Semiparametrically Efficient Rank-based Inference for Shape I. Optimal Rank-based Tests for Sphericity

We propose a class of rank-based procedures for testing that the shape matrix V of an elliptical distribution (with unspecified center of symmetry, scale, and radial density) has some fixed value V0; this includes, for V0 = Ik, the problem of testing for sphericity as an important particular case. The proposed tests are invariant under translations, monotone radial transformations, rotations, and reflections with respect to the estimated center of symmetry. They are valid without any moment assumption. For adequately chosen scores, they are locally asymptotically maximin (in the Le Cam sense) at given radial densities. They are strictly distribution-free when the center of symmetry is specified, and asymptotically so, when it has to be estimated. The multivariate ranks used throughout are those of the distances—in the metric associated with the null value V0 of the shape matrix—between the observations and the (estimated) center of the distribution. Local powers (against elliptical alternatives) and asymptotic relative efficiencies (AREs) are derived with respect to the adjusted Mauchly test (a modified version proposed by Muirhead and Waternaux (1980) of the Gaussian likelihood ratio procedure) or, equivalently, with respect to (an extension of) John (1971)’s test for sphericity. For Gaussian scores, these AREs are uniformly larger than one, irrespective of actual underlying radial densities. Necessary and/or sufficient conditions for consistency under nonlocal, possibly non-elliptical alternatives, are given. Small-sample performances are investigated via a Monte-Carlo study.