Random-Design Regression under Long-Range Dependent Errors

Abstract

We consider the random-design nonparametric regression model with long-range dependent errors that may also depend on the independent and identically distributed explanatory variables.Disclosing a smoothing dichotomy, we show that the ®nite-dimensional distributions of the Nadaraya±Watson kernel estimator of the regression function converge either to those of a degenerate process with completely dependent marginals or to those of a Gaussian white-noise process.The ®rst case occurs when the bandwidths are large enough in a speci®ed sense to allow long-range dependence to prevail.The second case is for bandwidths that are small in the given sense, when both the required norming sequence and the limiting process are the same as if the errors were independent.This conclusion is also derived for all bandwidths if the errors are short-range dependent.The borderline situation results in a limiting convolution of the two cases.The main results contrast with previous ®ndings for deterministic-design regression.