  We consider nonparametric estimation of $L_2$, Renyi-$\alpha$ and
Tsallis-$\alpha$ divergences between continuous distributions. Our approach is
to construct estimators for particular integral functionals of two densities
and translate them into divergence estimators. For the integral functionals,
our estimators are based on corrections of a preliminary plug-in estimator. We
show that these estimators achieve the parametric convergence rate of
$n^{-1/2}$ when the densities' smoothness, $s$, are both at least $d/4$ where
$d$ is the dimension. We also derive minimax lower bounds for this problem
which confirm that $s > d/4$ is necessary to achieve the $n^{-1/2}$ rate of
convergence. We validate our theoretical guarantees with a number of
simulations.
