Unform in Bandwith of the Conditional Distribution Function with Functional Explanatory Variable: The Case of Spatial Data with the K Nearest Neighbour Method

Wahiba Bouabsa

doi:10.15611/eada.2022.2.03

Abstract

In this paper the author introduced a new conditional distribution function estimator, in short (cdf), when the co-variables are functional in nature. This estimator is a mix of both procedures the k Nearest Neighbour method and the spatial functional estimation.

References

Almanjahie, I., Aissiri, K., Laksaci, A., and Chiker el Mezouar, Z. (2020). The k nearest neighbors smoothing of the relative-error regression with functional regressor. Communications in Statistics – Theory and Methods, 356(10), 1-14.
Search in Google Scholar Back to article
Attouch, M., Bouabsa, W., and Chiker el Mozoaur, Z. (2018). The k-nearest neighbors estimation of the conditional mode for functional data under dependency. International Journal of Statistics & Economics, 19(1), 48-60.
Search in Google Scholar Back to article
Attouch, M., and Bouabsa, W. (2013). The k-nearest neighbors estimation of the conditional mode for functional data. Rev. Roumaine Math. Pures Appl., 58(4), 393-415.
Search in Google Scholar Back to article
Attouch, M., Laksaci, A., and Messabihi, N. (2015). Nonparametric relative error regression for spatial random variables, Statistical Papers., 58(4), 987-1008.
Search in Google Scholar Back to article
Beirlant, J., Berlinet, A., and Biau, G. (2018). Higher order estimation at Lebesgue points. Ann. Inst. Statist. Math., 90(60), 651-677.
Search in Google Scholar Back to article
Biau, G., and Cadre, B. (2004). Nonparametric spatial prediction. Stat. Inference Stoch. Process, 7(9), 327-349.10.1023/B:SISP.0000049116.23705.88
Search in Google Scholar Back to article
Bosq, D. (2000). Linear processes in function spaces: Theory and applications. Lecture Notes in Statistics, (149).10.1007/978-1-4612-1154-9_8
Search in Google Scholar Back to article
Bouabsa, W. (2021). Nonparametric relative error estimation via functional regressor by the k Nearest neighbors smoothing under truncation random. Applications and Applied Mathematics: An International Journal (AAM), 16(1), 97-116.
Search in Google Scholar Back to article
Burba, F., Ferraty, F., and Vieu, P. (2009). k-nearest neighbour method in functional nonparametric regression. J. Nonparametr. Stat., 21(4), 453-469.10.1080/10485250802668909
Search in Google Scholar Back to article
Carbon, M., Francq, C., and Tran, L.T. (2007). Kernel regression estimation for random fields. J. Statist. Plann. Inference, 137(20), 778-798.10.1016/j.jspi.2006.06.008
Search in Google Scholar Back to article
Carbon, M., Tran, L.T and Wu, B. (1997). Kernel density estimation for random fields. Statist. Probab. Lett., 36(12), 115-125.10.1016/S0167-7152(97)00054-0
Search in Google Scholar Back to article
Cover, T. M. (1968). Estimation by the nearest neighbor rule. IEEE Trans. Inform. Theory, 7(14), 50-55.10.1109/TIT.1968.1054098
Search in Google Scholar Back to article
Collomb, G. (1981). Estimation non paramétrique de la régression: Revue bibliographique. Internat. Statist. Rev., 18(49), 75-93.10.2307/1403039
Search in Google Scholar Back to article
Cressie, N.A. (1991). Statistics for spatial data. New York: Wiley.
Search in Google Scholar Back to article
Dabo-Niang, S., and Yao, A. F. (2007). Kernel regression estimation for continuous spatial processes. Math. Methods Statist., 16(15), 298-317.10.3103/S1066530707040023
Search in Google Scholar Back to article
Devroye, L., Györfi, L., Krzyzak, A. and Lugosi, G. (1994). On the strong universal consistency of nearest neighbor regression function estimates. Ann. Statist., 20(22), 1371-1385.10.1214/aos/1176325633
Search in Google Scholar Back to article
Devroye, L. and Wagner, T. (1977). The strong uniform consistency of nearest neighbor density. Ann. Statist., 10(5), 536-540.10.1214/aos/1176343851
Search in Google Scholar Back to article
Devroye, L. and Wagner, T. (1982). Nearest neighbor methods in discrimination. In Classification, pattern recognition and reduction of dimensionality. Handbook of Statistics, (2).10.1016/S0169-7161(82)02011-2
Search in Google Scholar Back to article
Dony, J., and Einmahl, U. (2009). Uniformin Bandwidth consistency of Kernel regression estimators at a fixed point. High Dimensional Probability. V: The Luminy Volume, 53(5), 308-325.
Search in Google Scholar Back to article
Einmahl, U., and Mason, D. (2005). Uniform in bandwidth consistency of Kernel-type function estimators. The Annals of Statistics, 33(3), 1380-403.10.1214/009053605000000129
Search in Google Scholar Back to article
El Machkouri, M., and Stoica, R. (2010). Asymptotic normality of kernel estimates in a regression model for random fields. J Nonparametric Stat., 22(15), 955-971.10.1080/10485250903505893
Search in Google Scholar Back to article
Ferraty, F., and Vieu, P. (2006). Nonparametric functional data analysis. Theory and practice. New York: Springer-Verlag.
Search in Google Scholar Back to article
Ferraty, F., Rabhi, A. and Vieu, P. (2008). Estimation non-parametrique de la fonction de hasard avec variable explicative fonctionnelle. Rev. Roumaine Math. Pures Appl., 53(15), 1-18.
Search in Google Scholar Back to article
Guyon, X. (1987). Estimation d’un champ par pseudo-vraisemblance conditionnelle: Etude asymptotique et application au cas Markovien. Proceedings of the Sixth Franco-Belgian Meeting of Statisticians.
Search in Google Scholar Back to article
Györfi, L., Kohler, A., Krzyzak, and Walk, H. (2002). A distribution-free theory of nonparametric regression. New York: Springer.10.1007/b97848
Search in Google Scholar Back to article
Kara, L. Z., Laksaci, A., and Vieu, P. (2017). Data-driven kNN estimation in nonparametric functional data analysis. Journal of Multivariate Analysis, 153(85), 176-188.10.1016/j.jmva.2016.09.016
Search in Google Scholar Back to article
Kudraszow, L. and Vieu, P. (2013). Uniform consistency of kNN regressors for functional variables. Statistics and Probability Letters, 83(15), 1863-1870.10.1016/j.spl.2013.04.017
Search in Google Scholar Back to article
Laksaci, A., and Mechab, B. (2010). Estimation non parametrique de la fonction de hasard avec variable explicative fonctionelle: cas des données spaciales. Rev. Roumaine Math. Pures Appl., 55(1), 35-51.
Search in Google Scholar Back to article
Laloë, T. (2008). A k-nearest neighbor approach for functional regression. Statistics & Probability Letters, 78(10), 1189-1193.10.1016/j.spl.2007.11.014
Search in Google Scholar Back to article
Li, J., and Tran, L. T. (2007). Hazard rate estimation on random fields. J. Multivariate Anal., 98(15), 1337-1355.10.1016/j.jmva.2007.03.002
Search in Google Scholar Back to article
Li, J., and Tran, L. T. (2009). Nonparametric estimation of conditional expectation. J. Statist. Plann. Inference, 139(36), 164-175.10.1016/j.jspi.2008.04.023
Search in Google Scholar Back to article
Li, J. P. (1985). Strong convergence rates of error probability estimation in the nearest neighbor discrimination rule. J. Math., 15(5), 113-118.
Search in Google Scholar Back to article
Lian, H. (2011). Convergence of functional k-nearest neighbor regression estimate with functional responses. Electronic Journal of Statistics, 5(133), 31-40.10.1214/11-EJS595
Search in Google Scholar Back to article
Lu, Z. and Chen, X. (2004). Spatial kernel regression: weak consistency. Stat Probab Lett., 68(30), 125-136.10.1016/j.spl.2003.08.014
Search in Google Scholar Back to article
Moore, D., and Yackel, J. (1977). Consistency properties of nearest neighbor density function estimators. Ann. Statist., 20(5), 143-154.10.1214/aos/1176343747
Search in Google Scholar Back to article
Masry, E. (1986). Recursive probability density estimation for weakly dependent stationary processes. IEEE Trans. Inform. Theory., 32(16), 254-267.10.1109/TIT.1986.1057163
Search in Google Scholar Back to article
Ramsay, J. O., and Silverman, B. W. (2002). Applied functional data analysis. Methods and case studies. New York: Springer-Verlag.
Search in Google Scholar Back to article
Robinson, P. M. (2011). Asymptotic theory for nonparametric regression with spatial data. J Econom., 165(15), 5-19.10.1016/j.jeconom.2011.05.002
Search in Google Scholar Back to article
Tran, L.T. (1990). Kernel density estimation on random fields. J. Multivariate Anal., 34(25), 37-53.10.1016/0047-259X(90)90059-Q
Search in Google Scholar Back to article
Tran, T., Wehrens R., and Buydens, L. (2006). kNN-kernel density-based clustering for high-dimensional multivariate data. Comput. Statist. Data Anal., 51(81), 513-525.10.1016/j.csda.2005.10.001
Search in Google Scholar Back to article

Unform in Bandwith of the Conditional Distribution Function with Functional Explanatory Variable: The Case of Spatial Data with the K Nearest Neighbour Method

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