Skip to main content
SpringerLink
Log in

Estimation of Local Smoothness Coefficients for Continuous Time Processes

  • Published:
Statistical Inference for Stochastic Processes Aims and scope Submit manuscript

Abstract

We consider processes that satisfied a local Hölder condition with coefficient γ0. According to the sampling times of observations given by iδ n with i = 0,...,n − 1, we study two general classes of estimators for γ0. Their almost sure rates of convergence depend on asymptotic independence of the observed processes, on δ n and eventually on an extra parameter β0. Since this last parameter is in general unknown, we construct a family of preliminary estimators for β0 with their rates of almost sure convergence. Finally we present some numerical simulations in order to compare the behaviour of our various estimators.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Adler, R. J.: The Geometry of Random Fields, Wiley, New York, 1981.

    Google Scholar 

  2. Blanke, D. and Bosq, D.: Accurate rates of density estimators for continuous time processes, Statist. Probab. Lett. 33(2) (1997), 185-191.

    Google Scholar 

  3. Blanke, D. and Bosq, D.: A family of minimax rates for density estimators in continuous time, Stoch. Anal. Appl. 18(6) (2000), 871-900.

    Google Scholar 

  4. Blanke, D. and Pumo, B.: Optimal sampling for density estimation in continuous time, In revision for J. Time Ser. Anal. 1999.

  5. Bosq, D.: Nonparametric Statistics for stochastic Processes. Estimation and Prediction, Lecture Notes in Statist. 110, 2nd edn, Springer Verlag, New York, 1998.

    Google Scholar 

  6. Bosq, D. and Davydov, Y. A.: Local time and density estimation in continuous time, Math. Meth. Statist. 8 (1999), 22-45.

    Google Scholar 

  7. Bradley, R.: Approximation theorems for strongly mixing random variables, Michigan Math. J. 30 (1983), 69-81.

    Google Scholar 

  8. Chan, G., Hall, P. and Poskitt, D.: Periodogram-based estimators of fractal properties, Ann. Statist. 23(5) (1995), 1684-1711.

    Google Scholar 

  9. Constantine, A. G. and Hall, P.: Characterizing surface smoothness via estimation of effective fractal dimension, J. Raoy. Statist. Soc. Ser. B 56(1) (1994), 97-113.

    Google Scholar 

  10. Csáki, E. and Csörg?, M.: Inequalities for increments of stochastic processes and moduli of continuity, Ann. Probab. 20(2) (1992), 1031-1052.

    Google Scholar 

  11. Davydov, Y. A.: Convergence of distributions generated by stationary stochastic processes, Theor. Prob. Appl. 13 (1968), 691-696.

    Google Scholar 

  12. Feuerverger, A., Hall, P. and Wood, A.: Estimation of fractal index and fractal dimension of a Gaussian process by counting the number of level crossing, J. Time Ser. Anal. 15(6) (1994), 587-606.

    Google Scholar 

  13. Ibragimov, I. A. and Rozanov, Y. A.: Gaussian Random Processes, Springer Verlag, 1978.

  14. Istas, J. and Lang, G.: Quadratic variations and estimation of the local hölder index of a Gaussian process, Ann. Inst. H. Poincaré, Probab. Statist. 33(4) (1997), 407-436.

    Google Scholar 

  15. Kent, J. T. and Wood, A. T.: Estimating the fractal dimension of a locally self-similar Gaussian process by using increments, J. Roy. Statist. Soc. Ser. B 59(3) (1997), 679-699.

    Google Scholar 

  16. Kloeden, P. E. and Platen, E.: Numerical solution of stochastic differential equations. In: Applications of Mathematics, Stochastic Modelling and Applied Probability, vol. 23, 2nd edn, Springer Verlag, 1995.

  17. Peltier, R. F. and Lévy-Véhel, J.: A new method for estimating the parameter of fractional brownian motion, Technical Report 2396, I.N.R.I.A., November 1994.

  18. Philippe, A. and Thilly, E.: Identification of locally self-similar Gaussian process by using convex rearrangements, Technical Report, Publ. I.R.M.A. 53(4), Université des Sciences et Technologies de Lille, 2000.

  19. Sköld, M. and Hössjer, O.: On the asymptotic variance of the continuous-time kernel density estimator, Statist. Probab. Lett. 44(1) (1999), 97-106.

    Google Scholar 

  20. Skorohod, A. V.: Random processes with independent increments. In: Mathematics and Its Applications, vol. 47, Kluwer Academic Publisher, 1991.

  21. Taylor, C. C. and Taylor, S. J.: Estimating the dimension of a fractal, J. Roy. Statist. Soc. Ser. B 53 (1991), 353-364.

    Google Scholar 

  22. Viano, M. C., Deniau, C. and Oppenheim, G.: Continuous-time fractional ARMA processes, Statist. Probab. Lett. 21 (1994), 323-336.

    Google Scholar 

  23. Wong, E.: Some results concerning the zero-crossings of Gaussian noise, SIAM J. Appl. Math. 14(6) (1966), 1246-1254.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. L.M.A.H. (Université du Havre), 25 rue Philippe Lebon, BP 540, 76058, Le Havre Cedex, France

    D. Blanke

  2. L.S.T.A. (Université Paris 6), France

    D. Blanke

Authors
  1. D. Blanke
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Blanke, D. Estimation of Local Smoothness Coefficients for Continuous Time Processes. Statistical Inference for Stochastic Processes 5, 65–93 (2002). https://doi.org/10.1023/A:1013726523753

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/A:1013726523753

Search

Navigation