It is well known that, under standard regularity conditions, the maximum likelihood estimator (MLE) satisfies a central limit theorem and converges in distribution to a Gaussian random variable as the sample size grows. This paper strengthens this classical result by developing several stronger forms of asymptotic normality for the normalized MLE. With additional assumptions on the score, we first establish sub-Gaussian tail bounds and convergence of all moments for the normalized estimation error. We then prove an entropic central limit theorem for a smoothed version of the estimator, showing convergence in relative entropy to the limiting Gaussian law. When the Fisher information of the normalized estimate is bounded, or its density has bounded first derivative, we further show that the smoothing can be removed, yielding entropic normality of the MLE itself. The proofs develop auxiliary tools that may be of independent interest, including exponential consistency bounds, high-moment estimates, and entropy-control arguments for the estimator.

We thank the reviewers for their useful feedback and their time. We have corrected all the minor comments, as suggested. We now provide specific answers to each reviewer below. We thank the reviewer for their positive evaluation of our work and their comments. "All the theoretical contributions seem to me a bit marginal" Since the Sliced-Wasserstein distance is an average of one-6 We will explain these observations more explicitly to clarify our contributions.

Large Language Models that are evaluated on multiple metrics.

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