Advertisement

Ricerche di Matematica

, Volume 58, Issue 1, pp 103–127 | Cite as

On the asymptotic behavior of the parameter estimators for some diffusion processes: application to neuronal models

  • Maria Teresa Giraudo
  • Rosa Maria Mininni
  • Laura Sacerdote
Article

Abstract

We consider a sample \({\left\{T_n\right\}_{1\leq n\leq N}}\) of i.i.d. times and we interpret each item as the first-passage time (FPT) of a diffusion process through a constant boundary. The problem is to estimate the parameters characterizing the underlying diffusion process through the experimentally observable FPT’s. Recently in Ditlevsen and Lánský (Phys Rev E 71, 2005) and Ditlevsen and Lánský (Phys Rev E 73, 2006) closed form estimators have been proposed for neurobiological applications. Here we study the asymptotic properties (consistency and asymptotic normality) of the class of moment type estimators for parameters of diffusion processes like those in Ditlevsen and Lánský (Phys Rev E 71, 2005) and Ditlevsen and Lánský (Phys Rev E 73, 2006). Furthermore, to make our results useful for application instances we establish upper bounds for the rate of convergence of the empirical distribution of each estimator to the normal density. Applications are also considered by means of simulated experiments in a neurobiological context.

Keywords

Diffusion processes First-passage time Moment type estimators Asymptotic properties Rate of convergence Neuronal models 

Mathematics Subject Classification (2000)

62M05 62F12 60J60 62P10 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Aalen O.O., Gjessing H.K.: Survival models based on the Ornstein-Uhlenbeck process. Lifetime Data Anal. 10, 407–423 (2005)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Alili L., Patie P., Pedersen J.L.: Representation of the first hitting time densiy of an Ornstein- Uhlenbeck process. Stoch. Models 21, 967–980 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Bulsara A.R., Elston T.C., Doering C.R., Lowen S.B., Lindenberg K.: Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics. Phys. Rev. E 53(4), 3958–3969 (1996)CrossRefGoogle Scholar
  4. 4.
    Casella G., Berger R.L.: Statistical Inference, 2nd edn. Duxbury Advanced Series, USA (2002)Google Scholar
  5. 5.
    Cox J.C., Ingersoll J.E., Ross S.A.: A theory of the term structure of interest rates. Econometrica 53, 385–407 (1985)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Ditlevsen S.: A result on the FPT of an Ornstein-Uhlenbeck process. Statist. Probab. Lett. 77(18), 1744–1749 (2007)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ditlevsen S., Ditlevsen O.: Parameter estimation from observations of first-passage times of the Ornstein-Uhlenbeck process and the Feller process. Probab. Eng. Mech. 23(2–3), 170–179 (2008)CrossRefGoogle Scholar
  8. 8.
    Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Ornstein-Uhlenbeck neuronal model. Phys. Rev. E 71, Art. No. 011907 (2005)Google Scholar
  9. 9.
    Ditlevsen, S., Lánský, P.: Estimation of the input parameters in the Feller neuronal model. Phys. Rev. E 73, Art. No. 061910 (2006)Google Scholar
  10. 10.
    Ditlevsen, S., Lánský, P.: Parameters of stochastic diffusion processes estimated from observations of first-hitting times: application to the leaky integrate-and-fire neuronal model. Phys. Rev. E 76, Art. No. 041906 (2007)Google Scholar
  11. 11.
    Erdélyi A., Magnus W., Oberhettinger F., Tricomi F.G.: Higher transcendental functions. McGrawHill, New York (1953)Google Scholar
  12. 12.
    Feller, W.: Diffusion processes in genetics. In: Neyman, J. (ed.) Proceedings of the second Berkeley symposium on mathematical statistics and probability. University of California Press, Berkeley, pp. 227–246 (1951)Google Scholar
  13. 13.
    Giorno V., Lánský P., Nobile A.G., Ricciardi L.M.: Diffusion approximation and first-passage-time problem for a model neuron III. A birth and death proces approach. Biol. Cybern. 58, 387–404 (1988)zbMATHCrossRefGoogle Scholar
  14. 14.
    Giorno V., Nobile A.G., Ricciardi L.M., Sato S.: On the evaluation of first-passage-time probability densities via non singular integral equations. Adv. Appl. Probab. 21, 20–36 (1989)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Giraudo M.T., Sacerdote L.: An improved technique for the simulation of first passage times for diffusion processes. Commun. Stat. Simul. Comput. 28(4), 1135–1163 (1999)zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Giraudo M.T., Sacerdote L., Zucca C.: A Monte Carlo method for the simulation of first passage times of diffusion processes. Methodol. Comput. Appl. Probab. 3, 215–231 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Heath R.A., Willcox C.H.: A stochastic model for inter-keypress times in a typing task. Acta Psychol. (Amst.) 75(1), 13–39 (1990)CrossRefGoogle Scholar
  18. 18.
    Inoue J., Sato S., Ricciardi L.M.: On the parameter estimation for diffusion models of single neurons activites. Biol. Cybern. 73, 209–221 (1995)zbMATHCrossRefGoogle Scholar
  19. 19.
    Karlin S., Taylor H.M.: A second course in stochastic processes. Academic Press, Inc., London (1981)zbMATHGoogle Scholar
  20. 20.
    Linetsky V.: Computing hitting time densities for CIR and OU diffusions: applications to mean-reverting models. J. Comput. Finance 7, 1–22 (2004)MathSciNetGoogle Scholar
  21. 21.
    Leblanc B., Scaillet O.: Path dependent options on yields in the affine term structure. Finance Stoch. 2, 349–367 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  22. 22.
    Lánský P., Sacerdote L., Tomassetti F.: On the comparison of Feller and Ornstein-Uhlenbeck models for neural activity. Biol. Cybern. 73, 457–465 (1995)zbMATHCrossRefGoogle Scholar
  23. 23.
    Morris K.W., Szynal D.: On the convergence rate in the central limit theorem of some functions of the average of independent random variables. Prob. Math. Stat. 3(1), 85–95 (1982)zbMATHMathSciNetGoogle Scholar
  24. 24.
    Mullowney P., Iyengar S.: Parameter estimation for a leaky integrate-and-fire neuronal model from ISI data. J. Comput. Neurosci. 24(2), 179–194 (2008)CrossRefMathSciNetGoogle Scholar
  25. 25.
    Nobile A.G., Ricciardi L.M., Sacerdote L.: Exponential trends for a class of diffusion processes with steady state distribution. J. Appl. Prob. 22, 611–618 (1985)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Patie P.: A martingale associated to generalized Ornstein-Uhlenbeck processes and application to finance. Stoch. Process. Appl. 115(4), 593–607 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Ricciardi, L.M.: Diffusion processes and related topics in biology. Lecture Notes in Biomathematics, vol. 14. Springer, Berlin (1977)Google Scholar
  28. 28.
    Ricciardi L.M., Sacerdote L.: The Ornstein-Uhlenbeck process as a model for neuronal activity. Biol. Cybern. 35, 1–9 (1979)zbMATHCrossRefGoogle Scholar
  29. 29.
    Ricciardi L.M., Sato S.: First-passage-time density and moments of the Ornstein-Uhlenbeck process. J. Appl. Prob. 25, 43–57 (1988)zbMATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Ricciardi L.M., Di Crescenzo A., Giorno V., Nobile A.G.: An outline of theoretical and algorithm approaches to first passage time problems with applications to biological modeling. Math. Jpn. 50(2), 247–322 (1999)zbMATHMathSciNetGoogle Scholar
  31. 31.
    Shiryaev A.N.: Probability, 2nd ed. Springer, New York (1996)Google Scholar
  32. 32.
    Shreve S.E.: Stochastic calculus for finance II. Continuous-time models, Springer Finance. Springer, USA (2004)Google Scholar
  33. 33.
    Tuckwell H.C.: Introduction to theoretical neurobiology, vol. 2: Nonlinear and stochastic theories. Cambridge University Press, Cambridge (1988)Google Scholar
  34. 34.
    Vaugirard V.: A canonical first passage time model to pricing nature-linked bonds. Econ. Bull. 7(2), 1–7 (2004)Google Scholar

Copyright information

© Università degli Studi di Napoli "Federico II" 2009

Authors and Affiliations

  • Maria Teresa Giraudo
    • 1
  • Rosa Maria Mininni
    • 2
  • Laura Sacerdote
    • 1
  1. 1.Department of MathematicsUniversity of TorinoTorinoItaly
  2. 2.Department of MathematicsUniversity of BariBariItaly

Personalised recommendations