A Chernoff–Savage result for shape:On the non-admissibility of pseudo-Gaussian methods

Chernoff and Savage [Asymptotic normality and efficiency of certain non-parametric tests, Ann. Math. Statist. 29 (1958) 972–994] established that, in the context of univariate location models, Gaussian-score rank-based procedures uniformly dominate—in terms of Pitman asymptotic relative efficiencies—their pseudo-Gaussian parametric counterparts. This result, which had quite an impact on the success and subsequent development of rank-based inference, has been extended to many location problems, including problems involving multivariate and/or dependent observations. In this paper, we show that this uniform dominance also holds in problems for which the parameter of interest is the shape of an elliptical distribution. The Pitman non-admissibility of the pseudo-Gaussian maximum likelihood estimator for shape and that of the pseudo-Gaussian likelihood ratio test of sphericity follow.