Non-parametric estimation of residual moments and covariance

The aim of non-parametric regression is to model the behaviour of a response vector Y in terms of an explanatory vector X , based only on a finite set of empirical observations. This is usually performed under the additive hypothesis Y = f ( X )+ R , where f ( X )= ( Y | X ) is the true regression function and R is the true residual variable. Subject to a Lipschitz condition on f , we propose new estimators for the moments (scalar response) and covariance (vector response) of the residual distribution, derive their asymptotic properties and discuss their application in practical data analysis.