On Spatial Adaptive Estimation of Nonparametric Regression

Alexander Goldenshluger, Arkadi Nemirovski

Research output: Contribution to journalArticlepeer-review

Abstract

The paper is devoted to developing spatial adaptive estimates for restoring functions from noisy observations. We show that the traditional least square (piecewise polynomial) estimate equipped with adaptively adjusted window possesses simultaneously many attractive adaptive properties, namely, 1) it is near– optimal within ln n–factor for estimating a function (or its derivative) at a single point; 2) it is spatial adaptive in the sense that its quality is close to that one which could be achieved if smoothness of the underlying function was known in advance; 3) it is optimal in order (in the case of “strong” accuracy measure) or near–optimal within ln n–factor (in the case of “weak” accuracy measure) for estimating whole function (or its derivative) over wide range of the classes and global loss functions. We demonstrate that the “spatial adaptive abilities” of our estimate are, in a sense, the best possible. Besides this, our adaptive estimate is computationally efficient and demonstrates reasonable practical behavior.
Original languageEnglish
Pages (from-to)1-35
Number of pages35
JournalMathematical Methods of Statistics
Volume6
Issue number2
StatePublished - 1 Jan 1997

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