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The celebrated Nadaraya-Watson kernel estimator is among the most studiedmethod for nonparametric regression. A classical result is that its rate ofconvergence depends on the number of covariates and deteriorates quickly as thedimension grows, which underscores the "curse of dimensionality" and haslimited its use in high dimensional settings. In this article, we show thatwhen the true regression function is single or multi-index, the effects of thecurse of dimensionality may be mitigated for the Nadaraya-Watson kernelestimator. Specifically, we prove that with $K$-fold cross-validation, theNadaraya-Watson kernel estimator indexed by a positive semidefinite bandwidthmatrix has an oracle property that its rate of convergence depends on thenumber of indices of the regression function rather than the number ofcovariates. Intuitively, this oracle property is a consequence of allowing thebandwidths to diverge to infinity as opposed to restricting them all toconverge to zero at certain rates as done in previous theoretical studies. Ourresult provides a theoretical perspective for the use of kernel estimation inhigh dimensional nonparametric regression and other applications such as metriclearning when a low rank structure is anticipated. Numerical illustrations aregiven through simulations and real data examples.

An Oracle Property of The Nadaraya-Watson Kernel Estimator for High Dimensional Nonparametric Regression