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Articles

Recursive kernel regression estimation under α – mixing data

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Pages 8459-8475 | Received 19 Apr 2020, Accepted 25 Feb 2021, Published online: 18 Mar 2021

References

  • Ahmad, I., and P. E. Lin. 1976. Nonparametric sequential estimation of a multiple regression function. Bulletin of Mathematical Statistics 17 (1–2):63–75. doi:10.5109/13104.
  • Blum, J. R. 1954. Multidimensional stochastic approximation methods. The Annals of Mathematical Statistics 25 (4):737–44. doi:10.1214/aoms/1177728659.
  • Bosq, D. 1999. Nonparametric statistics for stochastic processes: Estimation and prediction. 2nd ed. New York: Springer Verlag.
  • Bosq, D., and D. Blanke. 2007. Inference and prediction in large dimensions, Wiley Series in Probability and Statistics. Chichester Dunod and Wiley Sons. https://onlinelibrary.wiley.com/doi/book/10.1002/9780470724033
  • Boukabour, S., and A. Masmoudi. 2020. Semiparametric Bayesian networks for continuous data. Communications in Statistics-Theory and Methods. Advance online publication. doi:10.1080/03610926.2020.1738486.
  • Bowman, A., and A. Azzalini. 1997. Applied smoothing techniques for data analysis: The kernel approach with S-Plus illustrations. Oxford University Press. https://doi.org/10.1007/s001800000033
  • Bradley, R. C. 2005. Introduction to strong mixing conditions. Vol. 2. Technical Report (March 2005 printing), Department of Mathematics, Indiana University, Bloomington. Bloomington: Custom Publishing of I.U.
  • Cai, Z. 1998. Asymptotic properties of Kaplan-Meier estimator for censored dependent data. Statistics & Probability Letters 37 (4):381–89. doi:10.1016/S0167-7152(97)00141-7.
  • Cai, Z. 2001. Estimating a distribution function for censored time series data. Journal of Multivariate Analysis. 78 (2):299–318. doi:10.1006/jmva.2000.1953.
  • Davies, I. 1973. Strong consistency of a sequential estimator of a probability density function. Bulletin of Mathematical Statistics 15 (3–4):49–54. doi:10.5109/13071.
  • Dedecker, J., P. Doukhan, G. Lang, J. R. Leon, S. Louhichi, and C. Prieur. 2007. Weak dependence with examples and applications. Lecture Notes in Statistics. Springer-Verlag. https://www.springer.com/gp/book/9780387699516
  • Deheuvels, P. 1973. Sur l’estimation sequentielle de la densite. C.R. Acad. Sci. Paris Ser. A-B 276:1119–21.
  • Devroye, L. 1979. On the pointwise and integral convergence of recursive kernel estimates of probability densities. Utilitas Math 15:113–28.
  • Devroye, L., and T. J. Wagner. 1980. On the L1 convergence of kernel estimators of regression functions with application in discrimination. Zeitschrift für Wahrscheinlichkeitstheorie und verwandte Gebiete 51 (1):15–25. doi:10.1007/BF00533813.
  • Doob, J. L. 1953. Stochastic processes. New York: John Wiley & Sons, Inc.; London: Chapman & Hall, Limited.
  • Doukhan, P. 1994. Lectures notes in statistics mixing: Properties and examples. 1st ed. Vol. 85, New York: Springer Verlag.
  • Dippon, J. 2003. Accelerated randomized stochastic optimization. The Annals of Statistics 31 (4):1260–81.
  • Duflo, M. 1996. Algorithmes stochastiques. In Collection applications of mathematics. Berlin: Springer.
  • Duflo, M. 1997. Random iterative models. In Collection applications of mathematics. Berlin: Springer.
  • Fabian, V. 1967. Stochastic approximation of minima with improved asymptotic speed. The Annals of Mathematical Statistics 38 (1):191–200. doi:10.1214/aoms/1177699070.
  • Györfi, L., A. Khöler, A. Krzyzak, and H. Walk. 2002. A distribution-free theory of nonparametric regression. New York: Springer-Verlag.
  • Hall, P., and C. C. Heyde. 1980. Martingale limit theory and its application. New York: Academic Press, Inc.
  • Hall, P., and P. Patil. 1994. On the efficiency of on-line density estimators. IEEE Transactions on Information Theory 40 (5):1504–12. doi:10.1109/18.333864.
  • 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.
  • Masry, E., and J. Fan. 1997. Local polynomial estimation of recursive function for mixing processes. Scandinavian Journal of Statistics. 24:165–79.
  • 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.
  • Mokkadem, A., M. Pelletier, and Y. Slaoui. 2009. The stochastic approximation method for the estimation of a multivariate probability density. Journal of Statistical Planning and Inference 139 (7):2459–78. doi:10.1016/j.jspi.2008.11.012.
  • Nadaraya, E. A. 1964. On estimating regression. Theory of Probability & Its Applications 9 (1):141–42.
  • Qin, Y. S. 1995. Asymptotic distribution of a recursive kernel estimator for a nonparametric regression function under dependent sampling. Mathematica Applicata 8:7–13.
  • Révész, P. 1973. Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I. Studia Scientiarum Mathematicarum Hungarica 8:391–98.
  • Révész, P. 1977. How to apply the method of stochastic approximation in the non-parametric estimation of a regression function. Series Statistics 8 (1):119–26. doi:10.1080/02331887708801361.
  • Rio, E. 2000. Théorie asymptotique des processus aléatoires faiblement dépendants. Mathématiques et applications. Vol. 31. New York: Springer-Verlag.
  • Robbins, H., and M. Monro. 1951. A stochastic approximation method. The Annals of Mathematical Statistics 22 (3):400–407. doi:10.1214/aoms/1177729586.
  • Roussas, G. G. 1990. Nonparametric regression estimation under mixing conditions. Stochastic Processes and Their Applications 36 (1):107–16. doi:10.1016/0304-4149(90)90045-T.
  • Roussas, G. G., and L. T. Tran. 1992. Asymptotic normality of the recursive kernel regression estimate under dependence conditions. The Annals of Statistics 20 (1):98–120. doi:10.1214/aos/1176348514.
  • Simonoff, J. S. 1996. Smoothing methods in statistics. New York: Springer.
  • Slaoui, Y. 2014a. Bandwidth selection for recursive kernel density estimators defined by stochastic approximation method. Journal of Probability and Statistics. 2014:1–11. doi:10.1155/2014/739640.
  • Slaoui, Y. 2014b. The stochastic approximation method for the estimation of a distribution function. Mathematical Methods of Statistics 23 (4):306–25. doi:10.3103/S1066530714040048.
  • Slaoui, Y. 2015. Plug-In Bandwidth selector for recursive kernel regression estimators defined by stochastic approximation method. Statistica Neerlandica. 69 (4):483–509. doi:10.1111/stan.12069.
  • Slaoui, Y. 2016a. 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.
  • Slaoui, Y. 2016b. On the choice of smoothing parameters for semi-recursive nonparametric hazard estimators. Journal of Statistical Theory and Practice 10 (4):656–72. doi:10.1080/15598608.2016.1214853.
  • 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.
  • Vilar, J. M., and J. A. Vilar. 2000. Recursive local polynomial regression under dependence conditions. Test 9:209–32.
  • Volkonskiĭ, V. A., and Y. Rozanov. 1959. A. Some limit theorems for random functions. I. (Russian) Teor. Veroyatnost. i Primenen 4: 186–207.
  • Wand, M. P., and S., M. C. Jones. 1995. Kernel smoothing. London: Chapman and Hall.
  • Wang, L., and H. Y. Liang. 2004. Strong uniform convergence of the recursive regression estimator under ψ-mixing conditions. Metrika 59 (3):245–61. doi:10.1007/s001840300282.
  • Watson, G. S. 1964. Smooth regression analysis. Sankhya A 26:359–72.
  • Wegman, J., and I. Davies. 1979. Remarks on some recursive estimators of a probability density. The Annals of Statistics 7 (2):316–27. doi:10.1214/aos/1176344616.
  • Wolverton, C., and T. J. Wagner. 1969. Asymptotically optimal discriminant functions for pattern classifcation. IEEE Transactions on Information Theory 15 (2):258–65. doi:10.1109/TIT.1969.1054295.
  • Yamato, H. 1971. Sequential estimation of a continuous probability density function and mode. Bulletin of Mathematical Statistics 14 (3–4):1–12. doi:10.5109/13049.

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