Locally Adaptive Semiparametric Estimation of the Mean and Variance Functions in Regression Models

Abstract

This article considers the estimation of a regression model with Gaussian errors, where the mean and the log variance are modeled as a linear combination of explanatory variables. We consider Bayesian variable selection priors and model averaging to obtain efficient estimators when the number of explanatory variables is large. To make the model semiparametric using this framework we allow explanatory variables to enter the mean and log variance models flexibly by representing a covariate effect as a linear combination of basis functions. Our methodology for estimating flexible effects is locally adaptive in the sense that it works well when the flexible effects vary rapidly in some parts of the predictor space but only slowly in other parts. The whole model is estimated using a Markov chain simulation method that samples the posterior distribution with coefficients in the mean model integrated out analytically and highly dependent parameters generated in blocks. The methodology in the paper is applied to a number of simulated and real examples and is shown to work well.