We propose a class of rank-based procedures for testing that the shape matrix$\mathbf{V}$ of an elliptical distribution (with unspecified center ofsymmetry, scale and radial density) has some fixed value ${\mathbf{V}}_0$; thisincludes, for ${\mathbf{V}}_0={\mathbf{I}}_k$, the problem of testing forsphericity as an important particular case. The proposed tests are invariantunder translations, monotone radial transformations, rotations and reflectionswith respect to the estimated center of symmetry. They are valid without anymoment assumption. For adequately chosen scores, they are locallyasymptotically maximin (in the Le Cam sense) at given radial densities. Theyare strictly distribution-free when the center of symmetry is specified, andasymptotically so when it must be estimated. The multivariate ranks usedthroughout are those of the distances--in the metric associated with the nullvalue ${\mathbf{V}}_0$ of the shape matrix--between the observations and the(estimated) center of the distribution. Local powers (against ellipticalalternatives) and asymptotic relative efficiencies (AREs) are derived withrespect to the adjusted Mauchly test (a modified version of the Gaussianlikelihood ratio procedure proposed by Muirhead and Waternaux [Biometrika 67(1980) 31--43]) or, equivalently, with respect to (an extension of) the testfor sphericity introduced by John [Biometrika 58 (1971) 169--174]. For Gaussianscores, these AREs are uniformly larger than one, irrespective of the actualradial density. Necessary and/or sufficient conditions for consistency undernonlocal, possibly nonelliptical alternatives are given. Finite sampleperformances are investigated via a Monte Carlo study.
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