Asymptotic theory and bias correction for the Wallace--Freeman estimator

TL;DR

This paper develops a comprehensive asymptotic theory for the Wallace--Freeman estimator, including bias correction and its relation to penalised likelihood, enhancing its theoretical foundation for parametric inference.

Contribution

It formulates the Wallace--Freeman estimator as a penalised M-estimator, deriving its asymptotic properties and explicit bias correction, extending classical bias formulas.

Findings

Established existence, consistency, and asymptotic normality of the estimator.

Derived the first-order bias correction term explicitly.

Extended Cox--Snell bias formula to the Wallace--Freeman estimator.

Abstract

The Wallace--Freeman estimator is a classical invariant point estimator whose large-sample properties have not been fully developed in a modern asymptotic framework. We show that the estimator can be formulated as a penalised M-estimator with a specific penalty weight, yielding a unified route to its asymptotic analysis. This representation allows us to establish existence, consistency, an asymptotic linear expansion, and asymptotic normality under standard regularity conditions. We further derive the first-order difference between the Wallace--Freeman estimator and the maximum likelihood estimator, and show that this induces an explicit $O(n^{-1})$ bias correction determined by the gradient of the penalty. As a consequence, the Cox--Snell bias formula for the maximum likelihood estimator extends naturally to the Wallace--Freeman estimator by the addition of a penalty-driven correction term. As an illustration, we derive the first-order bias of the Wallace--Freeman estimator for the Weibull model and show how the penalty modifies the corresponding maximum likelihood bias. These results place the Wallace--Freeman estimator within the general theory of penalised likelihood and provide a rigorous asymptotic basis for its use in parametric inference.