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April 2011 Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional
T. Tony Cai, Mark G. Low
Ann. Statist. 39(2): 1012-1041 (April 2011). DOI: 10.1214/10-AOS849


A general lower bound is developed for the minimax risk when estimating an arbitrary functional. The bound is based on testing two composite hypotheses and is shown to be effective in estimating the nonsmooth functional (1/n)∑|θi| from an observation YN(θ, In). This problem exhibits some features that are significantly different from those that occur in estimating conventional smooth functionals. This is a setting where standard techniques fail to yield sharp results.

A sharp minimax lower bound is established by applying the general lower bound technique based on testing two composite hypotheses. A key step is the construction of two special priors and bounding the chi-square distance between two normal mixtures. An estimator is constructed using approximation theory and Hermite polynomials and is shown to be asymptotically sharp minimax when the means are bounded by a given value M. It is shown that the minimax risk equals β2M2(log log n/log n)2 asymptotically, where β is the Bernstein constant.

The general techniques and results developed in the present paper can also be used to solve other related problems.


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T. Tony Cai. Mark G. Low. "Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional." Ann. Statist. 39 (2) 1012 - 1041, April 2011.


Published: April 2011
First available in Project Euclid: 8 April 2011

zbMATH: 1277.62101
MathSciNet: MR2816346
Digital Object Identifier: 10.1214/10-AOS849

Primary: 62G07
Secondary: 62620

Keywords: Best polynomial approximation , composite hypotheses , Hermite polynomial , ℓ1 norm , minimax lower bound , nonsmooth functional , Optimal rate of convergence

Rights: Copyright © 2011 Institute of Mathematical Statistics


Vol.39 • No. 2 • April 2011
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