NPMLE shown minimax optimal for univariate Gaussian mixtures
Original title: Minimax Optimality of the NPMLE for Gaussian Mixtures in Hellinger Distance
With compact support, a constrained NPMLE reaches the slightly faster rate log(n)/(n log log n).
The paper analyzes estimating univariate Gaussian mixture densities with the nonparametric maximum likelihood estimator from n i.i.d. samples. Unconstrained NPMLE is minimax optimal at log(n)/n in squared Hellinger distance when the mixing measure is sub-Gaussian. For compactly supported mixing measures, constraining the estimator to a compact interval is optimal at a slightly faster rate. The result is also applied to regret in Gaussian empirical Bayes with bounded priors.
- Proofs characterize local Hellinger bracketing entropy of finite Gaussian mixtures.
- A technical step bounds L2 bracketing entropy of bounded-degree polynomial balls.
- g-modeling based on the NPMLE is regret-optimal for bounded priors.
From the original article
1 min read at arxiv.orgWe 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.
By Shayan Hundrieser · arxiv.org
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