  We prove minimax bounds for estimating Gaussian location mixtures on
$\mathbb{R}^d$ under the squared $L^2$ and the squared Hellinger loss
functions. Under the squared $L^2$ loss, we prove that the minimax rate is
upper and lower bounded by a constant multiple of $n^{-1}(\log n)^{d/2}$. Under
the squared Hellinger loss, we consider two subclasses based on the behavior of
the tails of the mixing measure. When the mixing measure has a sub-Gaussian
tail, the minimax rate under the squared Hellinger loss is bounded from below
by $(\log n)^{d}/n$. On the other hand, when the mixing measure is only assumed
to have a bounded $p^{\text{th}}$ moment for a fixed $p > 0$, the minimax rate
under the squared Hellinger loss is bounded from below by $n^{-p/(p+d)}(\log
n)^{-3d/2}$. These rates are minimax optimal up to logarithmic factors.
