Oracle inequalities and nonparametric function estimation (Q1126819)

In nonparametric function estimation partial prior information about the unknown function is expressed by a family of models or estimators. In the present paper estimation of functions \(\mu\) observed in additive noise is considered; given a family of models for \(\mu\) the \textit{ideal risk} is introduced. The ideal risk is not attainable by an estimator depending on the data alone, but it is a useful benchmark and one seeks estimators that in an appropriate sense optimally mimick the ideal risk. For that purpose the author states oracle inequalities which bound the mean squared error of a given estimator in terms of the ideal risk. The survey concentrates on three settings: the James Stein estimator, soft thresholding, and complexity penalized least squares. To demonstrate that oracle inequalities are informative tools consequences for adaptive minimax estimation are described.