Robust Nonparametric Regression with Simultaneous Scale Curve Estimation

Abstract

Let $\{X_i, Y_i\}^n_{i=1} \subset \mathbb{R}^d \times \mathbb{R}$ be independent identically distributed random variables. If the conditional distribution $F(y \mid x)$ can be parametrized by $F(y \mid x) = F_0((y - m(x))/\sigma(x))$ with a fixed and known distribution $F_0$, the regression curve $m(x)$ and scale curve $\sigma(x)$ could be estimated by some parametric method. More generally, we assume that $F$ is unknown and consider nonparametric simultaneous $M$-type estimates of the unknown functions $m(x)$ and $\sigma(x)$, using kernel estimators for the conditional distribution function $F(y \mid x)$. We show pointwise consistency and asymptotic normality of these estimates. The rate of convergence is optimal in the sense of Stone (1980). The asymptotic bias term of this robust estimate turns out to be the same as for the linear Nadaraya-Watson kernel estimate.