The Properties of Functional Depth

• Activity Number: 66
• Type: Topic Contributed
• Date/Time: Sunday, August 9, 2015 : 4:00 PM to 5:50 PM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Alicia Nieto-Reyes* and Heather Battey
• Companies: Universidad de Cantabria and Princeton University
• Keywords: functional data ; multivariate statistics ; partial observability ; robustness ; statistical depth

Abstract

A formal definition of statistical depth for functional data on the basis of six properties is provided, recognising topological features such as continuity, smoothness and contiguity. Amongst our depth defining properties is one that addresses the delicate challenge of inherent partial observability of functional data, with fulfilment giving rise to a minimal guarantee on the performance of the empirical depth beyond the idealised and practically infeasible case of full observability. As an incidental product, functional depths satisfying our definition achieve the robustness that is commonly ascribed to depth, despite the absence of a formal guarantee in the multivariate definition of depth. We demonstrate the fulfilment or otherwise of our properties for six widely used functional depth proposals, thereby providing a systematic basis for selection of a depth function.