A note on Silvey's (1959) Theorem

We give an alternative proof of a Theorem by Silvey (1959) for the asymptotic covariance matrix of the maximum likelihood estimator under different types of constraints in the parameter space. We show that such constraints can be classified into three types: constraints that arise from the selection of a simplified parametric model (model constraints), from non-identifiability of the parameter space (identifiability constraints), or from the problem that without proper constraints a family of statistical models is not defined (basic constraints). The main idea of the new proof is to augment the model such that the parameter space without constraints becomes identifiable, and then to apply a Theorem by Aitchison and Silvey (1958) to the augmented model. Our approach clarifies how the different types of constraints can be treated in a unified way, and shows that the Theorem can be applied to a variety of statistical estimation problems for which asymptotic results have not been obtained so far. The Theorem is illustrated with examples from categorical data analysis and canonical correlation analysis.