Additive regression for predictors of various natures and possibly incomplete Hilbertian responses

Abstract

In this paper we consider a fully nonparametric additive regression model for responses and predictors of various natures. This includes the case of Hilbertian and incomplete (like censored or missing) responses, and continuous, nominal discrete and ordinal discrete predictors. We propose a backfitting technique that estimates this additive model, and establish the existence of the estimator and the convergence of the associated backfitting algorithm under minimal conditions. We also develop a general asymptotic theory for the estimator such as the rates of convergence and asymptotic distribution. We verify the practical performance of the proposed estimator in a simulation study. We also apply the method to various real data sets, including those for a density-valued response regressed on a mixture of continuous and nominal discrete predictors, for a compositional response regressed on a mixture of continuous and ordinal discrete predictors, and for a censored scalar response regressed on a mixture of continuous and nominal discrete predictors.