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

Abstract

This article proposes a Bayesian method for estimating a heteroscedastic regression model with Gaussian errors, where the mean and the log variance are modeled as linear combinations of explanatory variables. We use Bayesian variable selection priors and model averaging to make the estimation more efficient. The model is made semiparametric by allowing explanatory variables to enter the mean and log variance 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. Our article develops an efficient Markov chain Monte Carlo simulation method to sample from the posterior distribution and applies the methodology to a number of simulated and real examples.