Consistency of non-integrated depths for functional data

Irène Gijbels and Stanislav Nagy

Journal of Multivariate Analysis, 2015, vol. 140, issue C, 259-282

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 Φ-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 (search for similar items in EconPapers)
Date: 2015
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Citations: View citations in EconPapers (7)

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DOI: 10.1016/j.jmva.2015.05.012

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