Likelihood-Based Local Linear Estimation of the Conditional Variance Function

We consider estimation of mean and variance functions with kernel-weighted local polynomial fitting in a heteroscedastic nonparametric regression model. Our preferred estimators are based on a localized normal likelihood, using a standard local linear form for estimating the mean and a local log-linear form for estimating the variance. It is important to allow two bandwidths in this problem, separate ones for mean and variance estimation. We provide data-based methods for choosing the bandwidths. We also consider asymptotic results, and study and use them. The methodology is compared with a popular competitor and is seen to perform better for small and moderate sample sizes in simulations. A brief example is provided.