Asymptotic theory and first-order bias of the Wallace--Freeman estimator

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The Wallace--Freeman estimator is shown to be equivalent to a penalized likelihood criterion with a \(n^{-1}\) penalty weight in regular parametric models, establishing its asymptotic properties within standard theory; this equivalence also reveals an explicit \(O(n^{-1})\) bias difference from the maximum likelihood estimator. The findings are illustrated for the Weibull model, where the Wallace--Freeman penalty modifies the leading bias of the shape parameter's estimation.

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arXiv:2604.01568v2 Announce Type: replace Abstract: The Wallace--Freeman estimator is a classical minimum message length estimator whose relationship with likelihood-based asymptotic theory has not been fully developed. We show that, in regular parametric models, the Wallace--Freeman criterion is equivalent, up to constants, to a penalised likelihood criterion with penalty weight \(n^{-1}\). This representation places the estimator within the standard theory of penalised M-estimation and yields existence, consistency, an asymptotic linear expansion, and asymptotic normality under regularity conditions. We further derive the first-order difference between the Wallace--Freeman estimator and the maximum likelihood estimator, showing that it is an explicit \(O(n^{-1})\) shift determined by the gradient of the Wallace--Freeman penalty. Combining this expansion with the Cox--Snell formula gives a first-order bias expansion for the Wallace--Freeman estimator. The result clarifies its relationship with maximum likelihood, Jeffreys-prior penalisation, and Firth-type bias reduction. We illustrate the theory for the Weibull model, where the penalty modifies the leading bias of the maximum likelihood estimator of the shape parameter.