Real-time estimation for functional stochastic regression models

References

• Bosq D. Nonparametric statistics for stochastic processes. Estimation and prediction. New York: Springer-Verlag; 1998. (Lecture notes in statistics; 110).
   Google Scholar
• 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. 1986;23(1):77–89. doi: 10.1016/0304-4149(86)90017-7
   Web of Science ®Google Scholar
• Rosenblatt M. Conditional probability density and regression estimators. Multivariate Analysis, II. Proceeding of International Symposium; Dayton, OH; New York: Academic Press; 1969. p. 25–31.
   Google Scholar
• Stone CJ. Optimal global rates of convergence for nonparametric regression. Ann Statist. 1982;10(4):1040–1053. doi: 10.1214/aos/1176345969
   Web of Science ®Google Scholar
• Ramsay JO, Silverman BW. Functional data analysis. New York: Springer Series in Statistics; 2005.
   Google Scholar
• Ferraty F, Vieu P. Nonparametric modelling for functional data, methods, theory, applications and implementations. London: Springer-Verlag; 2006.
   Google Scholar
• Ferraty F, Romain Y. The Oxford handbook of functional data analysis. Oxford: Oxford University Press; 2011.
   Google Scholar
• Hu Y. Nonparametric estimation of variance function for functional data under mixing conditions. Comm. Statistics – Theory and Methods. 2013;42(10):1774–1786. doi: 10.1080/03610926.2011.599007
   Web of Science ®Google Scholar
• Ferraty F, Mas A, Vieu P. Nonparametric regression on functional data: inference and practical aspects. Aust N Z J Stat. 2007;49(3):267–286. doi: 10.1111/j.1467-842X.2007.00480.x
   Web of Science ®Google Scholar
• Ferraty F, Vieu P. Nonparametric models for functional data, with applications in regression, time series prediction and curve discrimination. J Nonparametr Stat. 2004;16:111–125. The International Conference on Recent Trends and Directions in Nonparametric Statistics. doi: 10.1080/10485250310001622686
   Web of Science ®Google Scholar
• Masry E. Nonparametric regression estimation for dependent functional data: asymptotic normality. Stochastic Process Appl. 2005;115:155–177. doi: 10.1016/j.spa.2004.07.006
   Web of Science ®Google Scholar
• Rosenblatt M. Uniform ergodicity and strong mixing. Z Wahrs Verw Gebiete. 1972;24:79–84. doi: 10.1007/BF00532465
   Google Scholar
• Gastwirth JL, Rubin H. The asymptotic distribution theory of the empiric cdf for mixing stochastic processes. Ann Statist. 1975;3(4):809–824. doi: 10.1214/aos/1176343184
   Web of Science ®Google Scholar
• Andrews DWK. Non-mixing autoregressive processes. J Appl Probab. 1984;21:930–934. doi: 10.2307/3213710
   Web of Science ®Google Scholar
• Chernick MR. A limit theorem for the maximum of autoregressive processes with uniform marginal distributions. Ann Probab. 1981;9:145–149. doi: 10.1214/aop/1176994514
   Web of Science ®Google Scholar
• Pham TD, Ran LT. Some mixing properties of time series models. Stochastic Process Appl. 1985;19297–303. doi: 10.1016/0304-4149(85)90031-6
   Web of Science ®Google Scholar
• Laïb N, Louani D. Nonparametric kernel regression estimation for functional stationary ergodic data: asymptotic properties. J Multivariate Anal. 2010;101(10):2266–2281. doi: 10.1016/j.jmva.2010.05.010
   Web of Science ®Google Scholar
• Laïb N, Louani D. Rates of strong consistencies of the regression function estimator for functional stationary ergodic data. J Statist Plann Inference. 2011;141(1):359–372. doi: 10.1016/j.jspi.2010.06.009
   Web of Science ®Google Scholar
• Bouzebda S, Chaouch M, Laïb N. Limiting law results for a class of conditional mode estimates for functional stationary ergodic data. Math Meth Stat. 2016; 25(3):168–195. doi: 10.3103/S1066530716030029
   Web of Science ®Google Scholar
• Delecroix M. Sur l'estimation et la prévision nonparamétrique des processes ergodiques. Doctorat d'état, Université des Sciences et Techniques de Lille Flandre-Artois; 1987.
   Google Scholar
• Krengel U. Ergodic theorms. Berlin: Walter de Gruyter & Co; 1985.
   Google Scholar
• Ling N, Liang L, Vieu P: Nonparametric regression estimation for functional stationary ergodic data with missing at random. J Statist Plann Inference. 2015; 162: 75–87. doi: 10.1016/j.jspi.2015.02.001
   Web of Science ®Google Scholar
• Cai TT, Wang L. Adaptive variance function estimation in heteroscedastic nonparametric regression. Ann Statist. 2008;36(5):2025–2054. doi: 10.1214/07-AOS509
   Web of Science ®Google Scholar
• Fan JQ, Yao Q. Efficient estimation of conditional variance functions in stochastic regression. Biometrika. 1998;85(3):645–660. doi: 10.1093/biomet/85.3.645
   Web of Science ®Google Scholar
• Laïb N. Kernel estimates of the mean and the volatility functions in a nonlinear autoregressive model with ARCH errors. J Statist Plann Inference. 2005;134:116–139. doi: 10.1016/j.jspi.2004.01.019
   Web of Science ®Google Scholar
• Hall P, Carroll RJ. Variance function estimation in regression, the effect of estimating the mean. J Roy Statist Soc Ser B-Methodol. 1989;51(1):3–14.
   Google Scholar
• Chaouch M. Volatility estimation in a nonlinear heteroscedastic functional regression model with martingale difference errors. J Multivariate Anal. 2019;170:129–148. doi: 10.1016/j.jmva.2018.11.008
   Web of Science ®Google Scholar
• Stadtmüller U, Tsybakov AB. Nonparametric recursive variance estimation. Statistics. 1995;27:55–63. doi: 10.1080/02331889508802510
   Web of Science ®Google Scholar
• Robbins R, Monro SA. A stochastic approximation method. Ann Statist. 1951;22(3):400–407. doi: 10.1214/aoms/1177729586
   Web of Science ®Google Scholar
• Révész P. Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I. Studia Sci Math Hungar. 1973;8:391–398.
   Google Scholar
• Amiri A. Recursive regression estimators with application to nonparametric prediction. J Nonparam Stat. 2012;24(1):169–186. doi: 10.1080/10485252.2011.626855
   Web of Science ®Google Scholar
• Györfi L, Kholer M, Krzyzak A, Walk H. A distribution-free theory of nonparametric regression. New York: Springer-Verlag; 2002.
   Google Scholar
• Wang L, Liang HY. Strong uniform convergence of the recursive regression estimator under ϕ-mixing conditions. Metrika. 2004;59(3):245–261. doi: 10.1007/s001840300282
   Web of Science ®Google Scholar
• Amiri A, Crambes C, Thiam B. Recursive estimation of nonparametric regression with functional covariate. Comp Stat and Data Anal. 2014;69:154–172. doi: 10.1016/j.csda.2013.07.030
   Web of Science ®Google Scholar
• Amiri A, Thiam B. Consistency of the recursive nonparametric regression estimation for dependent functional data. J Nonparam Statist. 2014;26(13):471–487. doi: 10.1080/10485252.2014.907406
   Google Scholar
• Rachdi M, Vieu Ph. Nonparametric regression for functional data: automatic smoothing parameter selection. J Statist Plann Inference. 2017;137:2784–2801. doi: 10.1016/j.jspi.2006.10.001
   Google Scholar
• Polyak BT. Introduction to optimization. New York: Optim. Software; 1987.
   Google Scholar