Smooth kNN Local Linear Estimation of the Conditional Distribution Function

Previous works were dedicated to the functional k-Nearest Neighbors (kNN) and the local linearity method estimations of a regression operator. In this paper, a sequence pair of (Xi,Yi)i=1,…,n of functional mixing observations are considered. We treat the local linear estimation of the cumulative function of Yi given functional input variable Xi. Precisely, we combine the kNN method with the local linear algorithm to construct a new and fast efficiency estimator of the conditional distribution function. The main purpose of this paper is to prove the strong convergence of the constructed estimator under mixing conditions. An application to the functional times series prediction is used to compare our proposed estimator with the existing competitive estimators, and show its efficiency and superiority.

[1]  F. Al-Awadhi,et al.  k‐Nearest neighbors local linear regression for functional and missing data at random , 2020, Statistica Neerlandica.

[2]  Frédéric Ferraty,et al.  Locally modelled regression and functional data , 2010 .

[3]  Mustapha Rachdi,et al.  Uniform consistency in number of neighbors of the kNN estimator of the conditional quantile model , 2021, Metrika.

[4]  Thomas Laloë.  A k-nearest neighbor approach for functional regression , 2008 .

[5]  Amparo Baíllo,et al.  Local linear regression for functional predictor and scalar response , 2009, J. Multivar. Anal..

[6]  Mustapha Rachdi,et al.  Data-driven kNN estimation in nonparametric functional data analysis , 2017, J. Multivar. Anal..

[7]  Mustapha Rachdi,et al.  Computational aspects of the kNN local linear smoothing for some conditional models in high dimensional statistics , 2021 .

[8]  Zhengyan Lin,et al.  Asymptotic normality of locally modelled regression estimator for functional data , 2016 .

[9]  I. Almanjahie,et al.  FDA: strong consistency of the kNN local linear estimation of the functional conditional density and mode , 2018, Journal of Nonparametric Statistics.

[10]  P. Vieu,et al.  k-Nearest Neighbour method in functional nonparametric regression , 2009 .

[11]  kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case , 2018, Comptes Rendus Mathematique.

[12]  Nadia L. Kudraszow,et al.  Uniform consistency of kNN regressors for functional variables , 2013 .

[13]  Jan G. De Gooijer,et al.  Nonparametric conditional predictive regions for time series , 2000 .

[14]  Mustapha Rachdi,et al.  Spatial modelization: Local linear estimation of the conditional distribution for functional data , 2013 .

[15]  Alain Berlinet,et al.  Local linear regression for functional data , 2007, 0710.5218.