A nearest neighbor estimate of the residual variance

Abstract

We study the problem of estimating the smallest achievable mean-squared error in regression function estimation. The problem is equivalent to estimating the second moment of the regression function of $Y$ on $X\in{\mathbb{R}} ^{d}$. We introduce a nearest-neighbor-based estimate and obtain a normal limit law for the estimate when $X$ has an absolutely continuous distribution, without any condition on the density. We also compute the asymptotic variance explicitly and derive a non-asymptotic bound on the variance that does not depend on the dimension $d$. The asymptotic variance does not depend on the smoothness of the density of $X$ or of the regression function. A non-asymptotic exponential concentration inequality is also proved. We illustrate the use of the new estimate through testing whether a component of the vector $X$ carries information for predicting $Y$.