Variance function estimation in multivariate nonparametric regression with fixed design

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established in the iid Gaussian case. Our work uses the approach that generalizes the one used in [A. Munk, Bissantz, T. Wagner, G. Freitag, On difference based variance estimation in nonparametric regression when the covariate is high dimensional, J. R. Stat. Soc. B 67 (Part 1) (2005) 19-41] for the constant variance case. As is the case when the number of dimensions d=1, and very much contrary to standard thinking, it is often not desirable to base the estimator of the variance function on the residuals from an optimal estimator of the mean. Instead it is desirable to use estimators of the mean with minimal bias. Another important conclusion is that the first order difference based estimator that achieves minimax rate of convergence in the one-dimensional case does not do the same in the high dimensional case. Instead, the optimal order of differences depends on the number of dimensions.