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Consistency of non-integrated depths for functional data

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  • Gijbels, Irène
  • Nagy, Stanislav

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.

Suggested Citation

  • Gijbels, Irène & Nagy, Stanislav, 2015. "Consistency of non-integrated depths for functional data," Journal of Multivariate Analysis, Elsevier, vol. 140(C), pages 259-282.
    • Handle: RePEc:eee:jmvana:v:140:y:2015:i:c:p:259-282
    DOI: 10.1016/j.jmva.2015.05.012
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    1. Sara López-Pintado & Ying Sun & Juan Lin & Marc Genton, 2014. "Simplicial band depth for multivariate functional data," Advances in Data Analysis and Classification, Springer;German Classification Society - Gesellschaft für Klassifikation (GfKl);Japanese Classification Society (JCS);Classification and Data Analysis Group of the Italian Statistical Society (CLADAG);International Federation of Classification Societies (IFCS), vol. 8(3), pages 321-338, September.
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    Cited by:

    1. Daniel Kosiorowski & Dominik Mielczarek & Jerzy P. Rydlewski, 2018. "Forecasting of a Hierarchical Functional Time Series on Example of Macromodel for the Day and Night Air Pollution in Silesia Region - A Critical Overview," Central European Journal of Economic Modelling and Econometrics, Central European Journal of Economic Modelling and Econometrics, vol. 10(1), pages 53-73, March.
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    3. Nagy, Stanislav & Gijbels, Irène & Hlubinka, Daniel, 2016. "Weak convergence of discretely observed functional data with applications," Journal of Multivariate Analysis, Elsevier, vol. 146(C), pages 46-62.
    4. Nagy, Stanislav & Ferraty, Frédéric, 2019. "Data depth for measurable noisy random functions," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 95-114.
    5. Nagy, Stanislav, 2017. "Integrated depth for measurable functions and sets," Statistics & Probability Letters, Elsevier, vol. 123(C), pages 165-170.
    6. Daniel Kosiorowski & Dominik Mielczarek & Jerzy. P. Rydlewski, 2017. "Forecasting of a Hierarchical Functional Time Series on Example of Macromodel for Day and Night Air Pollution in Silesia Region: A Critical Overview," Papers 1712.03797, arXiv.org.
    7. Daniel Kosiorowski & Dominik Mielczarek & Jerzy P. Rydlewski, 2017. "Aggregated moving functional median in robust prediction of hierarchical functional time series - an application to forecasting web portal users behaviors," Papers 1710.02669, arXiv.org, revised Jul 2018.

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