Integrated depth for functional data: statistical properties and consistency

¹ KU Leuven, Department of Mathematics and Leuven Statistics Research Center (LStat), Celestijnenlaan 200b, box 2400, 3001 Leuven, Belgium.
² Charles University in Prague, Faculty of Mathematics and Physics, Department of Probability and Math. Statistics, Sokolovská 83, Praha 8, 186 75 Czech Republic.
hlubinka@karlin.mff.cuni.cz

Received: 22 December 2014
Accepted: 24 February 2016

Abstract

Several depths suitable for infinite-dimensional functional data that are available in the literature are of the form of an integral of a finite-dimensional depth function. These functionals are characterized by projecting functions into low-dimensional spaces, taking finite-dimensional depths of the projected quantities, and finally integrating these projected marginal depths over a preset collection of projections. In this paper, a general class of integrated depths for functions is considered. Several depths for functional data proposed in the literature during the last decades are members of this general class. A comprehensive study of its most important theoretical properties, including measurability and consistency, is given. It is shown that many, but not all, properties of the integrated depth are shared with the finite-dimensional depth that constitutes its building block. Some pending measurability issues connected with all integrated depth functionals are resolved, a broad new notion of symmetry for functional data is proposed, and difficulties with respect to consistency results are identified. A general universal consistency result for the sample depth version, and for the generalized median, for integrated depth for functions is derived.