Flexible integrated functional depths

Germain Van Bever, Pauliina Ilmonen, Stanislav Nagy, Sami Helander, Lauri Viitasaari

Research output: Contribution to journalArticlepeer-review


This paper develops a new class of functional depths. A generic member of this class is coined Jth order kth moment integrated depth. It is based on the distribution of the cross-sectional halfspace depth of a function in the marginal evaluations (in time) of the random process. Asymptotic properties of the proposed depths are provided: we show that they are uniformly consistent and satisfy an inequality related to the law of the iterated logarithm. Moreover, limiting distributions are derived under mild regularity assumptions. The versatility displayed by the new class of depths makes them particularly amenable for capturing important features of functional distributions. This is illustrated in supervised learning, where we show that the corresponding maximum depth classifiers outperform classical competitors.

Original languageEnglish
Pages (from-to)673-701
Number of pages29
JournalBernoulli : a journal of mathematical statistics and probability
Issue number1
Publication statusPublished - Feb 2021


  • Asymptotics
  • Data depth
  • Functional data analysis
  • Integrated depths
  • Supervised classification


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