Nonparametric Inference Based on Depth for Multivariate Data

Statistical data depth is a nonparametric tool applicable to multivariate datasets in an attempt to generalize quantiles to complex data such as random vectors, random functions, or distributions on manifolds and graphs. The main idea is, for a general multivariate space M, to assign to a point x from M and a probability distribution P on M a real number D(x;P) characterizing how "centrally located" x is with respect to P. A point maximizing D(.;P) is then a generalization of the median to M-valued data, and the locus of points whose depth value is greater than a certain threshold constitutes the inner depth-quantile region corresponding to P.

In the thesis, we focus on data depth designed for infinite-dimensional spaces M and functional data. A review of depth functionals available in the literature is given. The emphasis of the exposition is put on the unification of these diverse concepts from the theoretical point of view. It is shown that most of the established depths fall into the general framework of projection-driven functionals of either integrated, or infimal type.

Based on the proposed methodology, characteristics and theoretical properties of all these depths can be evaluated simultaneously. The first part of the work is devoted to the investigation of these theoretical properties, mainly consistency and measurability, and conditions under which they can be guaranteed. It is shown that some of the most used well-established depths fail to meet these vital conditions, and hence cannot be considered suitable for statistical analysis. For the family of integrated depths for functional data, we present a comprehensive study of their most important theoretical properties, including a discussion on the desirable features that an infinite-dimensional depth functional should satisfy.

In the second part of the work we focus on extensions and applications of the proposed methodology. We deal with the important issues of discretely observed, and discontinuous functional data, and provide the theoretical background for the use of the data depth in these setups. Finally, a new depth capable of recognizing the shape properties of functional data is proposed, and studied.