Sub-Gaussian Concentration and Entropic Normality of the Maximum Likelihood Estimator

2026-05-11 · Source: stat.ML updates on arXiv.org · Field: Science & Research — Mathematics & Computational Sciences, Artificial Intelligence & Machine Learning · Depth: Expert, long

Summary

This paper strengthens the classical central limit theorem (CLT) for the maximum likelihood estimator (MLE) by establishing stronger forms of asymptotic normality. Under additional assumptions on the score function, the research first demonstrates sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. It then proves an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. Crucially, when the Fisher information of the normalized estimate is bounded or its density has a bounded first derivative, the smoothing step can be removed, leading to entropic normality of the MLE itself. The proofs introduce auxiliary tools, including exponential consistency bounds, high-moment estimates, and entropy-control arguments, which may have broader applicability.

Key takeaway

For AI Scientists and Research Scientists working with statistical inference, understanding these stronger forms of MLE asymptotic normality is crucial. Your models' reliability can be significantly enhanced by leveraging sub-Gaussian tail bounds and entropic convergence, particularly when dealing with large datasets. This work suggests that under specific regularity conditions, the MLE offers more robust guarantees than previously understood, impacting confidence intervals and hypothesis testing in high-stakes applications.

Key insights

MLEs exhibit stronger asymptotic normality, including sub-Gaussian tails and entropic convergence, under specific regularity conditions.

Principles

• Entropic convergence is stronger than distributional convergence.
• Bounded Fisher information enables direct entropic normality.
• Sub-Gaussianity implies convergence of all moments.

Method

The method involves establishing sub-Gaussian tail bounds, proving entropic convergence for a smoothed estimator, and then removing smoothing under bounded Fisher information or bounded first derivative conditions.

In practice

• Apply sub-Gaussian bounds for tighter error control.
• Use entropic normality for stronger statistical guarantees.
• Consider Fisher information bounds for direct MLE entropic convergence.

Topics

• Maximum Likelihood Estimator
• Sub-Gaussian Tail Bounds
• Entropic Central Limit Theorem
• Relative Entropy
• Fisher Information

Best for: AI Scientist, Research Scientist

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Editorial summary, takeaway, and curation by AIssential. Original article published by stat.ML updates on arXiv.org.