Semiparametric regression with localized Bregman divergence

Abstract This paper focuses on semiparametric regression based on minimizing the localized Bregman divergence. A local parametric model derived from the framework of the generalized linear model with multiple covariates and a linear predictor is utilized. The parameter vector included in the model is estimated under localization. The asymptotic behavior of both the locally estimated parameter vector and the induced regression estimator is investigated. Theoretical comparisons of estimators by using the divergence risk measure are also addressed. Further generalization, including a multivariate polynomial predictor, is explored, where Faa di Bruno's theorem concerning the derivative of a composition of multivariate functions is efficiently utilized. Simulations and application to a real dataset demonstrate that the proposed regression estimator works efficiently.