Effect of Mean on Variance Function Estimation in Nonparametric Regression

Variance function estimation in nonparametric regression is considered and the minimax rate of convergence is derived. We are particularly interested in the effect of the unknown mean on the estimation of the variance function. Our results indicate that, 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 more desirable to use estimators of the mean with minimal bias. In addition our results also correct the optimal rate claimed in the previous literature.