Skip to main content
Log in

On the Local Linear Modelization of the Conditional Distribution for Functional Data

  • Published:
Sankhya A Aims and scope Submit manuscript

Abstract

In this paper, we investigate the problem of the local linear estimation of the cumulative distribution function of a real random variable Y conditioned by a functional variable X (valued in an infinite dimensional space). The almost-complete and the mean square consistencies, with rates, of the constructed estimator are stated. We precise that the exact expression involved in the leading terms of the mean squared error is given. We point out, also, that the accuracy of our asymptotic results leads to interesting perspectives from a practical point of view. Thus, we discuss the features of our functional local modeling and the applicability of our asymptotic result on some statistical problems such as the choice of the smoothing parameters and the determination of confidence intervals. Moreover, a simulation study has been conducted in order to highlight, on a finite sample, the superiority of our method to the standard kernel method, in the functional framework.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  • Barrientos-Marin, J., Ferraty, F. and Vieu, P. (2010). Locally Modelled Regression and Functional Data. J. of Nonparametric Statistics 22, 617–632.

    Article  MathSciNet  MATH  Google Scholar 

  • Benhenni, K., Ferraty, F., Rachdi, M. and Vieu, P. (2007). Local smoothing regression with functional data. Computational Statistics 22, 353–369.

    Article  MathSciNet  Google Scholar 

  • Benhenni, K., Griche-Hedli, S. and Rachdi, M (2010). Estimation of the regression operator from functional fixed-design with correlated errors. J. Multivariate Anal. 101, 476–490.

    Article  MathSciNet  MATH  Google Scholar 

  • Boj, E., Delicado, P. and Fortiana, J. (2010). Distance-based local linear regression for functional predictors. Computational Statistics and Data Analysis 54, 429–437.

    Article  MathSciNet  MATH  Google Scholar 

  • Bosq, D. (2000) Linear Processes in Function Spaces: Theory and applications. Lecture Notes in Statistics, 149, Springer.

  • Baìllo, A. and Grané, A. (2009). Local linear regression for functional predictor and scalar response. J. of Multivariate Analysis 100, 102–111.

    Article  MATH  Google Scholar 

  • De Gooijer, J.G. and Gannoun, A. (2000). Nonparametric conditional predictive regions for time series. Computational Statistics & Data Analysis 33, 259–275.

    Article  MATH  Google Scholar 

  • Demongeot, J., Laksaci, A., Madani, F. and Rachdi, M. (2011) A fast functional locally modeled conditional density and mode for functional time-series. Recent Advances in Functional Data Analysis and Related Topics, Contributions to Statistics, 2011, Pages 85-90, doi:10.1007/978-3-7908-2736-1_13, Physica-Verlag/Springer.

  • Demongeot, J., Laksaci, A., Madani, F. and Rachdi, M. (2013). Functional data: local linear estimation of the conditional density and its application. Statistics 47, 26–44.

    Article  MathSciNet  MATH  Google Scholar 

  • El Methni, M. and Rachdi, M. (2011). Local weighted average estimation of the regression operator for functional data. Communications in Statistics-Theory and Methods 40, 3141–3153.

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, J. and Yao, Q. (2003). Nolinear Time Series: Nonparametric and Parametric Methods. Springer-Verlag, New York.

    Google Scholar 

  • Fan, J. (1992). Design-adaptive nonparametric regression. J. Amer. Statist. Assoc. 87, 998–1004.

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, J., Yao, Q. and Tong, H. (1996). Estimation of conditional densities and sensitivity measures in nonlinear dynamical systems. Biometrika 83, 189–206.

    Article  MathSciNet  MATH  Google Scholar 

  • Fan, J. and Gijbels, I. (1996). Local Polynomial Modelling and its Applications. Chapman & Hall, London.

    MATH  Google Scholar 

  • Fan, J. and Yim, T.-H. (2000). A cross-validation method for estimating conditional densities. Biometrika 91, 819–834.

    Article  MathSciNet  Google Scholar 

  • Ferraty, F., Laksaci, A., Tadj, A. and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables. J. of Statistical Planning and Inference 140, 335–352.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferraty, F., Van Keilegom, I. and Vieu, P. (2010). On the Validity of the Bootstrap in Non-Parametric Functional Regression. Scandinavian Journal of Statistics 37, 286–306.

    Article  MATH  Google Scholar 

  • Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical aspects. Aust. N. Z. J. Stat. 49, 267–286.

    Article  MathSciNet  MATH  Google Scholar 

  • Ferraty, F. and Vieu, P. (2006). Nonparametric functional data analysis: Theory and Practice. Springer Series in Statistics, New York.

    Google Scholar 

  • Ferraty, F., Laksaci, A. and Vieu, P. (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models. Stat. Inference Stoch. Process. 9, 47–76.

    Article  MathSciNet  MATH  Google Scholar 

  • Hall, P., Wolef, R.C.L. and Yao, Q. (1999). Methods for estimating a conditional distribution function. J. Amer. Statist. Assoc. 94, 154–163.

    Article  MathSciNet  MATH  Google Scholar 

  • Jirina, M. (1962). Conditional probabilities on σ-algebras with countable basis. Select. Transl. Math. Stat. Probab. 2, 79–86.

    MathSciNet  MATH  Google Scholar 

  • Jirina, M. (1959). On regular conditional probabilities. Czechoslovak Mathematical Journal 9, 445–451.

    MathSciNet  Google Scholar 

  • Laksaci, A., Madani, F. and Rachdi, M. (2013). Kernel conditional density estimation when the regressor is valued in a semi-metric space. Commun. in Statist.–Theory and Meth. 42, 3544–3570.

    Article  MathSciNet  MATH  Google Scholar 

  • Marron, J.S. and Wand, M.P. (1992). Exact mean integrated squared error. Ann. Statist. 20, 1919–1932.

    Article  MathSciNet  Google Scholar 

  • Ouassou, I. and Rachdi, M. (2010). Stein type estimation of the regression operator for functional data. Advances and Applications in Statistical Sciences 1, 233–250.

    MathSciNet  MATH  Google Scholar 

  • Polonik, W. and Yao, Q. (2008). Testing for multivariate volatility functions using minimum volume sets and inverse regression. Extended version of paper in Journal of Econometrics 147, 151–162.

    MathSciNet  Google Scholar 

  • Rachdi, M., Laksaci, A., Demongeot, J., Abdali, A. and Madani, F. (2014). Theoretical and practical aspects on the quadratic error in the local linear estimation of the conditional density for functional data. Computational Statistics and Data Analysis 73, 53–68.

    Article  MathSciNet  Google Scholar 

  • Rachdi, M. and Vieu, P. (2007). Nonparametric regression for functional data: automatic smoothing parameter selection. J. Stat. Plann. Inference. 137, 2784–2801.

    Article  MathSciNet  MATH  Google Scholar 

  • Ramsay, J.O. and Silverman, B.W. (2002). Applied functional data analysis: Methods and case studies. Springer Series in Statistics, New York.

    Google Scholar 

  • Ramsay, J.O. and Silverman, B.W. (2002). Functional data analysis. Springer Series in Statistics, New York.

    MATH  Google Scholar 

  • Uspensky, J.V. (1937). Introduction to Mathematical Probability. McGraw-Hill Book Company.

  • Yu, K. and Jones, M.C. (1997). A comparison of local constant and local linear regression quantile estimators. Computational Statistics and Data Analysis 25, 159–166.

    Article  MATH  Google Scholar 

  • Yu, K. and Jones, M. C. (1998). Local Linear Quantile Regression. J. Amer. Statist. Assoc. 93, 228–237.

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Ali Laksaci.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Demongeot, J., Laksaci, A., Rachdi, M. et al. On the Local Linear Modelization of the Conditional Distribution for Functional Data. Sankhya A 76, 328–355 (2014). https://doi.org/10.1007/s13171-013-0050-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s13171-013-0050-z

Keywords and phrases.

AMS (2000) subject classification.

Navigation