Share:


Local linear modelling of the conditional distribution function for functional ergodic data

    Somia Ayad Affiliation
    ; Ali Laksaci Affiliation
    ; Saâdia Rahmani   Affiliation
    ; Rachida Rouane Affiliation

Abstract

The focus of functional data analysis has been mostly on independent functional observations. It is therefore hoped that the present contribution will provide an informative account of a useful approach that merges the ideas of the ergodic theory and the functional data analysis by using the local linear approach. More precisely, we aim, in this paper, to estimate the conditional distribution function (CDF) of a scalar response variable given a random variable taking values in a semimetric space. Under the ergodicity assumption, we study the uniform almost complete convergence (with a rate), as well as the asymptotic normality of the constructed estimator. The relevance of the proposed estimator is verified through a simulation study.

Keyword : ergodic sata, functional data, local linear estimator, conditional distribution function, nonparametric estimation, asymptotic properties

How to Cite
Ayad, S., Laksaci, A., Rahmani, S., & Rouane, R. (2022). Local linear modelling of the conditional distribution function for functional ergodic data. Mathematical Modelling and Analysis, 27(3), 360–382. https://doi.org/10.3846/mma.2022.14909
Published in Issue
Aug 12, 2022
Abstract Views
142
PDF Downloads
181
Creative Commons License

This work is licensed under a Creative Commons Attribution 4.0 International License.

References

S. Ayad, A. Laksaci, S. Rahmani and R. Rouane. On the local linear modelization of the conditional density for functional and ergodic data. METRON, 78(2):237– 254, 2020. https://doi.org/10.1007/s40300-020-00174-6

A. Baíllo and A. Grané. Local linear regression for functional predictor and scalar response. Journal of Multivariate Analysis, 100(1):102–111, 2009. https://doi.org/10.1016/j.jmva.2008.03.008

J. Barrientos-Marin, F. Ferraty and P. Vieu. Locally modelled regression and functional data. Journal of Nonparametric Statistics, 22(5):617–632, 2010. https://doi.org/10.1080/10485250903089930

O. Bouanani, A. Laksaci, M. Rachdi and S. Rahmani. Asymptotic normality of some conditional nonparametric functional parameters in high-dimensional statistics. Behaviormetrika, 46(1):199–233, 2019. https://doi.org/10.1007/s41237-018-0057-9

M. Chaouch, N. Laïb and E. Ould-Saïd. Nonparametric M-estimation for right censored regression model with stationary ergodic data. Statistical Methodology, 33:234–255, 2016.

M-Y. Cheng, J. Fan and J.S Marron. On automatic boundary corrections. The Annals of Statistics, 25(4):1691–1708, 1997. https://doi.org/10.1214/aos/1031594737

M. Delecroix and A.C. Rosa. Nonparametric estimation of a regression function and its derivatives under an ergodic hypothesis. Journal of Nonparametric Statistics, 6(4):367–382, 1996. https://doi.org/10.1080/10485259608832682

J. Demongeot, A. Naceri, A. Laksaci and M. Rachdi. Local linear regression modelization when all variables are curves. Statistics & Probability Letters, 121:37– 44, 2017. https://doi.org/10.1016/j.spl.2016.09.021

M. El Methni and M. Rachdi. Local weighted average estimation of the regression operator for functional data. Communications in Statistics-Theory and Methods, 40(17):3141–3153, 2011. https://doi.org/10.1080/03610921003778209

J. Fan and I. Gijbels. Local Polynomial Modelling and Its Applications. Chapman & Hall/CRC, 1996.

J. Fan and Q. Yao. Nonlinear time series: nonparametric and parametric methods. Springer, New York, 2003.

F. Ferraty, A. Laksaci, A. Tadj and P. Vieu. Rate of uniform consistency for nonparametric estimates with functional variables. Journal of Statistical Planning and Inference, 140(2):335–352, 2010. https://doi.org/10.1016/j.jspi.2009.07.019

F. Ferraty and Y. Romain. The Oxford handbook of functional data analaysis. Oxford University Press, 2011.

F. Ferraty and P. Vieu. Nonparametric functional data analysis: theory and practice. Springer Science & Business Media, 2006. https://doi.org/10.1007/0-387-36620-2

L. Kara-Zaitri, A. Laksaci, M. Rachdi and P. Vieu. Uniform in bandwidth consistency for various kernel estimators involving functional data. Journal of Nonparametric Statistics, 29(1):85–107, 2017.

P. Kokoszka and M. Reimherr. Introduction to functional data analysis. CRC press, 2017. https://doi.org/10.1201/9781315117416

N. Laib and D. Louani. Rates of strong consistencies of the regression function estimator for functional stationary ergodic data. Journal of Statistical Planning and Inference, 141(1):359–372, 2011. https://doi.org/10.1016/j.jspi.2010.06.009

N. Laïb and E. Ould-Saïd. A robust nonparametric estimation of the autoregression function under an ergodic hypothesis. Canadian Journal of Statistics, 28(4):817–828, 2000. https://doi.org/10.2307/3315918

A. Laksaci, M. Rachdi and S. Rahmani. Spatial modelization: local linear estimation of the conditional distribution for functional data. Spatial Statistics, 6:1–23, 2013. https://doi.org/10.1016/j.spasta.2013.04.004

N. Ling, S. Meng and P. Vieu. Uniform consistency rate of kNN regression estimation for functional time series data. Journal of Nonparametric Statistics, 31(2):451–468, 2019. https://doi.org/10.1080/10485252.2019.1583338

Z. Zhou and Z. Lin. Asymptotic normality of locally modelled regression estimator for functional data. Journal of Nonparametric Statistics, 28(1):116–131, 2016. https://doi.org/10.1080/10485252.2015.1114112