The Annals of Statistics

Nonquadratic estimators of a quadratic functional

T. Tony Cai and Mark G. Low

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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.

Article information

Ann. Statist., Volume 33, Number 6 (2005), 2930-2956.

First available in Project Euclid: 17 February 2006

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Zentralblatt MATH identifier

Primary: 62G99: None of the above, but in this section
Secondary: 62F12: Asymptotic properties of estimators 62F35: Robustness and adaptive procedures 62M99: None of the above, but in this section

Besov balls Gaussian sequence model information bound minimax estimation quadratic functional quadratic estimators


Cai, T. Tony; Low, Mark G. Nonquadratic estimators of a quadratic functional. Ann. Statist. 33 (2005), no. 6, 2930--2956. doi:10.1214/009053605000000147.

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