The $k$ nearest neighbors local linear estimator of functional conditional density when there are missing data

Year 2022, Volume: 51 Issue: 3, 914 - 931, 01.06.2022

İbrahim Almanjahie , Wafaa Mesfer Laksaci Ali

https://doi.org/10.15672/hujms.796694

Cited By: 1

Abstract

Our key aim is to propose effective estimators for the conditional probability density of a scalar response variable given a functional co-variable, where the response variable is considered to have missing data at random. Such estimators are constructed by combining the approaches of the local linear method and the kernel nearest neighborhood. The main feature of this estimation is the possibility to model the missing phenomena. Under less restrictive conditions, we show the strong consistency of the proposed estimators. To assess the efficacy of the developed estimators, empirical analysis as well as real data analyses are performed.

Keywords

functional data analysis, small ball probability, local linear method, k nearest neighbors, conditional density, almost complete convergence

Supporting Institution

King Khalid University

Project Number

R.G.P.2/68/41.

Thanks

The authors are very grateful to the Deanship of Scientific Research at King Khalid University, Kingdom of Saudi Arabia for supporting and funding this work through the research groups program under the project number R.G.P.2/68/41.

References

• [1] I.M. Almanjahie, Z. Chikr Elmezouar, B.A. Bachir, and Z. Kaid, Spatial local linear estimation of the L-1-conditional quantiles for functional regressors, Comm. Statist. Theory Methods 49 (23), 5666-5685, 2020.
• [2] I.M. Almanjahie, Z. Chikr Elmezouar, A. Laksaci and M. Rachdi, kNN local linear estimation of the conditional cumulative distribution function: Dependent functional data case, C. R. Math. 356 (10), 1036-1039, 2018.
• [3] G. Aneiros Pérez, R. Cao and P. Vieu, Editorial on the special issue on functional data analysis and related topics, Comput. Statist. 34 (2), 447-450, 2019.
• [4] M. Attouch and F. Belabed, (2014), The k nearest neighbors estimation of the conditional hazard function for functional data, REVSTAT 12 (3), 273-297, 2014.
• [5] M. Attouch and W. Bouabça, The k-nearest neighbors estimation of the conditional mode for functional data, Roumaine Math. Pures Appl. 58 (4), 393-415, 2013.
• [6] A. Baìllo and A. Grané, Local linear regression for functional predictor and scalar response, J. Multivariate Anal. 100 (1), 102-111, 2009.
• [7] J. Barrientos-Marin, F. Ferraty and P. Vieu, Locally modelled regression and functional data, J. Nonparametr. Stat. 22 (5), 617-632, 2010.
• [8] A. Benchiha and Z. Kaid, Local linear estimate for functional regression with missing data at random, Int. J. Math. Stat. 19, 22-33, 2018.
• [9] E. Boj, P. Delicado and J. Fortiana, Distance-based local linear regression for functional predictors, Comput. Statist. Data Anal. 54 (2), 429-437, 2010.
• [10] F. Burbea, F. Ferraty and P. Vieu, k-nearest neighbor method in functional non- parametric regression, J. Nonparametr. Stat. 21 (4), 453-469, 2009.
• [11] Z. Chikr Elmezouar, I.M. Almanjahie, A. Laksaci and M. Rachdi, FDA: strong consistency of the kNN local linear estimation of the functional conditional density and mode, J. Nonparametr. Stat. 31 (1), 175-195, 2019.
• [12] G. Collomb, W. Härdle and S. Hassani, A note on prediction via estimation of the conditional mode function, J. Statist. Plann. Inference 15, 227-236, 1987.
• [13] S. Dabo-Niang, Z. Kaid and A. Laksaci, Asymptotic properties of the kernel estimate of spatial conditional mode when the regressor is functional, AStA Adv. Stat. Anal. 99 (2), 131-160, 2015.
• [14] J. Demongeot, A. Laksaci, F. Madani and M. Rachdi, Functional data: local linear estimation of the conditional density and its application, Statistics 47 (1), 26-44, 2013.
• [15] S. Efromovich, Missing and modified data in nonparametric estimation with R examples, in Monographs on Statistics and Applied Probability, 156, CRC Press, 2018.
• [16] M. Ezzahrioui and E. Ould Saïd, Some asymptotic results of a non-parametric conditional mode estimator for functional time-series data, Stat. Neerl. 64 (2), 171-201, 2010.
• [17] F. Ferraty, A. Laksaci and P. Vieu, Estimating some characteristics of the conditional distribution in nonparametric functional models, Stat. Inference Stoch. Process. 9 (1), 47-76, 2006
• [18] F. Ferraty, M. Sued and P. Vieu, Mean estimation with data missing at random for functional covariables, Statistics 47 (4), 688-706, 2013.
• [19] F. Ferraty and P. Vieu, Nonparametric Functional Data Analysis: Theory and Practice, Springer-Verlag, 2006.
• [20] L. Kara-Zaitri, A. Laksaci, M. Rachdi and P. Vieu, Data-driven kNN estimation for various problems involving functional data, J. Multivariate Anal. 153, 176-188, 2017.
• [21] N. Kudraszow, and P. Vieu, Uniform consistency of kNN regressors for functional variables, Statist. Probab. Lett. 83 (8), 1863-1870, 2013.
• [22] A. Laksaci, Quadratic error of the kernel estimator of conditional density when the regressor is functional, C. R. Math. Acad. Sci. Paris 345 (3), 171-175, 2007.
• [23] A. Laksaci and A. Yousfate, Functional estimate of Markov transition operator density: discrete time case, C. R. Math. Acad. Sci. Paris 334 (11), 1035-1038, 2002.
• [24] H. Lian, Convergence of functional k-nearest neighbor regression estimate with functional responses, Electron. J. Stat. 5, 31-40, 2011.
• [25] N. Ling, Y. Liu and P. Vieu, Nonparametric regression estimation for functional stationary ergodic data with missing at random, J. Statist. Plann. Inference 162, 75-87, 2015.
• [26] N. Ling, Y. Liu and P. Vieu, Conditional mode estimation for functional stationary ergodic data with responses missing at random, Statistics 50 (5), 991-1013, 2016.
• [27] N. Ling, and P. Vieu, Nonparametric modelling for functional data: selected survey and tracks for future, Statistics 52 (4), 934-949, 2018.
• [28] D. Louani, and E. Ould-Saïd, Asymptotic normality of kernel estimators of the conditional mode under strong mixing hypothesis, J. Nonparametr. Stat. 11 (4), 413-442, 1999.
• [29] E. Miquel Becker, J. Christensen, C.S. Frederiksen and V.K Haugaard, Front-face fluorescence spectroscopy and chemometrics in analysis of yogurt: rapid analysis of riboflavin, J. Dairy Sci. 86 (8), 2508-2515, 2003.
• [30] A. Quintela-Del-Río and P. Vieu, A nonparametric conditional mode estimate, J. Nonparametr. Stat. 8 (3), 253-266, 1997.
• [31] M. Rachdi, A. Laksaci, I.M Almanjahie, and Z. Chikr Elmezouar, FDA: theoretical and practical efficiency of the local linear estimation based on the kNN smoothing of the conditional distribution when there are missing data, J. Stat. Comput. Simul. 90 (8), 1479-1495, 2020.
• [32] M. Rachdi, A. Laksaci, J. Demongeot, A. Abdali and F. Madani, Theoretical and practical aspects of the quadratic error in the local linear estimation of the conditional density for functional data, Comput. Statist. Data Anal. 73, 53-68, 2014.