IDEAS home Printed from https://ideas.repec.org/a/spr/stmapp/v26y2017i1d10.1007_s10260-016-0356-9.html
   My bibliography  Save this article

Rate of uniform consistency for a class of mode regression on functional stationary ergodic data

Author

Listed:
  • Mohamed Chaouch

    (United Arab Emirates University)

  • Naâmane Laïb

    (Université de Paris 6
    EISTI)

  • Djamal Louani

    (Université de Paris 6
    Universite de Reims-Champagne-Ardenne)

Abstract

The aim of this paper is to study the asymptotic properties of a class of kernel conditional mode estimates whenever functional stationary ergodic data are considered. To be more precise on the matter, in the ergodic data setting, we consider a random elements (X, Z) taking values in some semi-metric abstract space $$E\times F$$ E × F . For a real function $$\varphi $$ φ defined on the space F and $$x\in E$$ x ∈ E , we consider the conditional mode of the real random variable $$\varphi (Z)$$ φ ( Z ) given the event “ $$X=x$$ X = x ”. While estimating the conditional mode function, say $$\theta _\varphi (x)$$ θ φ ( x ) , using the well-known kernel estimator, we establish the strong consistency with rate of this estimate uniformly over Vapnik–Chervonenkis classes of functions $$\varphi $$ φ . Notice that the ergodic setting offers a more general framework than the usual mixing structure. Two applications to energy data are provided to illustrate some examples of the proposed approach in time series forecasting framework. The first one consists in forecasting the daily peak of electricity demand in France (measured in Giga-Watt). Whereas the second one deals with the short-term forecasting of the electrical energy (measured in Giga-Watt per Hour) that may be consumed over some time intervals that cover the peak demand.

Suggested Citation

  • Mohamed Chaouch & Naâmane Laïb & Djamal Louani, 2017. "Rate of uniform consistency for a class of mode regression on functional stationary ergodic data," Statistical Methods & Applications, Springer;Società Italiana di Statistica, vol. 26(1), pages 19-47, March.
    • Handle: RePEc:spr:stmapp:v:26:y:2017:i:1:d:10.1007_s10260-016-0356-9
    DOI: 10.1007/s10260-016-0356-9
    as

    Download full text from publisher

    File URL: http://link.springer.com/10.1007/s10260-016-0356-9
    File Function: Abstract
    Download Restriction: Access to the full text of the articles in this series is restricted.
    File URL: https://libkey.io/10.1007/s10260-016-0356-9?utm_source=ideas
    LibKey link: if access is restricted and if your library uses this service, LibKey will redirect you to where you can use your library subscription to access this item
    ---><---

    As the access to this document is restricted, you may want to search for a different version of it.

    References listed on IDEAS

    as
    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. 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.
    3. Goia, Aldo & May, Caterina & Fusai, Gianluca, 2010. "Functional clustering and linear regression for peak load forecasting," International Journal of Forecasting, Elsevier, vol. 26(4), pages 700-711, October.
    4. M'hamed Ezzahrioui & Elias Ould Saïd, 2010. "Some asymptotic results of a non‐parametric conditional mode estimator for functional time‐series data," Statistica Neerlandica, Netherlands Society for Statistics and Operations Research, vol. 64(2), pages 171-201, May.
    5. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. M. D. Ruiz-Medina & D. Miranda & R. M. Espejo, 2019. "Dynamical multiple regression in function spaces, under kernel regressors, with ARH(1) errors," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 28(3), pages 943-968, September.
    2. Bouzebda, Salim & Chaouch, Mohamed, 2022. "Uniform limit theorems for a class of conditional Z-estimators when covariates are functions," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    3. Ibrahim M. Almanjahie & Zoulikha Kaid & Ali Laksaci & Mustapha Rachdi, 2022. "Estimating the Conditional Density in Scalar-On-Function Regression Structure: k -N-N Local Linear Approach," Mathematics, MDPI, vol. 10(6), pages 1-16, March.
    4. Chaouch, Mohamed, 2019. "Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 129-148.

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Ling, Nengxiang & Xu, Qian, 2012. "Asymptotic normality of conditional density estimation in the single index model for functional time series data," Statistics & Probability Letters, Elsevier, vol. 82(12), pages 2235-2243.
    2. Ling, Nengxiang & Wang, Chao & Ling, Jin, 2016. "Modified kernel regression estimation with functional time series data," Statistics & Probability Letters, Elsevier, vol. 114(C), pages 78-85.
    3. Liu, Qiaojing & Zhao, Shoujiang, 2013. "Pointwise and uniform moderate deviations for nonparametric regression function estimator on functional data," Statistics & Probability Letters, Elsevier, vol. 83(5), pages 1372-1381.
    4. Mohamed Chaouch & Salah Khardani, 2015. "Randomly censored quantile regression estimation using functional stationary ergodic data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(1), pages 65-87, March.
    5. 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.
    6. Geenens, Gery, 2015. "Moments, errors, asymptotic normality and large deviation principle in nonparametric functional regression," Statistics & Probability Letters, Elsevier, vol. 107(C), pages 369-377.
    7. Attaoui, Said & Laksaci, Ali & Ould Said, Elias, 2011. "A note on the conditional density estimate in the single functional index model," Statistics & Probability Letters, Elsevier, vol. 81(1), pages 45-53, January.
    8. Bouzebda, Salim & Chaouch, Mohamed, 2022. "Uniform limit theorems for a class of conditional Z-estimators when covariates are functions," Journal of Multivariate Analysis, Elsevier, vol. 189(C).
    9. Goia, Aldo, 2012. "A functional linear model for time series prediction with exogenous variables," Statistics & Probability Letters, Elsevier, vol. 82(5), pages 1005-1011.
    10. Zhiyong Zhou & Zhengyan Lin, 2016. "Asymptotic normality of locally modelled regression estimator for functional data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 28(1), pages 116-131, March.
    11. Kamal Boukhetala & Jean-François Dupuy, 2019. "Modélisation Stochastique et Statistique Book of Proceedings," Post-Print hal-02593238, HAL.
    12. 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.
    13. Igor S. Borisov & Yuliana Yu. Linke & Pavel S. Ruzankin, 2021. "Universal weighted kernel-type estimators for some class of regression models," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 84(2), pages 141-166, February.
    14. Laksaci, Ali & Lemdani, Mohamed & Ould-Sad, Elias, 2009. "A generalized L1-approach for a kernel estimator of conditional quantile with functional regressors: Consistency and asymptotic normality," Statistics & Probability Letters, Elsevier, vol. 79(8), pages 1065-1073, April.
    15. Akkal Fatima & Kadiri Nadia & Rabhi Abbes, 2021. "Asymptotic Normality of Conditional Density and Conditional Mode in the Functional Single Index Model," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 25(1), pages 1-24, March.
    16. Oussama Bouanani & Saâdia Rahmani & Ali Laksaci & Mustapha Rachdi, 2020. "Asymptotic normality of conditional mode estimation for functional dependent data," Indian Journal of Pure and Applied Mathematics, Springer, vol. 51(2), pages 465-481, June.
    17. Aneiros, Germán & Cao, Ricardo & Fraiman, Ricardo & Genest, Christian & Vieu, Philippe, 2019. "Recent advances in functional data analysis and high-dimensional statistics," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 3-9.
    18. Chaouch, Mohamed, 2019. "Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors," Journal of Multivariate Analysis, Elsevier, vol. 170(C), pages 129-148.
    19. Joydeep Chowdhury & Probal Chaudhuri, 2020. "Convergence rates for kernel regression in infinite-dimensional spaces," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 72(2), pages 471-509, April.
    20. Krebs, Johannes T.N., 2019. "The bootstrap in kernel regression for stationary ergodic data when both response and predictor are functions," Journal of Multivariate Analysis, Elsevier, vol. 173(C), pages 620-639.

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:spr:stmapp:v:26:y:2017:i:1:d:10.1007_s10260-016-0356-9. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Sonal Shukla or Springer Nature Abstracting and Indexing (email available below). General contact details of provider: http://www.springer.com .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.