Testing Composite Hypotheses, Hermite Polynomials, and Optimal Estimation of a Nonsmooth Functional
Tony Cai and Mark Low
A sharp minimax lower bound is established by applying the general lower bound technique based on testing two fuzzy 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 β_{*} ^{2} M^{2} (loglog n)^{2}/(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.
Cai, T. & Low, M. (2004).
Minimax estimation of linear functionals over nonconvex parameter spaces.
The Annals of Statistics 32, 552 - 576.
Cai, T. & Low, M. (2005).
Non-quadratic estimators of a quadratic functional.
The Annals of Statistics 33, 2930-2956.