An Upper Bound of the Bias of Nadaraya-Watson Kernel Regression under Lipschitz Assumptions
This work provides improved theoretical guarantees for nonparametric regression, which is incremental but useful in fields requiring hard error bounds.
The paper tackles the problem of deriving a finite-sample bias bound for the Nadaraya-Watson kernel regression estimator, extending beyond asymptotic analysis to handle finite bandwidths, discontinuous derivatives, multidimensional domains, and known function bounds, resulting in a tighter upper bound.
The Nadaraya-Watson kernel estimator is among the most popular nonparameteric regression technique thanks to its simplicity. Its asymptotic bias has been studied by Rosenblatt in 1969 and has been reported in a number of related literature. However, Rosenblatt's analysis is only valid for infinitesimal bandwidth. In contrast, we propose in this paper an upper bound of the bias which holds for finite bandwidths. Moreover, contrarily to the classic analysis we allow for discontinuous first order derivative of the regression function, we extend our bounds for multidimensional domains and we include the knowledge of the bound of the regression function when it exists and if it is known, to obtain a tighter bound. We believe that this work has potential applications in those fields where some hard guarantees on the error are needed