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Nonparametric estimation of the mode of a distribution of random curves

Author

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  • Th. Gasser
  • P. Hall
  • B. Presnell

Abstract

Motivated by the need to develop meaningful empirical approximations to a ‘typical’ data value, we introduce methods for density and mode estimation when data are in the form of random curves. Our approach is based on finite dimensional approximations via generalized Fourier expansions on an empirically chosen basis. The mode estimation problem is reduced to a problem of kernel‐type multivariate estimation from vector data and is solved using a new recursive algorithm for finding the empirical mode. The algorithm may be used as an aid to the identification of clusters in a set of data curves. Bootstrap methods are employed to select the bandwidth.

Suggested Citation

  • 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.
    • Handle: RePEc:bla:jorssb:v:60:y:1998:i:4:p:681-691
    DOI: 10.1111/1467-9868.00148
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    Cited by:

    1. Dabo-Niang, S. & Guillas, S. & Ternynck, C., 2016. "Efficiency in multivariate functional nonparametric models with autoregressive errors," Journal of Multivariate Analysis, Elsevier, vol. 147(C), pages 168-182.
    2. Liu, Jicai & Zhang, Riquan & Zhao, Weihua & Lv, Yazhao, 2013. "A robust and efficient estimation method for single index models," Journal of Multivariate Analysis, Elsevier, vol. 122(C), pages 226-238.
    3. Kadiri Nadia & Rabhi Abbes & Bouchentouf Amina Angelika, 2018. "Strong uniform consistency rates of conditional quantile estimation in the single functional index model under random censorship," Dependence Modeling, De Gruyter, vol. 6(1), pages 197-227, November.
    4. Salim Bouzebda & Amel Nezzal & Tarek Zari, 2022. "Uniform Consistency for Functional Conditional U -Statistics Using Delta-Sequences," Mathematics, MDPI, vol. 11(1), pages 1-39, December.
    5. Obereder, Andreas & Scherzer, Otmar & Kovac, Arne, 2007. "Bivariate density estimation using BV regularisation," Computational Statistics & Data Analysis, Elsevier, vol. 51(12), pages 5622-5634, August.
    6. Sultana Didi & Salim Bouzebda, 2022. "Wavelet Density and Regression Estimators for Continuous Time Functional Stationary and Ergodic Processes," Mathematics, MDPI, vol. 10(22), pages 1-37, November.
    7. Hamza Daoudi & Zouaoui Chikr Elmezouar & Fatimah Alshahrani, 2023. "Asymptotic Results of Some Conditional Nonparametric Functional Parameters in High-Dimensional Associated Data," Mathematics, MDPI, vol. 11(20), pages 1-24, October.
    8. Dabo-Niang, Sophie & Ferraty, Frederic & Vieu, Philippe, 2007. "On the using of modal curves for radar waveforms classification," Computational Statistics & Data Analysis, Elsevier, vol. 51(10), pages 4878-4890, June.
    9. Salim Bouzebda & Inass Soukarieh, 2022. "Non-Parametric Conditional U -Processes for Locally Stationary Functional Random Fields under Stochastic Sampling Design," Mathematics, MDPI, vol. 11(1), pages 1-69, December.
    10. Cristhian Leonardo Urbano-Leon & Manuel Escabias & Diana Paola Ovalle-Muñoz & Javier Olaya-Ochoa, 2023. "Scalar Variance and Scalar Correlation for Functional Data," Mathematics, MDPI, vol. 11(6), pages 1-20, March.
    11. Lihong Wang, 2020. "Nearest neighbors estimation for long memory functional data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 29(4), pages 709-725, December.
    12. 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.
    13. Hsu, Chih-Yuan & Wu, Tiee-Jian, 2013. "Efficient estimation of the mode of continuous multivariate data," Computational Statistics & Data Analysis, Elsevier, vol. 63(C), pages 148-159.
    14. 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.
    15. 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.
    16. Ferraty, F. & Vieu, P., 2003. "Curves discrimination: a nonparametric functional approach," Computational Statistics & Data Analysis, Elsevier, vol. 44(1-2), pages 161-173, October.
    17. 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.
    18. Said Attaoui, 2014. "Strong uniform consistency rates and asymptotic normality of conditional density estimator in the single functional index modeling for time series data," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 98(3), pages 257-286, July.
    19. 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).
    20. Salim Bouzebda & Boutheina Nemouchi, 2023. "Weak-convergence of empirical conditional processes and conditional U-processes involving functional mixing data," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 33-88, April.
    21. Laib, Naâmane & Louani, Djamal, 2010. "Nonparametric kernel regression estimation for functional stationary ergodic data: Asymptotic properties," Journal of Multivariate Analysis, Elsevier, vol. 101(10), pages 2266-2281, November.

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