Tyler shape depth

Abstract

SummaryIn many problems from multivariate analysis, the parameter of interest is a shape matrix: a normalized version of the corresponding scatter or dispersion matrix. In this article we propose a notion of depth for shape matrices that involves data points only through their directions from the centre of the distribution. We refer to this concept as 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 due to Tyler (1987). Besides estimation, shape depth, like its Tyler antecedent, also allows hypothesis testing on shape. Its main benefit, however, lies in the ranking of the shape matrices it provides, the practical relevance of which is illustrated by applications to principal component analysis and 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, and the existence of a deepest shape matrix, and we 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.