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Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance

Shayan Hundrieser, Tudor Manole · 2026 · arXiv (Cornell University)

DOI: 10.48550/arxiv.2610.30109 · cited by 0 works (OpenAlex) · open-access PDF

Chinese

我们分析了使用非参数最大概率估计器(NPMLE)基于 $n$独立和相同分布的样品估计单变加斯基混合密度的问题。我们表明,未限制的NPMLE在平方Hellinger距离达到最低最优率 $\log(n)/n$,当真正的混合尺寸是sub-Gaussian时。此外,我们表明,当真正的混合尺寸具有紧凑的支持时,NMPLE被限制在紧凑的间隔内也达到最低最优率,这是略快的,并且是 $\log(n)/n$的最高速度。我们的证据是对最终加斯基混合物类别的Hellinger本地混合尺寸的特征。我们的一项关键技术创新是将 $L^$2的混合尺寸严格绑定到

Machine translation (Meta M2M100); the English original follows.

Abstract

We analyze the problem of estimating univariate Gaussian mixture densities using the nonparametric maximum likelihood estimator (NPMLE) based on $n$ independent and identically distributed samples. We show that the unconstrained NPMLE achieves the minimax optimal rate $\log(n)/n$ in squared Hellinger distance, when the true mixing measure is sub-Gaussian. In addition, we show that when the true mixing measure has compact support, the NMPLE constrained to a compact interval also achieves the minimax optimal rate, which is slightly faster and of order $\log(n)/(n \log \log n)$. Our proof consists of sharply characterizing the local Hellinger bracketing entropy of the class of finite Gaussian mixtures. One of our key technical innovations is to sharply bound the $L^2$ bracketing entropy of balls of bounded-degree polynomials, which may be of independent interest. As an application, we establish the regret-optimality of $g$-modeling procedures based on the NPMLE in the Gaussian Empirical Bayes problem with bounded priors.

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Topics: Mathematics, Hellinger distance, Minimax, Entropy (arrow of time), Mixing (physics), Gaussian

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