Optimal adaptive estimation of a quadratic functional

Details Optimal adaptive estimation of a quadratic functional Optimal adaptive estimation of a quadratic functional

2007-02-23

arxiv.org/abs/math/0702682v1

Annals of Statistics 2006, Vol. 34, No. 5, 2298-2325

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053606000000849 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053606000000849

Adaptive estimation of a quadratic functional over both Besov and $L_p$ balls is considered. A collection of nonquadratic estimators are developed which have useful bias and variance properties over individual Besov and $L_p$ balls. An adaptive procedure is then constructed based on penalized maximization over this collection of nonquadratic estimators. This procedure is shown to be optimally rate adaptive over the entire range of Besov and $L_p$ balls in the sense that it attains certain constrained risk bounds.

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