Geodesic Hypothesis Testing for Comparing Location Parameters in Elliptical Populations

Abstract

In this paper we study the geometry of the differentiable manifold associated with two samples of symmetric distributions in the real line equipped with the Fisher information as Riemannian metric. Expressions for the entries of the information matrix are obtained under different assumptions. The geodesic or Rao distance induced by this geometry is used to construct asymptotic parametrization-invariant testing procedures for comparing location parameters. As special cases, we obtain new asymptotic tests for the two sample Behrens-Fisher and Fieller-Creasy problems. Testing equality of several location parameters is also considered. It is shown that when scale parameters are equal, the geodesic test statistic is a strictly monotone increasing function of the Wald statistic. Empirical results for the Student-t distribution provide evidence that the geodesic test statistic has good sampling properties in terms of level and power.