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Strong consistency and rates for recursive probability density estimators of stationary processes

Author

Listed:
  • Masry, Elias
  • Györfi, László

Abstract

Let {Xj}j = - [infinity][infinity] be a vector-valued stationary process with a first-order univariate probability density f on Rd. We consider the recursive estimation of f(x) from n observations {Xj}j=1n which need not be independent. For processes {Xj}j = - [infinity][infinity] which are asymptotically uncorrelated, we establish sharp rates for the almost sure convergence of kernel-type estimators fn(x).

Suggested Citation

  • Masry, Elias & Györfi, László, 1987. "Strong consistency and rates for recursive probability density estimators of stationary processes," Journal of Multivariate Analysis, Elsevier, vol. 22(1), pages 79-93, June.
    • Handle: RePEc:eee:jmvana:v:22:y:1987:i:1:p:79-93
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    Citations

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    Cited by:

    1. Carbon, Michel & Tran, Lanh Tat & Wu, Berlin, 1997. "Kernel density estimation for random fields (density estimation for random fields)," Statistics & Probability Letters, Elsevier, vol. 36(2), pages 115-125, December.
    2. Lacour, Claire, 2008. "Nonparametric estimation of the stationary density and the transition density of a Markov chain," Stochastic Processes and their Applications, Elsevier, vol. 118(2), pages 232-260, February.
    3. Senoussi, R., 2000. "Uniform iterated logarithm laws for martingales and their application to functional estimation in controlled Markov chains," Stochastic Processes and their Applications, Elsevier, vol. 89(2), pages 193-211, October.
    4. P. Cattiaux & José R. León & C. Prieur, 2015. "Recursive estimation for stochastic damping hamiltonian systems," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 27(3), pages 401-424, September.
    5. Michel Carbon, 2005. "Frequency Polygons for Random Fields," Working Papers 2005-04, Center for Research in Economics and Statistics.
    6. Hallin, Marc & Lu, Zudi & Tran, Lanh T., 2004. "Kernel density estimation for spatial processes: the L1 theory," Journal of Multivariate Analysis, Elsevier, vol. 88(1), pages 61-75, January.
    7. Lu, Zudi & Chen, Xing, 2004. "Spatial kernel regression estimation: weak consistency," Statistics & Probability Letters, Elsevier, vol. 68(2), pages 125-136, June.
    8. Marc Hallin & Lanh Tran, 1996. "Kernel density estimation for linear processes: Asymptotic normality and optimal bandwidth derivation," Annals of the Institute of Statistical Mathematics, Springer;The Institute of Statistical Mathematics, vol. 48(3), pages 429-449, September.
    9. Sophie Dabo-Niang & Anne-Françoise Yao, 2013. "Kernel spatial density estimation in infinite dimension space," Metrika: International Journal for Theoretical and Applied Statistics, Springer, vol. 76(1), pages 19-52, January.
    10. Michel Carbon, 2008. "Asymptotic Normality of Frequency Polygons for Random Fields," Working Papers 2008-09, Center for Research in Economics and Statistics.
    11. Carbon, Michel & Garel, Bernard & Tran, Lanh Tat, 1997. "Frequency polygons for weakly dependent processes," Statistics & Probability Letters, Elsevier, vol. 33(1), pages 1-13, April.

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