support. The new model is studied and illustrated through simulations and example analyses. ? KW: Baseline distribution; Canonical link; Density-ratio model; Exponential tilting; Linear exponential family; Natural exponential family; Quasi-likelihood.