Minimax bounds for estimating multivariate Gaussian location mixtures

Understand

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 $(\log n)^{d}/n$.