The Annals of Statistics

Minimax estimation of linear functionals over nonconvex parameter spaces

T. Tony Cai and Mark G. Low

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Abstract

The minimax theory for estimating linear functionals is extended to the case of a finite union of convex parameter spaces. Upper and lower bounds for the minimax risk can still be described in terms of a modulus of continuity. However in contrast to the theory for convex parameter spaces rate optimal procedures are often required to be nonlinear. A construction of such nonlinear procedures is given. The results developed in this paper have important applications to the theory of adaptation.

Article information

Source
Ann. Statist., Volume 32, Number 2 (2004), 552-576.

Dates
First available in Project Euclid: 28 April 2004

Permanent link to this document
https://projecteuclid.org/euclid.aos/1083178938

Digital Object Identifier
doi:10.1214/009053604000000094

Mathematical Reviews number (MathSciNet)
MR2060169

Zentralblatt MATH identifier
1048.62054

Subjects
Primary: 62G99: None of the above, but in this section
Secondary: 62F12: Asymptotic properties of estimators 62C20: Minimax procedures 62M99: None of the above, but in this section

Keywords
Constrained risk inequality linear functionals minimax estimation modulus of continuity nonparametric functional estimation white noise model

Citation

Cai, T. Tony; Low, Mark G. Minimax estimation of linear functionals over nonconvex parameter spaces. Ann. Statist. 32 (2004), no. 2, 552--576. doi:10.1214/009053604000000094. https://projecteuclid.org/euclid.aos/1083178938


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