Tyler shape depth

In many problems from multivariate analysis, including principal component analysis, canonical correlation analysis, and the problem of testing for sphericity, the parameter of interest is a shape matrix, that is, a normalized version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices. The concept is of a sign nature in the sense that it involves data points only through their directions from the center of the distribution. We use the terminology "Tyler shape depth" since the resulting estimator of shape, namely the deepest shape matrix, is the median-based counterpart of the celebrated M-estimator of shape from Tyler (1987a). Beyond estimation, shape depth, like its Tyler antecedent, also allows to conduct hypothesis testing on shape. The main benefit of shape depth, however, lies in the resulting ranking of shape matrices it provides, whose practical relevance is illustrated in two applications: the first one in robust principal component analysis and the second one in shape-based outlier detection. To fully grasp the proposed concept, we study the invariance, quasi-concavity and continuity properties of Tyler shape depth, as well as the topological and boundedness properties of the corresponding depth regions. We also study existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko-Cantelli-type result and establish the almost sure consistency of the deepest shape matrix estimator.