Optimal variance estimation without estimating the mean function

We study the least squares estimator in the residual variance estimation context. We show that the mean squared differences of paired observations are asymptot-ically normally distributed. We further establish that, by regressing the mean squared differences of these paired observations on the squared distances between paired covariates via a simple least squares procedure, the resulting variance esti-mator is not only asymptotically normal and root-n consistent, but also reaches the optimal bound in terms of estimation variance. We also demonstrate the advantage of the least squares estimator in comparison with existing methods in terms of the second order asymptotic properties