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On spatially adaptive estimation of nonparametric regression. (English) Zbl 0892.62018
Summary: We develop spatially adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses many attractive adaptive properties, namely, it is near-optimal within log $$n$$-factor for estimating a function (or its derivative) at a single point; it is spatially adaptive in the sense that its accuracy is close to the one which could be achieved if the smoothness of the underlying function were known in advance; it is optimal in order (for a “strong” accuracy measure) or near-optimal within a log $$n$$-factor (for a “weak” accuracy measure) for estimating the entire function (or its derivative) over a wide range of classes and global loss functions. We demonstrate that the “spatial adaptive abilities” of our estimate are, in a sense, the best possible ones. Besides this, our adaptive estimate is computationally efficient and demonstrates a reasonable practical behavior.

##### MSC:
 62G07 Density estimation 62C20 Minimax procedures in statistical decision theory 62G20 Asymptotic properties of nonparametric inference