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Nonparametric estimation of a surrogate density function in infinite-dimensional spaces

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  • Frédéric Ferraty
  • Nadia Kudraszow
  • Philippe Vieu

Abstract

A density function is generally not well defined in functional data context, but we can define a surrogate of a probability density, also called pseudo-density, when the small ball probability can be approximated by the product of two independent functions, one depending only on the centre of the ball. The aim of this paper is to study two kernel methods for estimating a surrogate probability density for functional data. We present asymptotic properties of these estimators: the convergence in probability and their rates. Simulations are given, including a functional version of smoother bootstrap selection of the parameters of the estimate.

Suggested Citation

  • Frédéric Ferraty & Nadia Kudraszow & Philippe Vieu, 2012. "Nonparametric estimation of a surrogate density function in infinite-dimensional spaces," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 24(2), pages 447-464.
    • Handle: RePEc:taf:gnstxx:v:24:y:2012:i:2:p:447-464
    DOI: 10.1080/10485252.2012.671943
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    References listed on IDEAS

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    1. Masry, Elias, 2005. "Nonparametric regression estimation for dependent functional data: asymptotic normality," Stochastic Processes and their Applications, Elsevier, vol. 115(1), pages 155-177, January.
    2. Cuevas, Antonio & Febrero, Manuel & Fraiman, Ricardo, 2006. "On the use of the bootstrap for estimating functions with functional data," Computational Statistics & Data Analysis, Elsevier, vol. 51(2), pages 1063-1074, November.
    3. Th. Gasser & P. Hall & B. Presnell, 1998. "Nonparametric estimation of the mode of a distribution of random curves," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 60(4), pages 681-691.
    4. Peter Hall, 2002. "Estimating and depicting the structure of a distribution of random functions," Biometrika, Biometrika Trust, vol. 89(1), pages 145-158, March.
    5. M'hamed Ezzahrioui & Elias Ould-Saïd, 2008. "Asymptotic normality of a nonparametric estimator of the conditional mode function for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 20(1), pages 3-18.
    6. Frédéric Ferraty & Ali Laksaci & Philippe Vieu, 2006. "Estimating Some Characteristics of the Conditional Distribution in Nonparametric Functional Models," Statistical Inference for Stochastic Processes, Springer, vol. 9(1), pages 47-76, May.
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    Cited by:

    1. Bongiorno, E.G. & Goia, A. & Vieu, P., 2020. "Estimating the complexity index of functional data: Some asymptotics," Statistics & Probability Letters, Elsevier, vol. 161(C).
    2. Sigve Hovda, 2014. "Using pseudometrics in kernel density estimation," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 26(4), pages 669-696, December.
    3. Aubin, Jean-Baptiste & Bongiorno, Enea G. & Goia, Aldo, 2022. "The correction term in a small-ball probability factorization for random curves," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    4. Bongiorno, Enea G. & Goia, Aldo, 2016. "Classification methods for Hilbert data based on surrogate density," Computational Statistics & Data Analysis, Elsevier, vol. 99(C), pages 204-222.
    5. Enea G. Bongiorno & Aldo Goia & Philippe Vieu, 2019. "Modeling functional data: a test procedure," Computational Statistics, Springer, vol. 34(2), pages 451-468, June.

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