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On Strong Consistency of Kernel Estimators Under Dependence Assumptions

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Mathematical Statistics and Probability Theory

Abstract

Some considerations on the uniform consistency of the kernel estimator of a density and of a regression function are made for certain dependent samples.

*partially supported by a NSF grant.

**partially supported by a Taft grant.

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References

  • Azuma, K. (1967). Weighted sums of certain dependent random variables. Tohoku Math. J. 19, 357–367.

    Article  MathSciNet  MATH  Google Scholar 

  • Bosq, D . (1975). Inegalité de Bernstein pour les processus stationnaires et mélangeants, Applications, C.R. Acad. Sci. Paris, 281, A, 1095–1098.

    MathSciNet  MATH  Google Scholar 

  • Chow, Y. S. (1966). Some convergence theorems for independent random variables. Ann. Math. Statist. 37, 1482–1493.

    Article  MathSciNet  MATH  Google Scholar 

  • Collomb, G. (1979). Conditions necessaires et sufficients de convergence uniform d’un estimateur de la régression, estimation des dérivées de la régression. C.R. Acad. Sci. Paris, 288, Ser. A, 161–164.

    MathSciNet  MATH  Google Scholar 

  • Collomb, G. (1984). Propriétès de convergence presque complete du prédicteur à noyau. Z. Wahr. verw.Gebiete 66, 441–460.

    Article  MathSciNet  MATH  Google Scholar 

  • Collomb, G., and Härdle, W. (1986). Strong uniform convergence rates in robust nonparametric time series analysis and prediction: kernel regression estimation from dependent observations. Stoch. Proc. and its Application (to appear).

    Google Scholar 

  • Deheuvels, P . (1974). Conditions necessaire et suffisantes de convergence ponctuelle presque sure et uniforme presque sure des estimateurs de la densite. C.R. Acad. Sci. Paris, Ser. A 278, 1217–1220.

    MathSciNet  MATH  Google Scholar 

  • Delecroix, M . (1980). Sur l’estimation des densités d’un processus stationaire a temps continue. Pulb. L’Inst. Statist. Univ. Paris 25, 17–40.

    MathSciNet  MATH  Google Scholar 

  • Devroye, L.P., Wagner, T.L. (1976). Nonparametric discrimination and density estimations, Technical Report 183, University of Texas, Austin.

    Google Scholar 

  • Eberlein, E. (1986). On strong invariance principle under dependence assumptions, Ann. of Prob., 14, 260–271.

    Article  MathSciNet  MATH  Google Scholar 

  • Kahane, J.P . (1986). Some random series of functions, Cambridge Studies in Advanced Mathematics 5.

    Google Scholar 

  • Komlos, J., Major, P., and Tusnady, G. (1975). An approximation of partial sums of independent random variables and the sample distribution functions. I. Z. Wahr. verw. Gebiete 32, 111–131.

    Article  MathSciNet  MATH  Google Scholar 

  • Härdle, W. and Marron, S. (1985). Optimal bandwidth selection in non-parametric regression function estimation, Ann. of Statistics 13, 1465–1482.

    Article  MATH  Google Scholar 

  • Ibragimov, I.A . (1962). Some limit theorem for stationary processes, Theory Prob. Appl. 7, 349–382.

    Article  MATH  Google Scholar 

  • Iosifescu, M., Theodorescu, R. (1969). Random Processes and Learning, Springer-Verlag.

    MATH  Google Scholar 

  • Nadaraja, E . (1964). On regression estimators, Theory Probab. Appl. 9, 157–159.

    MathSciNet  Google Scholar 

  • Parzen, E . (1962). On estimation of a probability density function and mode, Ann. Math. Statist. 33, 1065–1076.

    Article  MathSciNet  MATH  Google Scholar 

  • Peligrad, M. (1986). Properties of uniform consistency of the kernel estimator of a density and of a regression function under dependence assumptions. Preprint.

    Google Scholar 

  • Philipp, W . (1977). A functional law of the iterated logarithm for empirical distributions functions of weakly dependent random variables, Ann. Probab. 5, 319–350.

    Article  MathSciNet  MATH  Google Scholar 

  • Prakasa, Rao, B.L.S. (1983). Nonparametric functional estimation, Academic Press, New York.

    MATH  Google Scholar 

  • Rosenblatt, M . (1956). Remarks on some nonparametric estimates of a density function, Ann. Math. Statist. 27, 832–837.

    Article  MathSciNet  MATH  Google Scholar 

  • Rosenblatt, M. (1971). Markov processes, structure and asymptotic behavior, Springer-Verlag, Berlin.

    MATH  Google Scholar 

  • Roussas, G.G. (1969). Nonparametric estimation in Markov processes, Ann. Inst. Statist. Math. 21, 73–87.

    Article  MathSciNet  MATH  Google Scholar 

  • Rüschendorff, L. (1977). Consistency of estimators for multivariate density functions and for the mode. Sankhya Ser. A 39, 243–250.

    MathSciNet  Google Scholar 

  • Samur, J . (1984). Convergence of sums of mixing triangular arrays of random vectors with stationary rows, Ann. Probab. 12, 390–426.

    Article  MathSciNet  MATH  Google Scholar 

  • Serfling, R.J. (1969). Probability inequalities and convergence properties for sums of multiplicative random variables, Rep. M151. Florida State Dept. of Statist., Tallahasse, Florida.

    Google Scholar 

  • Silverman, B.W. (1978). Weak and strong uniform consistency of the kernel estimate of a density function and its derivatives, Ann. Statist. 6, 177–184 (Add. 8, 1175-1176 (1980)).

    Google Scholar 

  • Stout, W. (1974). Almost sure convergence. Academic Press.

    MATH  Google Scholar 

  • Watson, G.S. (1964). Smooth regression analysis, Sankhya Ser. A., 26, 359–372.

    MathSciNet  MATH  Google Scholar 

  • Yoshihara, K . (1978). Probability inequalities for sums of absolutely regular processes and their applications, Z. Wahr. verw. Gebiete, 43, 319–329.

    Article  MathSciNet  MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Cincinnati, Cincinnati, Ohio, 45221-0025, USA

    J. Deddens, M. Peligrad & T. Yang

Authors
  1. J. Deddens
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  2. M. Peligrad
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  3. T. Yang
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Editor information

Editors and Affiliations

  1. University of Cologne, Germany

    P. Bauer

  2. University of Agriculture, Vienna, Austria

    F. Konecny

  3. Technical University, Vienna, Austria

    W. Wertz

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© 1987 D. Reidel Publishing Company, Dordrecht, Holland

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Deddens, J., Peligrad, M., Yang, T. (1987). On Strong Consistency of Kernel Estimators Under Dependence Assumptions. In: Bauer, P., Konecny, F., Wertz, W. (eds) Mathematical Statistics and Probability Theory. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-3965-3_4

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  • DOI: https://doi.org/10.1007/978-94-009-3965-3_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-8259-4

  • Online ISBN: 978-94-009-3965-3

  • eBook Packages: Springer Book Archive

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