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.

翻译：以 $mathbb{R ⁇ d$ 和 平方 Hellinger 损失 函数 来估计 Gausian 位置 混合物 $mathb{R ⁇ d$ 和 平方 Hellinger 损失 。 在 折方 $ $2$ 损失 中, 我们证明迷你Max 利率是 以 $ +% 1} (\ log n) = d/2} 美元 常数乘以 $ +% 1 (\ log n) (log n) = d} / 美元 。 在平方 Hellinger 损失 中, 我们根据 平方 Hellinger 损失 的尾巴行为, 我们考虑根据 平方 Hellinger 损失 的尾巴方两个小类。 当混合措施的尾巴巴/ (p+) 尾巴 和 平方 Hellinger 损失 的底部, 平方 Hellinger 损失的迷你 率是 = n\\ 3d/2} 最优的 log 。