Tyler shape depth

In many problems from multivariate analysis, 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 that 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 M-estimator of shape of Tyler (1987). Beyond estimation, shape depth, like its Tyler antecedent, also allows hypothesis testing on shape. Its main benefit, however, lies in the ranking of shape matrices it provides, whose practical relevance is illustrated in principal component analysis and in shape-based outlier detection. We study the invariance, quasi-concavity and continuity properties of Tyler shape depth, the topological and boundedness properties of the corresponding depth regions, existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. Finally, we derive a Glivenko-Cantelli-type result and establish almost sure consistency of the deepest shape matrix estimator.