  Nondegenerate covariance, correlation and spectral density matrices are
necessarily symmetric or Hermitian and positive definite. The main contribution
of this paper is the development of statistical data depths for collections of
Hermitian positive definite matrices by exploiting the geometric structure of
the space as a Riemannian manifold. The depth functions allow one to naturally
characterize most central or outlying matrices, but also provide a practical
framework for inference in the context of samples of positive definite
matrices. First, the desired properties of an intrinsic data depth function
acting on the space of Hermitian positive definite matrices are presented.
Second, we propose two computationally fast pointwise and integrated data depth
functions that satisfy each of these requirements and investigate several
robustness and efficiency aspects. As an application, we construct depth-based
confidence regions for the intrinsic mean of a sample of positive definite
matrices, which is applied to the exploratory analysis of a collection of
covariance matrices associated to a multicenter research trial.
