Generalized regression estimation for continuous time processes with values in functional spaces

Bertrand Maillot; Christophe Chesneau

Commentationes Mathematicae Universitatis Carolinae (2021)

  • Volume: 62, Issue: 4, page 461-481
  • ISSN: 0010-2628

Abstract

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We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the convergence of the conditional distribution function as well as conditional quantiles. Note that a parametric rate of convergence in probability is reached while working with a naive kernel.

How to cite

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Maillot, Bertrand, and Chesneau, Christophe. "Generalized regression estimation for continuous time processes with values in functional spaces." Commentationes Mathematicae Universitatis Carolinae 62.4 (2021): 461-481. <http://eudml.org/doc/298030>.

@article{Maillot2021,
abstract = {We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the convergence of the conditional distribution function as well as conditional quantiles. Note that a parametric rate of convergence in probability is reached while working with a naive kernel.},
author = {Maillot, Bertrand, Chesneau, Christophe},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {continuous time process; regression function estimation; conditional distribution function},
language = {eng},
number = {4},
pages = {461-481},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Generalized regression estimation for continuous time processes with values in functional spaces},
url = {http://eudml.org/doc/298030},
volume = {62},
year = {2021},
}

TY - JOUR
AU - Maillot, Bertrand
AU - Chesneau, Christophe
TI - Generalized regression estimation for continuous time processes with values in functional spaces
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2021
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 62
IS - 4
SP - 461
EP - 481
AB - We consider two continuous time processes; the first one is valued in a semi-metric space, while the second one is real-valued. In some sense, we extend the results of F. Ferraty and P. Vieu in ``Nonparametric models for functional data, with application in regression, time-series prediction and curve discrimination'' (2004), by establishing the convergence, with rates, of the generalized regression function when a real-valued continuous time response is considered. As corollaries, we deduce the convergence of the conditional distribution function as well as conditional quantiles. Note that a parametric rate of convergence in probability is reached while working with a naive kernel.
LA - eng
KW - continuous time process; regression function estimation; conditional distribution function
UR - http://eudml.org/doc/298030
ER -

References

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  1. Bosq D., Vitesses optimales et superoptimales des estimateurs fonctionnels pour les processus à temps continu, C. R. Acad. Sci. Paris Sér. I Math. 317 (1993), no. 11, 1075–1078 (French). MR1249792
  2. Bosq D., 10.1007/978-1-4612-1718-3, Estimation and Prediction, Lecture Notes in Statistics, 110, Springer, New York, 1998. MR1640691DOI10.1007/978-1-4612-1718-3
  3. Bosq D., 10.1007/978-1-4612-1154-9_8, Theory and Applications, Lecture Notes in Statistics, 149, Springer, New York, 2000. MR1783138DOI10.1007/978-1-4612-1154-9_8
  4. Bradley R. C., 10.1214/154957805100000104, Probab. Surv. 2 (2005), 107–144. MR2178042DOI10.1214/154957805100000104
  5. Cardot H., Ferraty F., Sarda P., 10.1016/S0167-7152(99)00036-X, Statist. Probab. Lett. 45 (1999), no. 1, 11–22. MR1718346DOI10.1016/S0167-7152(99)00036-X
  6. Cardot H., Ferraty F., Sarda P., Spline estimators for the functional linear model, Statist. Sinica 13 (2003), no. 3, 571–591. MR1997162
  7. Castellana J. V., Leadbetter M. R., On smoothed probability density estimation for stationary processes, Stochastic Process. Appl. 21 (1986), no. 2, 179–193. MR0833950
  8. 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. 23 (1986), no. 1, 77–89. MR0866288
  9. Dabo-Niang S., Density estimation in a separable metric space, Ann. I.S.U.P. 47 (2003), no. 1–2, 3–21. MR2002884
  10. Dabo-Niang S., 10.1080/10485250310001624837, The International Conf. on Recent Trends and Directions in Nonparametric Statistics, J. Nonparametr. Stat. 16 (2004), no. 1–2, 171–186. MR2053068DOI10.1080/10485250310001624837
  11. Demongeot J., Laksaci A., Madani F., Rachdi M., A fast functional locally modeled conditional density and mode for functional time-series, Recent Advances in Functional Data Analysis and Related Topics, Contrib. Statist., Physica Verlag, Springer, Heidelberg, 2011, 85–90. MR2815565
  12. Ferraty F., Goia A., Vieu P., 10.1007/BF02595710, Test 11 (2002), no. 2, 317–344. MR1947601DOI10.1007/BF02595710
  13. Ferraty F., Laksaci A., Tadj A., Vieu P., 10.1214/11-EJS600, Electron. J. Stat. 5 (2011), 159–171. MR2786486DOI10.1214/11-EJS600
  14. Ferraty F., Rabhi A., Vieu P, Conditional quantiles for dependent functional data with application to the climatic El Niño phenomenon, Sankhyā 67 (2005), no. 2, 378–398. MR2208895
  15. Ferraty F., Vieu P., 10.1080/10485250310001622686, The International Conf. on Recent Trends and Directions in Nonparametric Statistics, J. Nonparametr. Stat. 16 (2004), no. 1–2, 111–125. MR2053065DOI10.1080/10485250310001622686
  16. Ferraty F., Vieu P., Nonparametric Functional Data Analysis, Theory and Practice, Springer Series in Statistics, Springer, New York, 2006. MR2229687
  17. Frank I. E., Friedman J. H., 10.1080/00401706.1993.10485033, Technometrics 35 (1993), no. 2, 109–135. DOI10.1080/00401706.1993.10485033
  18. Grunig R., Probabilités conditionnelles régulières sur des tribus de type non dénombrable, Ann. Inst. H. Poincaré Sect. B (N.S.) 2 (1965/1966), no. 3, 227–229 (French). MR0196799
  19. Jiřina M., 10.21136/CMJ.1954.100124, Czechoslovak. Math. J. 4(79) (1954), 372–380. MR0069416DOI10.21136/CMJ.1954.100124
  20. Jiřina M., 10.21136/CMJ.1959.100368, Czechoslovak. Math. J. 9(84) (1959), 445–451. MR0115202DOI10.21136/CMJ.1959.100368
  21. Krzy.zak A., Pawlak M., 10.1016/0378-3758(87)90065-6, J. Statist. Plann. Inference 16 (1987), no. 2, 159–166. MR0895756DOI10.1016/0378-3758(87)90065-6
  22. Masry E., 10.1016/j.spa.2004.07.006, Stochastic Process. Appl. 115 (2005), no. 1, 155–177. MR2105373DOI10.1016/j.spa.2004.07.006
  23. Ramsay J. O., Silverman B. W., Functional Data Analysis, Springer Series in Statistics, Springer, New York, 2005. MR2168993
  24. Rio E., Théorie asymptotique des processus aléatoires faiblement dépendants, Mathématiques & Applications, 31, Springer, Berlin, 2000 (French). MR2117923
  25. Rosenblatt M., Conditional probability density and regression estimators, in Multivariate Analysis II, Proc. Second Internat. Sympos., Dayton, Ohio, 1968, Academic Press, New York, 1969, pages 25–31. MR0254987
  26. Roussas G. G., 10.1016/0304-4149(90)90045-T, Stochastic Process. Appl. 36 (1990), no. 1, 107–116. Zbl0699.62038MR1075604DOI10.1016/0304-4149(90)90045-T
  27. Stone C. J., 10.1214/aos/1176345969, Ann. Statist. 10 (1982), no. 4, 1040–1053. MR0673642DOI10.1214/aos/1176345969
  28. Watson G. S., Smooth regression analysis, Sankhyā Ser. A 26 (1964), 359–372. MR0185765

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