Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance

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 L² 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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BibTeX RIS

BibTeX

@misc{hundrieser2026minimax,
  title = {{Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance}},
  author = {Shayan Hundrieser and Tudor Manole},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.30109},
  eprint = {2610.30109},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.30109},
}

RIS

TY  - UNPB
TI  - Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance
AU  - Shayan Hundrieser
AU  - Tudor Manole
PY  - 2026
DA  - 2026/10/21
PB  - arXiv
DO  - 10.48550/arxiv.2610.30109
UR  - https://arxiv.org/abs/2610.30109
AB  - 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.
ER  -

Abstract

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