Nonquadratic estimators of a quadratic functional

Details Nonquadratic estimators of a quadratic functional Nonquadratic estimators of a quadratic functional

2006-03-06

arxiv.org/abs/math/0603134v1

Annals of Statistics 2005, Vol. 33, No. 6, 2930-2956

Mathematics

Statistics Theory

Published at http://dx.doi.org/10.1214/009053605000000147 in the Annals of Statistics (http://www.imstat.org/aos/) by the Inst

Scientific paper

10.1214/009053605000000147

Estimation of a quadratic functional over parameter spaces that are not quadratically convex is considered. It is shown, in contrast to the theory for quadratically convex parameter spaces, that optimal quadratic rules are often rate suboptimal. In such cases minimax rate optimal procedures are constructed based on local thresholding. These nonquadratic procedures are sometimes fully efficient even when optimal quadratic rules have slow rates of convergence. Moreover, it is shown that when estimating a quadratic functional nonquadratic procedures may exhibit different elbow phenomena than quadratic procedures.

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