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. Conditions under which that efficacy does not depend upon basepoint and direction are derived. It leads to the extension of Pitman asymptotic relative efficiency (ARE) to location parameters in group models. 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 asymptotic relative efficiencies coincide for location and regression group models. To construct optimal score rank statistics, under somewhat stonger conditions (namely, in a two point homogeneous space), we introduce a notion of rank and sign, and further show that the rank is independent of the sign. If possible, I might introduce a class of optimal score rank statistics (in the sense of Pitman ARE) for directional data. Application to a common problem in palaeomagnetism, the proposed test would be much more robust and efficient than the existing well-known fold test.