Title: Data depth for discontinuous functions and random sets
Authors: Stanislav Nagy - Charles University (Czech Republic) [presenting]
Abstract: Data depth is a mapping, which to a point in a multivariate vector space $s\in S$ and a probability measure $P$ on $S$ assigns a number $D(s;P)$ describing how central $s$ is with respect to $P$, in an attempt to generalize quantiles to multivariate data. For $S$ having infinite dimension, depth is typically considered only for $S$ being the Banach space of continuous functions over a compact interval. We explore possibilities of extension of known depth functionals beyond this simplest setting. We discuss definitions, theoretical properties, and consistency/measurability issues connected with a straightforward generalization of commonly used depth functionals towards multidimensional random mappings which may lack continuity, be observed discretely, or be contaminated with additive noise.