We consider the asymptotic properties of the nearest neighbor estimation for functional data. Under α-mixing and some regularity assumptions, we investigate the consistency and asymptotic normality of the nearest neighbor estimation for the nonparametric regression models with functional data. For the empirical data analysis, we consider three different distributions of the errors. The results show that the advantages of the nearest neighbor estimation lie in its easy computation, robustness and good performance under finite sample.
Wen Jie XING, Li Hong WANG.
Nearest Neighbor Estimation for Functional Data. Acta Mathematica Sinica, Chinese Series, 2014, 57(2): 339-350 https://doi.org/10.12386/A2014sxxb0034
{{custom_sec.title}}
{{custom_sec.title}}
{{custom_sec.content}}
参考文献
[1] Araujo A., Gine E., The Cental Limit Theorem for Real and Banach Valued Random Variables, Wiley, New York, 1980. [2] Attouch M., Laksaci A., Ould-Said E., Asymptotic distribution of robust estimator for functional nonparametric models, Commun. Statist. Theor. Meth., 2009, 38: 1317-1335. [3] Boente G., Fraiman R., Asymptotic distribution of robust estimators for nonparametric models from mixing process, Ann. Statist., 1990, 18: 891-906. [4] Boente G., Fraiman R., Robust nonparametric regression estimation for dependent observations, Ann. Statist., 1989, 17: 1242-1256. [5] Boente G., Fraiman R., Strong order of convergence and asymptotic distribution of nearest neighbor density estimates from dependent observations, Sankhya Ser. A, 1991, 53: 891-906. [6] Bosq D., Nonparametric Statistics for Stochastic Processes: Estimation and Prediction, Lecture Notes in Statistics, Second ed., Vol. 110, Springer, Berlin, 1998. [7] Bradley R. C., Introduction to Strong Mixing Conditions, Vol. 3, Indiana University Publications, 2005. [8] Chen J., Zhang L., Asymptotic properties of nonparametric M-estimation for mixing functional data, J. Statist. Plan. Infer., 2009, 139: 533-546. [9] Collomb G., Estimation de la Regression par la Methode des k Points les Plus Proches Avec Noyau: Quelques Propietes de Convergence Ponctuelle, Lect. Notes in Mathematics, Vol. 831, 159-175, Springer, Berlin, 1980. [10] Collomb G., Härdle W., Strong uniform convergence rates in robust nonparametric time series analysis and prediction: Kernel regression estimation from dependent observations, Stochastic Process. Appl., 1984, 23: 77-89. [11] Fan J., Yao Q., Nonlinear Time Series, Springer, New York, 2003. [12] Ferraty F., Vieu P., Nonparametric Models for Functional Data with Application in Regression, Time Series Prediction and Curve Discrimination, Presented at the International Conference on Advances and Trends in Nonparametric Statistics, Crete, Greece, 2002. [13] Ferraty F., Vieu P., Nonparametric Functional Data Analysis, Springer, New York, 2006. [14] Gasser T., Hall P., Presnell B., Nonparametric estimation of the mode of a distribution of random curves, J. Roy. Statist. Soc. Ser. B, 1998, 60: 681-691. [15] Härdle W., Lütkepohl H., Chen R., A review of nonparametric time series analysis, Internat. Statist. Rev., 1997, 21: 49-72. [16] Ibragimov I. A., Linnik Y. V., Independent and Stationary Sequence of Random Variables, Wolters-Noordhoof, 1971. [17] Laksaci A., Lemdani M., Ould-Said E., L1-norm Kernel Estimator of Conditioal Quantile for Functional Regressors: Consistency and Asymptotic Normality, Manuscript, 2007. [18] Li J., Tran L. T., Nonparametric estimation of conditional expectation, J. Statist. Plan. Infer., 2009, 139: 164-175. [19] Mack Y. P., Local properties of k-NN regression estimates, SIAM J. Algebraic Discrete Methods, 1981, 2: 311-323. [20] Masry E., Nonparametric regression estimation for dependent functional data, Stochastic Process. Appl., 2005, 115: 155-177. [21] Masry E., Fan J., Local polynomial estimation of regression functions for mixing processes, Scand. J. Statist., 1997, 24: 165-179. [22] Pham T. D., Tran L. T., Some mixing properties of time series models, Stochastic Process. Appl., 1985, 19: 297-303. [23] Ramsay J., Silverman B., Applied Functional Data Analysis: Methods and Case Studies, Springer, New York, 2002. [24] Robinson P. M., Hypothesis testing in semiparametric and nonparametric models for econometric time series, Rev. Econom. Stud., 1989, 56: 511-534. [25] Robinson P. M., Nonparametric estimators for times series, J. Time Ser. Anal., 1983, 4: 185-207. [26] Robinson P. M., Time series residuals with application to probability dentisy estimation, J. Time Ser. Anal., 1987, 8: 329-344. [27] Roussas G. G., Nonparametric estimation of the distribution function of a Markov process, Ann. Math. Statist., 1969, 40: 1386-1400. [28] Roussas G. G., Nonparametric regression under mixing conditions, Stochastic Precess. Appl., 1990, 36: 107-116. [29] Roussas G. G., Tran L., Asymptotic normality of the recursive kernel regression estimate under dependence conditions, Ann. Statist., 1992, 20: 98-120. [30] Tran L. T., Nonparametric function estimation for time series by local average estimators, Ann. Statist., 1993, 21: 1040-1057. [31] Tran L. T., Yakowitz S., Nearest neighbor estimators for random fields, J. Multivariate Anal., 1993, 24: 23-46. [32] Truong Y. K., Stone C. J., Nonparametric function estimation involving time series, Ann. Statist., 1992, 20: 77-97. [33] Vakashnia N. N., Tarieladze V. I., Chobanyan S. A., Probability Distributions on Banach Space, D. Reidel Publishing Co., Boston, 1987. [34] Watson G. S., Smooth regression analysis, Sankhya Ser. A, 1964, 26: 359-372. [35] Yakowitz S. J., Nearest-neighbor methods for time series analysis, J. Time Ser. Anal., 1987, 8: 235-247. [36] Yakowitz S. J., Nonparametric density estimation, prediction and regression for Markov sequences, J. Amer. Statist. Assoc., 1985, 80: 215-221.