On difference-based variance estimation in nonparametric regression when the covariate is high dimensional

We consider the problem of estimating the noise variance in homoscedastic nonparametric regression models. For low dimensional covariates "t"  is an element of  <openface>R</openface>-super-"d", "d"&equals;1, 2, difference-based estimators have been investigated in a series of papers. For a given length of such an estimator, difference schemes which minimize the asymptotic mean-squared error can be computed for "d"&equals;1 and "d"&equals;2. However, from numerical studies it is known that for finite sample sizes the performance of these estimators may be deficient owing to a large finite sample bias. We provide theoretical support for these findings. In particular, we show that with increasing dimension "d" this becomes more drastic. If "d"&ges;4, these estimators even fail to be consistent. A different class of estimators is discussed which allow better control of the bias and remain consistent when "d"&ges;4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary. Copyright 2005 Royal Statistical Society.