** Tony Cai **

Department of Statistics. The Wharton School.

University of Pennsylvania

In this talk I will discuss some recent work on optimal estimation of nonsmooth functionals. These problems exhibit some interesting features that are significantly different from those that occur in estimating conventional smooth functionals. This is a setting where standard techniques fail. I will discuss a newly developed general minimax lower bound technique that is based on testing two fuzzy hypotheses and illustrate the ideas by focusing on the problem of optimal estimation of the l_1 norm of a high dimensional normal mean vector. An estimator is constructed using approximation theory and Hermite polynomials and is shown to be asymptotically sharp minimax. This is joint work with Mark Low.