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 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 subGaussian 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.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.00444
 Bibcode:
 2020arXiv201200444K
 Keywords:

 Mathematics  Statistics Theory