Consistency of non-integrated depths for functional data

Abstract

In the analysis of functional data, the concept of data depth is of importance. Strong consistency of a sample version of a data depth is among the basic statistical properties that need to hold.

In this paper we discuss consistency properties of three popular types of functional depth: the band depth, the half-region depth and the infimal depth. The latter is a special case of the recently introduced general class of Phi-depths.

All three considered depth functions are of a non-integrated type. Counterexamples illustrate some problems with consistency results for these data depths.

The main contribution of this paper consists of providing sufficient conditions for consistency of these non-integrated data depths to hold.

Keywords

• Band depth
• Functional data
• Glivenko-Cantelli theorem
• Half-region depth
• Infimal depth
• Uniform consistency