Beyond the delta method

TL;DR

This paper develops an explicit asymptotic expansion for the maximum likelihood estimator (MLE) and similar estimators, providing insights into their non-asymptotic behavior and accommodating non-Gaussian limits through an implicit function theorem approach.

Contribution

It introduces a computable asymptotic development for implicit estimators using an implicit function theorem, extending analysis beyond Gaussian limits.

Findings

Provides explicit asymptotic expansions for MLE and similar estimators.

Analyzes non-asymptotic behavior and deviations from limits.

Applicable to non-Gaussian limit scenarios.

Abstract

We give an asymptotic development of the maximum likelihood estimator (MLE), or any other estimator defined implicitly, in a way which involves the limiting behavior of the score and its higher-order derivatives. This development, which is explicitly computable, gives some insights about the non-asymptotic behavior of the renormalized MLE and its departure from its limit. We highlight that the results hold whenever the score and its derivative converge, including to non Gaussian limits. Our approach is based on an asymptotic implicit function theorem, inspired from perturbative approaches.