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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

Japanese

我々は、非パラメトリック最大確率推定器(NPMLE)を使用してユニバリアットガウシア混合密度を推定する問題を分析する $n$独立および同様に分散サンプルに基づいて。我々は、非制限NPMLEは、真の混合測量がサブガウシアである場合、平方ヘリンガウシア距離で最小最適率$\log(n)/n$を達成することを示しています。さらに、我々は、真の混合測量がコンパクトなサポートを有するとき、コンパクトな間隔に制約されたNMPLEはまた、わずかに速く、$\log(n)/n$の最小最適率を達成することを示しています。我々の証拠は、最終的なガ

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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