Explicit Formula for Asymptotic Higher Moments of the Nadaraya-Watson Estimator

The Nadaraya-Watson estimator is certainly the most popular nonparametric regression estimator. The asymptotic bias and variance of this estimator, say m̂(x), are well known. Nevertheless, its higher moments are rarely mentioned in the literature. In this paper, explicit formulas for asymptotic higher moments, such as E((m̂(x) – m(x))γ) or E((m̂(x) – E(m̂(x)))γ), for γ any positive integer, are derived and illustrated by some examples. In particular, explicit asymptotic expressions for the Lγ -errors of m̂(x), for any γ, are shown. These results also allow one to give alternative proofs for the asymptotic normality and a Large Deviation Principle for the estimator. Other kernel regression estimators are also briefly discussed.