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 R-d, d=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=1 and d=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 dgreater than or equal to4, 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 dgreater than or equal to4. These estimators are compared numerically with kernel-type estimators (which are asymptotically efficient), and some guidance is given about when their use becomes necessary.