Asymptotics of Maximum Likelihood without the LLN or CLT or Sample Size Going to Infinity

TL;DR

This paper demonstrates that the maximum likelihood estimator is approximately normally distributed under broad conditions without relying on traditional asymptotic assumptions like sample size going to infinity, using advanced convergence theories.

Contribution

It introduces a novel asymptotic analysis of MLE that does not depend on LLN, CLT, or increasing sample size, combining LAN, LAMN, and Cramér theories.

Findings

MLE is approximately normal without sample size assumptions

Convergence in law of the log likelihood function is key

Results apply to one-step and iterative Newton estimators

Abstract

If the log likelihood is approximately quadratic with constant Hessian, then the maximum likelihood estimator (MLE) is approximately normally distributed. No other assumptions are required. We do not need independent and identically distributed data. We do not need the law of large numbers (LLN) or the central limit theorem (CLT). We do not need sample size going to infinity or anything going to infinity. Presented here is a combination of Le Cam style theory involving local asymptotic normality (LAN) and local asymptotic mixed normality (LAMN) and Cram\'er style theory involving derivatives and Fisher information. The main tool is convergence in law of the log likelihood function and its derivatives considered as random elements of a Polish space of continuous functions with the metric of uniform convergence on compact sets. We obtain results for both one-step-Newton estimators and Newton-iterated-to-convergence estimators.