Statistical Inference for Functions: A Depth Approach

Abstract

Many fields provide naturally functions as observations. We propose a way to make inference for these functional data based on the idea of depth. First, we introduce a definition of depth for functions and analyse its properties. Moreover, we consider the finite dimensional version, which is an alternative to existing definitions of depth. Second, we define bands containing a fixed proportion of the deepest observations and use them to construct a scale function to describe dispersion. Finally, we present a rank test based on the depth ordering of the functions.