Two-time-scale nonparametric recursive regression estimator for independent functional data

References

• Amiri, A., C. Crambes, and B. Thiam. 2014. Recursive estimation of nonparametric regression with functional covariate. Computational Statistics & Data Analysis 69:154–72. doi:10.1016/j.csda.2013.07.030.
   Web of Science ®Google Scholar
• Aneiros, G., and P. Vieu. 2006. Semi-functional partial linear regression. Statistics & Probability Letters 76 (11):1102–10. doi:10.1016/j.spl.2005.12.007.
   Web of Science ®Google Scholar
• Aneiros, G., and P. Vieu. 2015. Partial linear modelling with multi-functional covariates. Computational Statistics 30 (3):647–71. doi:10.1007/s00180-015-0568-8.
   Web of Science ®Google Scholar
• Benatia, D., M. Carrasco, and J. P. Florens. 2017. Functional linear regression with functional response. Journal of Econometrics 201 (2):269–91. doi:10.1016/j.jeconom.2017.08.008.
   Web of Science ®Google Scholar
• Benhenni, K., S. Hedli-Griche, and M. Rachdi. 2010. Estimation of the regression operator from functional fixed-design with correlated errors. Journal of Multivariate Analysis. 101 (2):476–90. doi:10.1016/j.jmva.2009.09.019.
   Web of Science ®Google Scholar
• Bhatnagar, S., M. C. Fu, S. I. Marcus, and S. Bhatnagar. 2001. Two timescale algorithms for simulation optimization of hidden Markov models. Iie Transactions 33 (3):245–58. doi:10.1080/07408170108936826.
   Web of Science ®Google Scholar
• Bojanic, R., and E. Seneta. 1973. A unified theory of regularly varying sequences. Mathematische Zeitschrift 134 (2):91–106. doi:10.1007/BF01214468.
   Web of Science ®Google Scholar
• Borkar, V. S. 1997. Stochastic approximation with two time scales. Systems & Control Letters 29 (5):291–4. doi:10.1016/S0167-6911(97)90015-3.
   Web of Science ®Google Scholar
• Cai, T. T., and P. Hall. 2006. Prediction in functional linear regression. The Annals of Statistics 34 (5):2159–79. doi:10.1214/009053606000000830.
   Web of Science ®Google Scholar
• Chen, D., P. Hall, and H. G. Müller. 2011. Single and multiple index functional regression models with nonparametric link. The Annals of Statistics 39 (3):3458–86. doi:10.1214/11-AOS882.
   Web of Science ®Google Scholar
• Chiou, J. M., H. G. Müller, J. L. Wang, and J. R. Carey. 2003. A functional multiplicative effects model for longitudinal data, with application to reproductive histories of female medflies. Statist. Sinica 13:1119–33.
   PubMed Web of Science ®Google Scholar
• Cuevas, A., M. Febrero, and R. Fraiman. 2002. Linear functional regression: The case of fixed design and functional response. Canadian Journal of Statistics 30 (2):285–300. doi:10.2307/3315952.
   Web of Science ®Google Scholar
• Cuevas, A., M. Febrero, and R. Fraiman. 2006. On the use of the bootstrap for estimating functions with functional data. Computational Statistics & Data Analysis 51 (2):1063–74. doi:10.1016/j.csda.2005.10.012.
   Web of Science ®Google Scholar
• Dalal, G., B. Szörényi, G. Thoppe, and S. Mannor. 2018. Finite sample analysis of two-timescale stochastic approximation with applications to reinforcement learning. Proceedings of Machine Learning Research 75:1–35.
   Google Scholar
• Delaigle, A., and I. Gijbels. 2004. Practical bandwidth selection in deconvolution kernel density estimation. Computational Statistics & Data Analysis 45 (2):249–67. doi:10.1016/S0167-9473(02)00329-8.
   Web of Science ®Google Scholar
• Ferraty, F.,. A. Mas, and P. Vieu. 2007. Nonparametric regression on functional data: Inference and practical aspects. Australian & New Zealand Journal of Statistics 49 (3):267–86. doi:10.1111/j.1467-842X.2007.00480.x.
   Web of Science ®Google Scholar
• Ferraty, F.,. I. Van Keilegom, and P. Vieu. 2009. On the validity of the bootstrap in non-parametric functional regression. Scandinavian Journal of Statistics 37 (2):286–306. doi:10.1111/j.1467-9469.2009.00662.x.
   Web of Science ®Google Scholar
• Ferraty, F., and P. Vieu. 2002. The functional nonparametric model and application to spectrometric data. Computational Statistics 17 (4):545–64. doi:10.1007/s001800200126.
   Web of Science ®Google Scholar
• Ferraty, F., and P. Vieu. 2004. Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination. Journal of Nonparametric Statistics 16 (1-2):111–25. doi:10.1080/10485250310001622686.
   Web of Science ®Google Scholar
• Ferraty, F., and P. Vieu. 2006. Nonparametric functional data analysis: Theory and practice. Springer Series in Statistics, Springer, NewYork.
   Google Scholar
• Galambos, J., and E. Seneta. 1973. Regularly varying sequences. Proceedings of the American Mathematical Society 41 (1):110–6. doi:10.1090/S0002-9939-1973-0323963-5.
   Web of Science ®Google Scholar
• Goia, A., and P. Vieu. 2015. A partitioned single functional index model. Computational Statistics 30:676–92.
   Web of Science ®Google Scholar
• Hall, P., and J. L. Horowitz. 2007. Methodology and convergence rates for functional linear regression. The Annals of Statistics 35 (1):70–91. doi:10.1214/009053606000000957.
   Web of Science ®Google Scholar
• Härdle, W., and J. S. Marron. 1991. Bootstrap simultaneous error bars for nonparametric regression. Annals of Statistics 16:1696–708.
   Google Scholar
• Helali, S., and Y. Slaoui. 2020. Estimation of a distribution function using Lagrange polynomials with Tchebychev-Gauss points. Statistics and Its Interface 13 (3):399–410. doi:10.4310/SII.2020.v13.n3.a9.
   Web of Science ®Google Scholar
• Imaizumi, M., and K. Kato. 2018. PCA-based estimation for functional linear regression with functional responses. Journal of Multivariate Analysis. 163:15–36. doi:10.1016/j.jmva.2017.10.001.
   Web of Science ®Google Scholar
• Jmaei, A., Y. Slaoui, and W. Dellagi. 2017. Recursive distribution estimators defined by stochastic approximation method using Bernstein polynomials. Journal of Nonparametric Statistics 29 (4):792–805. doi:10.1080/10485252.2017.1369538.
   Web of Science ®Google Scholar
• Kara, L. Z., A. Laksaci, M. Rachdi, and P. Vieu. 2017. Data-Driven kNN estimation in nonparametric functional data analysis. Journal of Multivariate Analysis. 153:176–88. doi:10.1016/j.jmva.2016.09.016.
   Web of Science ®Google Scholar
• Konda, V. R., and V. S. Borkar. 1999. Actor-critic–type learning algorithms for markov decision processes. SIAM Journal on Control and Optimization 38 (1):94–123. doi:10.1137/S036301299731669X.
   Web of Science ®Google Scholar
• Konda, V. R., and J. N. Tsitsiklis. 2004. Convergence rate of linear two-time-scale stochastic approximation. Ann. of Appl. Prob ab 14:796–819.
   Web of Science ®Google Scholar
• Lian, H. 2011. Functional partial linear model. Journal of Nonparametric Statistics 23 (1):115–28. doi:10.1080/10485252.2010.500385.
   Web of Science ®Google Scholar
• Lian, H. 2012. Convergence of nonparametric functional regression estimates with functional responses. Electronic Journal of Statistics 6:1373–91.
   Web of Science ®Google Scholar
• Ling, N., and P. Vieu. 2018. Nonparametric modelling for functional data: Selected survey and tracks for future. Statistics 52 (4):934–49. doi:10.1080/02331888.2018.1487120.
   Web of Science ®Google Scholar
• Masry, E. 2005. Nonparametric regression estimation for dependent functional data: Asymptotic normality. Stochastic Processes and Their Applications 115 (1):155–77. doi:10.1016/j.spa.2004.07.006.
   Web of Science ®Google Scholar
• Mokkadem, A., and M. Pelletier. 2006. Convergence rate and averaging of nonlinear twotime-scale stochastic approximation algorithms. The Annals of Applied Probability 16 (3):1671–702. doi:10.1214/105051606000000448.
   Web of Science ®Google Scholar
• Mokkadem, A., and M. Pelletier. 2007. A companion for the Kiefer-Wolfowitz-Blum stochastic approximation algorithm. The Annals of Statistics 35 (4):1749–72. doi:10.1214/009053606000001451.
   Web of Science ®Google Scholar
• Mokkadem, A., M. Pelletier, and Y. Slaoui. 2009. Revisiting Révész’s stochastic approximation method for the estimation of a regression function. Latin American Journal of Probability and Mathematical Statistics 6:63–114.
   Google Scholar
• Nadaraya, E. A. 1965. On estimating regression. Theory of Probability & Its Applications 10 (1):186–90. doi:10.1137/1110024.
   Google Scholar
• Politis, D. N., and J. P. Romano. 1994. Limit theorems for weakly dependent Hilbert space valued random variables with application to the stationarity bootstrap. Statist. Sinica 4:461–76.
   Web of Science ®Google Scholar
• Ramsay, J. O., and B. W. Silverman. 2002. Applied functional data analysis: Methods and case studies. Springer, New York.
   Google Scholar
• Raña, P., G. Aneiros, and J. Vilar. 2015. Detection of outliers in functional time series. Environmetrics 26 (3):178–91. doi:10.1002/env.2327.
   Web of Science ®Google Scholar
• Raña, P., G. Aneiros, J. M. Vilar, and P. Vieu. 2016. Bootstrap confidence intervals in functional nonparametric regression under dependence. Electron. J. Stat 10:1973–99.
   Web of Science ®Google Scholar
• Révész, P. 1973. Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I. Studia Scientiarum Mathematicarum Hungarica 8:391–8.
   Google Scholar
• Sam, A., and E. Campitelli. 2020. Package “rsoi”. https://github.com/boshek/rsoi, https://boshek.github.io/rsoi/, R package version 0.5.2.
   Google Scholar
• Serfling, R. J. 1980. Approximation theorems of mathematical statistics. Wiley, New York.
   Google Scholar
• Shang, H. L. 2018. Bootstrap methods for stationary functional time series. Statistics and Computing 28 (1):1–10. doi:10.1007/s11222-016-9712-8.
   Web of Science ®Google Scholar
• Slaoui, Y. 2016. Optimal bandwidth selection for semi-recursive kernel regression estimators. Statistics and Its Interface 9 (3):375–88. doi:10.4310/SII.2016.v9.n3.a11.
   Web of Science ®Google Scholar
• Slaoui, Y. 2019. Wild bootstrap bandwidth selection of recursive nonparametric relative regression for independent functional data. Journal of Multivariate Analysis. 173:417–37.
   Web of Science ®Google Scholar
• Slaoui, Y. 2020. Recursive nonparametric regression estimation for independent functional data. Statistica Sinica 30:417–37.
   Web of Science ®Google Scholar
• Slaoui, Y., and A. Jmaei. 2019. Recursive density estimators based on Robbins-Monro’s scheme and using Bernstein polynomials. Statistics and Its Interface 12 (3):439–55. doi:10.4310/SII.2019.v12.n3.a8.
   Web of Science ®Google Scholar
• Slaoui, Y. 2006. Application des méthodes d’approximations stochastiques à l’estimation de la densité et de la régression. (Doctoral dissertation).
   Google Scholar
• Watson, G. S. 1964. Smooth regression analysis. Sankhya A 26:359–72.
   Google Scholar
• Zhang, Q., and J. Hu. 2017. A two-time-scale adaptive search algorithm for global optimization. Winter Simulation Conference (WSC) 2069–79. doi: 10.1109/WSC.2017.8247940.
   Google Scholar