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A general lower bound is developed for the minimax risk when estimating anarbitrary functional. The bound is based on testing two composite hypothesesand is shown to be effective in estimating the nonsmooth functional${\frac{1}{n}}\sum|\theta_i|$ from an observation $Y\sim N(\theta,I_n)$. Thisproblem exhibits some features that are significantly different from those thatoccur in estimating conventional smooth functionals. This is a setting wherestandard techniques fail to yield sharp results. A sharp minimax lower bound isestablished by applying the general lower bound technique based on testing twocomposite hypotheses. A key step is the construction of two special priors andbounding the chi-square distance between two normal mixtures. An estimator isconstructed using approximation theory and Hermite polynomials and is shown tobe asymptotically sharp minimax when the means are bounded by a given value$M$. It is shown that the minimax risk equals $\beta_*^2M^2({\frac{\log\logn}{\log n}})^2$ asymptotically, where $\beta_*$ is the Bernstein constant. Thegeneral techniques and results developed in the present paper can also be usedto solve other related problems.

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