Modeling Heteroscedasticity In The Single-Index Model With The Dirichlet Process
George Karabatsos · 2008
The single-index model is a nonparametric regression approach that has seen many ap-plications. The model avoids the curse of dimensionality by reducing the p-dimensional predictor to a univariate single-index (a linear combination of p regression coefficients and covariates), and provides a flexible alternative to ordinary linear regression. In the model, each observed continuous response has mean equal to an unknown (link) func-tion of the single-index, and the errors in regression have common variance. In this paper, a novel Bayesian heteroscedastic single-index model is introduced, where the link function is modeled by splines, and the distribution of the error variances (over observations) is modeled nonparametrically by a Dirichlet Process prior. The spline coefficients are regularized with a ridge prior with parameter assigned a hyperprior. Methods of Gibbs sampling and adaptive Metropolis-Hastings sampling are presented for posterior inference and goodness-of-fit analysis of the heteroscedastic model. The model is illustrated through the analysis of real and simulated data. Compared to the homoscedastic single-index model, the heteroscedastic model demonstrated superior predictive accuracy in each of the three real data sets.