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On the consistency of bootstrap methods in separable Hilbert spaces

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  • González-Rodríguez, Gil
  • Colubi, Ana

Abstract

Hilbert spaces are frequently used in statistics as a framework to deal with general random elements, specially with functional-valued random variables. The scarcity of common parametric distribution models in this context makes it important to develop non-parametric techniques, and among them, bootstrap has already proved to be specially valuable. The aim is to establish a methodology to derive consistency results for some usual bootstrap methods when working in separable Hilbert spaces. Naive bootstrap, bootstrap with arbitrary sample size, wild bootstrap, and more generally, weighted bootstrap methods, including double bootstrap and bootstrap generated by deterministic weights with the particular case of delete −h jackknife, will be proved to be consistent by applying the proposed methodology. The main results concern the bootstrapped sample mean, however since many usual statistics can be written in terms of means by considering suitable spaces, the applicability is notable. An illustration to show how to employ the approach in the context of a functional regression problem is included.

Suggested Citation

  • González-Rodríguez, Gil & Colubi, Ana, 2017. "On the consistency of bootstrap methods in separable Hilbert spaces," Econometrics and Statistics, Elsevier, vol. 1(C), pages 118-127.
    • Handle: RePEc:eee:ecosta:v:1:y:2017:i:c:p:118-127
    DOI: 10.1016/j.ecosta.2016.11.001
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    References listed on IDEAS

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    1. Frédéric Ferraty & Ingrid Van Keilegom & Philippe Vieu, 2010. "On the Validity of the Bootstrap in Non‐Parametric Functional Regression," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 37(2), pages 286-306, June.
    2. Kosorok, Michael R., 2003. "Bootstraps of sums of independent but not identically distributed stochastic processes," Journal of Multivariate Analysis, Elsevier, vol. 84(2), pages 299-318, February.
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    4. Gabrys, Robertas & Kokoszka, Piotr, 2007. "Portmanteau Test of Independence for Functional Observations," Journal of the American Statistical Association, American Statistical Association, vol. 102, pages 1338-1348, December.
    5. 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.
    6. Wang, Shaoli & Huang, Mian & Wu, Xing & Yao, Weixin, 2016. "Mixture of functional linear models and its application to CO2-GDP functional data," Computational Statistics & Data Analysis, Elsevier, vol. 97(C), pages 1-15.
    7. Yao, Fang & Muller, Hans-Georg & Wang, Jane-Ling, 2005. "Functional Data Analysis for Sparse Longitudinal Data," Journal of the American Statistical Association, American Statistical Association, vol. 100, pages 577-590, June.
    8. González-Rodríguez, Gil & Colubi, Ana & Gil, María Ángeles, 2012. "Fuzzy data treated as functional data: A one-way ANOVA test approach," Computational Statistics & Data Analysis, Elsevier, vol. 56(4), pages 943-955.
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    Cited by:

    1. Colubi, Ana & Ramos-Guajardo, Ana Belén, 2023. "Fuzzy sets and (fuzzy) random sets in Econometrics and Statistics," Econometrics and Statistics, Elsevier, vol. 26(C), pages 84-98.
    2. Ivanović, Blagoje & Milošević, Bojana & Obradović, Marko, 2020. "Comparison of symmetry tests against some skew-symmetric alternatives in i.i.d. and non-i.i.d. setting," Computational Statistics & Data Analysis, Elsevier, vol. 151(C).
    3. González–Rodríguez, Gil & Colubi, Ana & González–Manteiga, Wenceslao & Febrero–Bande, Manuel, 2024. "A consistent test of equality of distributions for Hilbert-valued random elements," Journal of Multivariate Analysis, Elsevier, vol. 202(C).
    4. Kokoszka, Piotr & Oja, Hanny & Park, Byeong & Sangalli, Laura, 2017. "Special issue on functional data analysis," Econometrics and Statistics, Elsevier, vol. 1(C), pages 99-100.

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