Variance Function Estimation in Multivariate Nonparametric Regression

Variance function estimation in multivariate nonparametric regression is considered and the minimax rate of convergence is established. Our work uses the approach that generalizes the one used in Munk et al (2005) for the constant variance case. As is the case when the number of dimensions d = 1, and very much contrary to the common practice, 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 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.