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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

Indonesian

Kami menganalisis masalah memperkirakan kepadatan campuran Gaussian univariate menggunakan estimator probabilitas maksimum non-parametrik (NPMLE) berdasarkan sampel yang independen $n$ dan didistribusikan secara identik. Kami menunjukkan bahwa NPMLE yang tidak terbatas mencapai tingkat optimal minimum $\log(n)/n$ pada jarak Hellinger persegi, ketika ukuran campuran yang sebenarnya adalah sub-Gaussian. Selain itu, kami menunjukkan bahwa ketika ukuran campuran yang sebenarnya memiliki dukungan yang kompak, NMPLE yang terbatas pada interval yang kompak juga mencapai tingkat optimal minimum, yang sedikit lebih cepat dan dari $\log(n)/n$ pada jarak Hellinger persegi. Bukti kami terdiri dari karakterisasi yang tajam dari bracketing Hellinger lokal dari entropy kelas campuran Gaussian finite. Salah satu inovasi teknis utama kami adalah

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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