  We consider the problem of minimax estimation of the entropy of a density
over Lipschitz balls. Dropping the usual assumption that the density is bounded
away from zero, we obtain the minimax rates $(n\ln n)^{-s/(s+d)} + n^{-1/2}$
for $0<s\leq 2$ for densities supported on $[0,1]^d$, where $s$ is the
smoothness parameter and $n$ is the number of independent samples. We
generalize the results to densities with unbounded support: given an Orlicz
functions $\Psi$ of rapid growth (such as the sub-exponential and sub-Gaussian
classes), the minimax rates for densities with bounded $\Psi$-Orlicz norm
increase to $(n\ln n)^{-s/(s+d)} (\Psi^{-1}(n))^{d(1-d/p(s+d))} + n^{-1/2}$,
where $p$ is the norm parameter in the Lipschitz ball. We also show that the
integral-form plug-in estimators with kernel density estimates fail to achieve
the minimax rates, and characterize their worst case performances over the
Lipschitz ball.
  One of the key steps in analyzing the bias relies on a novel application of
the Hardy-Littlewood maximal inequality, which also leads to a new inequality
on the Fisher information that may be of independent interest.
