Minimax bounds for estimating multivariate Gaussian location mixtures

Abstract

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 optimal rate is upper and lower bounded by a constant multiple of $n^{-1}{(logn)}^{d∕2}$. 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 ${(logn)}^d∕n$, which implies that the optimal minimax rate is between ${(logn)}^d∕n$ and the upper bound ${(logn)}^{d+1}∕n$ obtained by [11]. 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 $n^{-p∕(p+d)}{(logn)}^{-3d∕2}$. This rate is minimax optimal up to logarithmic factors.