Sequential Kernel Estimation of a Multivariate Regression Function

This paper presents a sequential estimation procedure for an unknown multivariate regression function. Observed regressors and noises of the model are supposed to be dependent and form sequences of dependent vectors and numbers respectively. Two types of estimators are considered. Both estimators are constructed on the basis of Nadaraya–Watson kernel estimators. First, sequential estimators with given bias and mean square error are defined. According to the sequential approach the duration of observations is a special stopping time. Then on the basis of these estimators, truncated sequential estimators of a regression function are constructed on a time interval of a fixed length. At the same time the variance of these estimators is also bounded by a non-asymptotic bound. Together with finite-sample, asymptotic properties of the presented estimators are investigated. It is shown, in particular, that by the appropriately chosen bandwidths both estimators have optimal (as compared to the case of independent data) rates of convergence.