Multiresolution local polynomial regression: A new approach to pointwise spatial adaptation

In nonparametric local polynomial regression the adaptive selection of the scale parameter (window size/bandwidth) is a key problem. Recently new efficient algorithms, based on Lepski's approach, have been proposed in mathematical statistics for spatially adaptive varying scale denoising. A common feature of these algorithms is that they form test-estimates yˆh different by the scale h∈H and special statistical rules are exploited in order to select the estimate with the best pointwise varying scale. In this paper a novel multiresolution (MR) local polynomial regression is proposed. Instead of selection of the estimate with the best scale h a nonlinear estimate is built using all of the test-estimates yˆh. The adaptive estimation consists of two steps. The first step transforms the data into noisy spectrum coefficients (MR analysis). On the second step, this noisy spectrum is filtered by the thresholding procedure and used for estimation (MR synthesis).