Approximation of Smooth Multivariate Regression Functions by Universal Kernel Estimators

We study universal locally constant kernel estimators in the classical nonparametric regression problem, where a multivariate regression function should be recovered from observations of its noisy values in some available tuple of fixed or random points (a tuple of regressors). Earlier these kernel estimators were studied only in the case of continuous multivariate regression functions. A distinctive feature of universal nuclear estimators is the presence of quite weak, fairly simple, and minimal (in a sense) conditions on the regressors which are universal relative to the stochastic nature of these quantities. In particular, in the case of a continuous regression function, for the uniform consistency of these kernel estimators, it is sufficient to require only the property of asymptotically (with increasing volume of observations) dense filling of the domain of the regression function by the regressors. We show that, under the additional smoothness assumption on the function, the accuracy of uniform approximation can be improved, where, as above, the regressors should only satisfy the above fairly general condition in terms of data density.