Variance estimation in nonparametric regression with jump discontinuities

Variance estimation is an important topic in nonparametric regression. In this paper, we propose a pairwise regression method for estimating the residual variance. Specifically, we regress the squared difference between observations on the squared distance between design points, and then estimate the residual variance as the intercept. Unlike most existing difference-based estimators that require a smooth regression function, our method applies to regression models with jump discontinuities. Our method also applies to the situations where the design points are unequally spaced. Finally, we conduct extensive simulation studies to evaluate the finite-sample performance of the proposed method and compare it with some existing competitors.