Asymptotic bias reduction of maximum likelihood estimates via penalized likelihoods with differential geometry

TL;DR

This paper introduces a novel method for reducing the asymptotic bias of maximum likelihood estimates using penalized likelihoods and differential geometry, applicable to various statistical models.

Contribution

It develops a new bias reduction technique based on solving a quasi-linear PDE with differential geometry, extending MLE accuracy in complex models.

Findings

Effective bias reduction in generalized linear models

Improved estimates in linear mixed-effects models

Application to location-scale families demonstrates versatility

Abstract

A procedure for asymptotic bias reduction of maximum likelihood estimates of generic estimands is developed. The estimator is realized as a plug-in estimator, where the parameter maximizes the penalized likelihood with a penalty function that satisfies a quasi-linear partial differential equation of the first order. The integration of the partial differential equation with the aid of differential geometry is discussed. Applications to generalized linear models, linear mixed-effects models, and a location-scale family are presented.