Convergence of functional \(k\)-nearest neighbor regression estimate with functional responses. (English) Zbl 1274.62291

Summary: Let \((X_{1},Y_{1}),\ldots ,(X_{n},Y_{n})\) be independent and identically distributed random elements taking values in \(\mathcal F\times \mathcal H\), where \(\mathcal F\) is a semi-metric space and \(\mathcal H\) is a separable Hilbert space. We investigate the rates of strong (almost sure) convergence of the \(k\)-nearest neighbor estimate. We give two convergence results assuming a finite moment condition and exponential tail condition on the noises respectively, with the latter requiring less stringent conditions on \(k\) for convergence.


62G08 Nonparametric regression and quantile regression
62G20 Asymptotic properties of nonparametric inference


fda (R)
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[1] Antoch, J., Prchal, L., De Rosa, M. R. and Sarda, P. (2008). Functional linear regression with functional response: Application to prediction of electricity consumption. In, Functional and Operatorial Statistics ( S. DaboNiang and F. Ferraty, eds.) 23-29. · doi:10.1007/978-3-7908-2062-1_4
[2] Bosq, D. (2000)., Linear processes in function spaces: theory and applications . Springer Verlag. · Zbl 0962.60004 · doi:10.1007/978-1-4612-1154-9
[3] Burba, F., Ferraty, F. and Vieu, P. (2009). k-Nearest Neighbour method in functional nonparametric regression., Journal of Nonparametric Statistics 21 453-469. · Zbl 1161.62017 · doi:10.1080/10485250802668909
[4] Cover, T. M. (1968). Estimation by the nearest neighbor rule., IEEE Transactions on Information Theory 14 50-55. · Zbl 0157.49404 · doi:10.1109/TIT.1968.1054098
[5] Cover, T. M. and Hart, P. E. (1967). Nearest neighbor pattern classification., IEEE Transactions on Information Theory 13 21-27. · Zbl 0154.44505 · doi:10.1109/TIT.1967.1053964
[6] Delsol, L. (2009). Advances on asymptotic normality in non-parametric functional time series analysis., Statistics 43 13-33. · Zbl 1278.62052 · doi:10.1080/02331880802184961
[7] Ferraty, F., Keilegom, I. and Vieu, P. (2010). On the Validity of the Bootstrap in Non-Parametric Functional Regression., Scandinavian Journal of Statistics 37 286-306. · Zbl 1223.62042 · doi:10.1111/j.1467-9469.2009.00662.x
[8] Ferraty, F., Mas, A. and Vieu, P. (2007). Nonparametric regression on functional data: Inference and practical aspects., Australian & New Zealand Journal of Statistics 49 267-286. · Zbl 1136.62031
[9] Ferraty, F. and Vieu, P. (2002). The functional nonparametric model and application to spectrometric data., Computational Statistics 17 545-564. · Zbl 1037.62032 · doi:10.1007/s001800200126
[10] Ferraty, F. and Vieu, P. (2004). Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination., Journal of nonparametric statistics 16 111-125. · Zbl 1049.62039 · doi:10.1080/10485250310001622686
[11] Ferraty, F. and Vieu, P. (2006)., Nonparametric functional data analysis: theory and practice . Springer series in statistics . Springer, New York, NY. · Zbl 1119.62046 · doi:10.1007/0-387-36620-2
[12] Ferraty, F., Laksaci, A., Tadj, A. and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables., Journal of Statistical Planning and Inference 140 335-352. · Zbl 1177.62044 · doi:10.1016/j.jspi.2009.07.019
[13] Fix, E. and Hodges, J. L. (1989). Discriminatory analysis. nonparametric discrimination: consistency properties., International Statistical Review 57 238-247. · Zbl 0715.62080 · doi:10.2307/1403797
[14] Ledoux, M. and Talagrand, M. (1991)., Probability in Banach spaces: isoperimetry and processes . Springer-Verlag, Berlin; New York. · Zbl 0748.60004
[15] Lian, H. (2007). Nonlinear functional models for functional responses in reproducing kernel Hilbert spaces., Canadian Journal of Statistics-Revue Canadienne De Statistique 35 597-606. · Zbl 1142.62020 · doi:10.1002/cjs.5550350410
[16] Parzen, E. (1962). On the estimation of a probability density function and mode., Annals of Mathematical Statistics 33 1065-1076. · Zbl 0116.11302 · doi:10.1214/aoms/1177704472
[17] Pollard, D. (1984)., Convergence of stochastic processes . Springer-Verlag, New York. · Zbl 0544.60045
[18] Preda, C. (2007). Regression models for functional data by reproducing kernel Hilbert spaces methods., Journal of Statistical Planning and Inference 137 829-840. · Zbl 1104.62043 · doi:10.1016/j.jspi.2006.06.011
[19] Ramsay, J. O. and Silverman, B. W. (2005)., Functional data analysis , 2nd ed. Springer series in statistics . Springer, New York. · Zbl 1079.62006
[20] Rosenblatt, M. (1956). Remarks on some nonparametric estimates of density function., Annals of Mathematical Statistics 27 832-837. · Zbl 0073.14602 · doi:10.1214/aoms/1177728190
[21] van der Geer, S. A. (2000)., Applications of empirical process theory . Cambridge University Press, Cambridge. · Zbl 0953.62049
[22] van der Vaart, A. W. and Van Zanten, J. H. (2008). Rates of contraction of posterior distributions based on Gaussian process priors., Annals of Statistics 36 1435-1463. · Zbl 1141.60018 · doi:10.1214/009053607000000613
[23] van der Vaart, A. W. and Wellner, J. A. (1996)., Weak convergence and empirical processes . Springer series in statistics . Springer, New York. · Zbl 0862.60002
[24] Yang, Y. H. and Barron, A. (1999). Information-theoretic determination of minimax rates of convergence., Annals of Statistics 27 1564-1599. · Zbl 0978.62008 · doi:10.1214/aos/1017939142
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