This paper considers naive and wild bootstrap procedures to construct pointwise confidence intervals for a nonparametric regression function when the predictor is of functional nature and when the data are dependent. Assuming $\alpha$-mixing conditions on the sample, the asymptotic validity of both procedures is obtained. A simulation study shows promising results when finite sample sizes are used, while an application to electricity demand data illustrates its usefulness in practice.
References
[1] Aneiros-Pérez, G. and Vieu, P. (2008). Nonparametric time series prediction: A semi-functional partial linear modeling., J Multivariate Anal, 99 834–857.[1] Aneiros-Pérez, G. and Vieu, P. (2008). Nonparametric time series prediction: A semi-functional partial linear modeling., J Multivariate Anal, 99 834–857.
[2] Aneiros, G., Vilar, J.M., Cao, R., and Muñoz-San-Roque, A. (2013). Functional prediction for the residual demand in electricity spot markets., IEEE Trans Power Syst, 28 4201–4208.[2] Aneiros, G., Vilar, J.M., Cao, R., and Muñoz-San-Roque, A. (2013). Functional prediction for the residual demand in electricity spot markets., IEEE Trans Power Syst, 28 4201–4208.
[3] Antoch, J., Prchal, L., De Rosa, M.R., and Sarda, P. (2010). Electricity comsuption prediction with functional linear regression using spline estimators., J Appl Statist, 37 2027–2041.[3] Antoch, J., Prchal, L., De Rosa, M.R., and Sarda, P. (2010). Electricity comsuption prediction with functional linear regression using spline estimators., J Appl Statist, 37 2027–2041.
[4] Bosq, D. (1998)., Nonparametric Statistics for Stochastic Processes: Estimation and Prediction (2nd edition). Lecture Notes in Statistics, 110. Springer-Verlag.[4] Bosq, D. (1998)., Nonparametric Statistics for Stochastic Processes: Estimation and Prediction (2nd edition). Lecture Notes in Statistics, 110. Springer-Verlag.
[8] Collomb, G. (1984). Propriétés de convergence presque complète du prédicteur à noyau. (French) [Almost complete convergence properties of kernel predictors], Z. Wahrsch. Verw. Gebiete, 66(3) 441–460.[8] Collomb, G. (1984). Propriétés de convergence presque complète du prédicteur à noyau. (French) [Almost complete convergence properties of kernel predictors], Z. Wahrsch. Verw. Gebiete, 66(3) 441–460.
[12] Fernández-de-Castro, B., Guillas, S., and González-Manteiga, W. (2005). Functional samples and bootstrap for predicting sulfur dioxide levels., Technometrics, 47 212–222.[12] Fernández-de-Castro, B., Guillas, S., and González-Manteiga, W. (2005). Functional samples and bootstrap for predicting sulfur dioxide levels., Technometrics, 47 212–222.
[13] Ferraty, F., Laksaci, A., Tadj, A., and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables., J. Statist. Plann. Inference, 140 335–352.[13] Ferraty, F., Laksaci, A., Tadj, A., and Vieu, P. (2010). Rate of uniform consistency for nonparametric estimates with functional variables., J. Statist. Plann. Inference, 140 335–352.
[14] Ferraty, F., Laksaci, A., and Vieu, P. (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models., Statistical Inference for Stochastic Processes, 9 47–76.[14] Ferraty, F., Laksaci, A., and Vieu, P. (2006). Estimating some characteristics of the conditional distribution in nonparametric functional models., Statistical Inference for Stochastic Processes, 9 47–76.
[15] Ferraty, F., Mas, A., and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical apects., Aust. N. Z. J. Stat., 49 267–286.[15] Ferraty, F., Mas, A., and Vieu, P. (2007). Nonparametric regression on functional data: inference and practical apects., Aust. N. Z. J. Stat., 49 267–286.
[16] Ferraty, F., Rabhi, A., and Vieu, P. (2005). Conditional quantiles for dependent functional data with application to the climatic El Niño phenomenon., Sankhya, 67(2) 378–398. MR2208895[16] Ferraty, F., Rabhi, A., and Vieu, P. (2005). Conditional quantiles for dependent functional data with application to the climatic El Niño phenomenon., Sankhya, 67(2) 378–398. MR2208895
[17] Ferraty, F., Van Keilegom, I., and Vieu, P. (2010). On the validity of the bootstrap in non-parametric functional regression., Scand. J. Statist., 37 286–306.[17] Ferraty, F., Van Keilegom, I., and Vieu, P. (2010). On the validity of the bootstrap in non-parametric functional regression., Scand. J. Statist., 37 286–306.
[19] Ferraty, F. and Vieu P. (2004). Nonparametric models for functional data, with application in regression, time–series prediction and curve discrimination., Nonparameric Statistics, 16: 111–125. MR2053065 10.1080/10485250310001622686[19] Ferraty, F. and Vieu P. (2004). Nonparametric models for functional data, with application in regression, time–series prediction and curve discrimination., Nonparameric Statistics, 16: 111–125. MR2053065 10.1080/10485250310001622686
[23] Györfi, L., Härdle, W., Sarda, P., and Vieu, P. (1989)., Nonparametric curve estimation from time series. Lecture Notes in Statistics, 60. Springer-Verlag, Berlin.[23] Györfi, L., Härdle, W., Sarda, P., and Vieu, P. (1989)., Nonparametric curve estimation from time series. Lecture Notes in Statistics, 60. Springer-Verlag, Berlin.
[28] Hsing, T. and Eubank, R. (2015)., Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester.[28] Hsing, T. and Eubank, R. (2015)., Theoretical foundations of functional data analysis, with an introduction to linear operators. Wiley Series in Probability and Statistics. John Wiley & Sons, Chichester.
[30] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality., Stochastic Process. Appl., 115 155–177.[30] Masry, E. (2005). Nonparametric regression estimation for dependent functional data: asymptotic normality., Stochastic Process. Appl., 115 155–177.
[31] Pesin, Y.B. (1993). On rigourous mathematicak definitions of correlation dimension and generalized spectrum for dimensions., Journal of Statistical Physic, 71 529–547.[31] Pesin, Y.B. (1993). On rigourous mathematicak definitions of correlation dimension and generalized spectrum for dimensions., Journal of Statistical Physic, 71 529–547.