On an Asymptotic Distribution for the MLE

TL;DR

This paper introduces a refined asymptotic distribution for the maximum likelihood estimator in exponential family models, providing more accurate approximations under weaker conditions than traditional normal approximations.

Contribution

It derives a new asymptotic distribution for MLE that improves upon the normal approximation and applies the technique to determine the exact distribution of the weighted likelihood bootstrap.

Findings

New asymptotic distribution for MLE in exponential families

Refinement of normal approximation similar to Edgeworth expansion

Exact distribution of weighted likelihood bootstrap obtained

Abstract

The paper presents a novel asymptotic distribution for a mle when the log--likelihood is strictly concave in the parameter for all data points; for example, the exponential family. The new asymptotic distribution can be seen as a refinement of the usual normal asymptotic distribution and is comparable to an Edgeworth expansion. However, it is obtained with weaker conditions than even those for asymptotic normality. The same technique is then used to find the exact distribution of the weighted likelihood bootstrap sampler.