On depth measures and dual statistics. A methodology for dealing with general data

A general depth measure, based on the use of one-dimensional linear continuous projections, is proposed. The applicability of this idea in different statistical setups (including inference in functional data analysis, image analysis and classification) is discussed. A special emphasis is made on the possible usefulness of this method in some statistical problems where the data are elements of a Banach space. The asymptotic properties of the empirical approximation of the proposed depth measure are investigated. In particular, its asymptotic distribution is obtained through U-statistics techniques. The practical aspects of these ideas are discussed through a small simulation study and a real-data example.