Tyler Shape Depth

In many problems from multivariate analysis (principal component analysis, testing for sphericity, etc.), the parameter of interest is a shape matrix, that is, a normalised version of the corresponding scatter or dispersion matrix. In this paper, we propose a depth concept for shape matrices which 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 depth-based counterpart of the celebrated M-estimator of shape from Tyler (1987). We in- vestigate 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 study existence of a deepest shape matrix and prove Fisher consistency in the elliptical case. We derive a Glivenko-Cantelli-type result and establish the almost sure consistency of the deepest shape matrix estimator. We also consider depth-based tests for shape and investigate their finite-sample per- formances through simulations. Finally, we illustrate the practical relevance of the proposed depth concept on a real data example.