The Annals of Statistics

Nonquadratic estimators of a quadratic functional

T. Tony Cai and Mark G. Low

Full-text: Open access

Abstract

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

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

Dates
First available in Project Euclid: 17 February 2006

Permanent link to this document
https://projecteuclid.org/euclid.aos/1140191679

Digital Object Identifier
doi:10.1214/009053605000000147

Mathematical Reviews number (MathSciNet)
MR2253108

Zentralblatt MATH identifier
1085.62055

Subjects
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

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

Citation

Cai, T. Tony; Low, Mark G. Nonquadratic estimators of a quadratic functional. Ann. Statist. 33 (2005), no. 6, 2930--2956. doi:10.1214/009053605000000147. https://projecteuclid.org/euclid.aos/1140191679


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