Asymptotic relative Pitman efficiency in group models

Abstract

The notion of asymptotic efficacy due to Hannan for multivariate statistics in a location problem is reformulated for manifolds. The matrices used in Hannan's definition are reformulated as Riemannian metrics on a manifold and hence are seen not to depend upon the particular parameterization of the manifold used to make the calculations. Conditions under which that efficacy does not depend upon basepoint and direction are derived. This leads to the extension of Pitman asymptotic relative efficiency to location parameters in group models. Under stronger conditions, that of a two-point homogeneous space, we introduce a notion of rank and sign and show that, under the null distribution, the sign is uniformly distributed on a suitably defined sphere and that the rank is independent of the sign. This work generalizes previous definitions of Neeman and Chang, Hössjer and Croux. For group models, a definition of a regression group model is given. Unlike the usual linear model, a location model is not a subcase of a regression group model. Nevertheless, it is shown that the Riemannian metrics for the regression model can be derived from those of the location model and hence, in many cases, the asymptotic relative efficiencies coincide for group and location models. As examples, rank score statistics for spherical and Procrustes regressions are derived. The Procrustes regression model arises in problems of image registration.