Efficient error variance estimation in non‐parametric regression

Abstract

Summary Error variance estimation plays a key role in the analysis of homogeneous non‐parametric regression models. For a random design model, most methods in the literature for error variance estimation assume the independence between the predictor variable X and the error ε. In this work, we derive the optimal semi‐parametric efficiency bound for the error variance σ2=varfalse(ϵfalse) without such an independence assumption. A residual‐based efficient estimator for σ2 is proposed and its asymptotic normality is established. An extensive simulation study is conducted, which shows that our proposed estimator works very favourably against competitors. A simple real‐data example is also presented.