Towards insensitivity of Nadaraya–Watson estimators with respect to design correlation

The consistency of Nadaraya–Watson estimators in nonparametric regression is proved without using traditional conditions for dependence of design elements (regressors). A design can be either fixed and not necessarily regular, or random and not necessarily satisfying classical correlation conditions for the design elements. A new characterization of the dependence of design elements is proposed, in terms of which sufficient conditions are formulated for both pointwise and uniform consistency of Nadaraya–Watson estimators.