The Annals of Statistics

Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional

T. Tony Cai and Mark G. Low

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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.

Article information

Ann. Statist., Volume 39, Number 2 (2011), 1012-1041.

First available in Project Euclid: 8 April 2011

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G07: Density estimation
Secondary: 62620

Best polynomial approximation ℓ1 norm composite hypotheses Hermite polynomial minimax lower bound nonsmooth functional optimal rate of convergence


Cai, T. Tony; Low, Mark G. Testing composite hypotheses, Hermite polynomials and optimal estimation of a nonsmooth functional. Ann. Statist. 39 (2011), no. 2, 1012--1041. doi:10.1214/10-AOS849.

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