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Consider a Gaussian nonparametric regression problem having both an unknownmean function and unknown variance function. This article presents a class ofdifference-based kernel estimators for the variance function. Optimalconvergence rates that are uniform over broad functional classes and bandwidthsare fully characterized, and asymptotic normality is also established. We alsoshow that for suitable asymptotic formulations our estimators achieve theminimax rate.

Variance estimation in nonparametric regression via the difference sequence method