Newest papers

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

Korean

우리는 $n$ 독립적이고 동일하게 분산된 샘플을 기반으로 비파라미터적 최대 확률 추정기 (NPMLE)를 사용하여 유니버리어한 가우시아 혼합 밀도를 추정하는 문제를 분석합니다. 우리는 비제한 NPMLE가 진정한 혼합 측정기가 sub-Gaussian인 경우 사각형 헬링거 거리에 $\log(n)/n$의 최소 최적 비율을 달성한다는 것을 보여줍니다. 또한 진정한 혼합 측정이 컴팩트한 지원을 받았을 때 NMPLE는 컴팩트한 간격으로 제한된 최소 최적 비율을 달성합니다. 이는 약간 빠르며 $\log(n)/n$의 최소 비율입니다. 우리의 증거는 최적화 된 가우시아 혼합 클래스

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.

Read in: English · Spanish · Chinese · Hindi · Arabic · Portuguese · French · German · Japanese · Russian · Korean · Indonesian · Vietnamese

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

Sources: OpenAlex (CC0), arXiv, Crossref. Agent Exchange