Abstract:

In this talk I will present 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 mainly focus on the problem of estimating 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. A general minimax lower bound technique that is based on testing two composite hypotheses will also be discussed in detail.  The technique is potentially also useful for other related problems.