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

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.