Minimax bounds for estimating multivariate Gaussian location mixtures
Minimax bounds for estimating multivariate Gaussian location mixtures
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DOI：
10.1214/21-ejs1975
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发表时间：
2020-12
影响因子：
1.1
通讯作者：
Arlene K. H. Kim;Adityanand Guntuboyina
中科院分区：
文献类型：
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作者：
Arlene K. H. Kim;Adityanand Guntuboyina
在平方的L^2和平方的Hellinger损失函数下，我们证明了在$Mathbb{R}^d上估计高斯位置混合的极小极大界.在损失平方为$L^2$的情况下，证明了极小极大率是由$n^-1}(\logn)^{d/2}$的常数倍数构成的上下界。在平方Hellinger损失下，我们考虑了基于混合测度尾部行为的两个子类。当混合测度具有亚高斯尾时，平方Hellinger损失下的极小极大速率从下有界于$(\logn)^{d}/n$。另一方面，当假设混合度量对于固定的$p>；0$只有一个有界的$p^{\Text{th}}$时，平方Hellinger损失下的极小极大速率从下有界于$n^{-p/(p+d)}(\logn)^{-3d/2}$.这些比率是极小极大最优的，直到对数因子。
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.