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

Enes Makalic, Daniel F. Schmidt

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.

Asymptotic theory and bias correction for 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 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.

Paper Structure

This paper contains 9 sections, 4 theorems, 61 equations.

Table of Contents

  1. Introduction
  2. Minimum message length and the Wallace--Freeman approximation
  3. The Wallace--Freeman estimator
  4. The Wallace--Freeman estimator as a penalised M-estimator
  5. Bias of the Wallace--Freeman estimator
  6. Weibull distribution example
  7. Proof of Theorem \ref{['thm:wf87']}
  8. Proof of Theorem \ref{['thm:wf87:penm']}
  9. Proof of Theorem \ref{['thm:wf87:bias']}

Key Result

Theorem 3.1

Let $P^* \subset \mathcal{X}^n$ denote a data-space cell of an optimal SMML partition, and let $\bm{\theta} \in \Theta \subseteq \mathbb{R}^d$ be the corresponding assertion. Assume that: Under these assumptions, the Wallace--Freeman codelength approximation to strict minimum message length eqn:smml is uniformly for $\bm{\theta}$ in compact subsets of $\operatorname{int}(\Theta)$. $\blacktriangl

Theorems & Definitions (4)

  • Theorem 3.1: Local derivation of the Wallace--Freeman codelength
  • Theorem 3.2: Properties of the Wallace--Freeman estimator
  • Theorem 3.3: Bias of the Wallace--Freeman estimator
  • Corollary 3.4