Invariance, Haar Measures, and Equivariant Estimators

Abstract

The ring certainly looked like stone, but it felt harder than steel and heavier than lead. And the circle of it was twisted. If she ran a finger along one edge, it would go around twice, inside as well as out; it only had one edge. She ran her finger along that edge twice, just to convince herself. Robert Jordan, The Dragon Reborn, Book III of the Wheel of Time. 9.1 Invariance principles Invariance can be seen as a notion introduced in frequentist settings to restrict the range of acceptable estimators sufficiently so that an optimal esti-mator can be derived. From this point of view, it appears as an alternative to unbiasedness, and is thus similarly at odds with the Bayesian paradigm. However, invariance can also be justified on a non-decision-theoretic heuris-tic, namely, that estimators should meet some consistency requirements under a group of transformations, and it is thus of interest to consider this notion. Moreover, optimal (equivariant) estimators are always Bayes or generalized Bayes estimators. The corresponding measures can then be considered as noninformative priors induced by the invariance structure. Therefore, a Bayesian study of invariance is appealing, not because classical optimality once more relies on Bayesian estimators, but mainly because of the connection between invariance structures and the derivation of non-informative distributions. A first version of the invariance principle is to consider that the properties of a statistical procedure should not depend on the unit of measurement. If x and θ are measured in unit u 1 , and if y and η are the transforms of x and θ for the new unit u 2 , then an estimator δ 2 (y) of η should then correspond to the estimator δ 1 (x) of θ by the same change of unit. Of course, the notion of unit of measurement is to be understood in a general sense: for instance, it can indicate the choice of a particular scale (cm vs. m)-and estimators should be scale-invariant-the choice of a particular