论文标题
最小值界限用于估计多元高斯位置混合物
Minimax bounds for estimating multivariate Gaussian location mixtures
论文作者
论文摘要
我们证明了在平方$ l^2 $下的$ \ mathbb {r}^d $上估算高斯位置混合物的最小值界限和平方hellinger损失功能。在平方的$ l^2 $损失下，我们证明了最小值是上层和下限的，由$ n^{ - 1}的常数倍数（\ log n）^{d/2} $。在平方的Hellinger损失下，我们根据混合度量尾巴的行为考虑了两个子类。当混合度量具有次高斯的尾巴时，平方hellinger损失下的最小速率从下面限制为$（\ log n）^{d}/n $。另一方面，当仅假定混合度量具有有界的$ p^{\ text {th}} $时刻，对于固定的$ p> 0 $时，平方hellinger损失下的最小值速率是$ n^{ - p/（p/（p+d）}（p/p+d）}（\ log n）^{ - 3D/2/2} $。这些速率是最佳对数因素的最佳选择。
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.
