  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
${\frac{1}{n}}\sum|\theta_i|$ from an observation $Y\sim N(\theta,I_n)$. 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 $\beta_*^2M^2({\frac{\log\log
n}{\log n}})^2$ asymptotically, where $\beta_*$ is the Bernstein constant. The
general techniques and results developed in the present paper can also be used
to solve other related problems.
