On adaptive local polynomial approximation with varying bandwidth

The local polynomial approximation (LPA) of noisy data is considered with the new adaptive procedure for varying bandwidth selection. The algorithm is simple to implement and nearly optimal within In N factor in the point-wise risk for estimating the function and its derivatives. The adaptive varying bandwidth enables the algorithm to be spatial adaptive over a wide range of the classes of functions in the sense that its quality is close to that which one could achieve if smoothness of the estimated function was known in advance. It is shown that the cross-validation adjustment of the threshold parameter of the algorithm significantly improves its accuracy. In particular, simulation demonstrates that the adaptive algorithm with the adjusted threshold parameter performs better than the wavelet estimators.