LAMN property for hidden processes : the case of integrated diffusions

Arnaud Gloter; Emmanuel Gobet

Annales de l'I.H.P. Probabilités et statistiques (2008)

  • Volume: 44, Issue: 1, page 104-128
  • ISSN: 0246-0203

Abstract

top
In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of X.

How to cite

top

Gloter, Arnaud, and Gobet, Emmanuel. "LAMN property for hidden processes : the case of integrated diffusions." Annales de l'I.H.P. Probabilités et statistiques 44.1 (2008): 104-128. <http://eudml.org/doc/77957>.

@article{Gloter2008,
abstract = {In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of X.},
author = {Gloter, Arnaud, Gobet, Emmanuel},
journal = {Annales de l'I.H.P. Probabilités et statistiques},
keywords = {diffusion processes; parametric estimation; LAMN property; Malliavin calculus; non-markovian data; local asymptotic mixed normality; integrated diffusion processes; likelihood ratio; conditional information},
language = {eng},
number = {1},
pages = {104-128},
publisher = {Gauthier-Villars},
title = {LAMN property for hidden processes : the case of integrated diffusions},
url = {http://eudml.org/doc/77957},
volume = {44},
year = {2008},
}

TY - JOUR
AU - Gloter, Arnaud
AU - Gobet, Emmanuel
TI - LAMN property for hidden processes : the case of integrated diffusions
JO - Annales de l'I.H.P. Probabilités et statistiques
PY - 2008
PB - Gauthier-Villars
VL - 44
IS - 1
SP - 104
EP - 128
AB - In this paper we prove the Local Asymptotic Mixed Normality (LAMN) property for the statistical model given by the observation of local means of a diffusion process X. Our data are given by ∫01X(s+i)/n dμ(s) for i=0, …, n−1 and the unknown parameter appears in the diffusion coefficient of the process X only. Although the data are neither markovian nor gaussian we can write down, with help of Malliavin calculus, an explicit expression for the log-likelihood of the model, and then study the asymptotic expansion. We actually find that the asymptotic information of this model is the same one as for a usual discrete sampling of X.
LA - eng
KW - diffusion processes; parametric estimation; LAMN property; Malliavin calculus; non-markovian data; local asymptotic mixed normality; integrated diffusion processes; likelihood ratio; conditional information
UR - http://eudml.org/doc/77957
ER -

References

top
  1. O. E. Barndorff-Nielsen and N. Shephard. Non-Gaussian Ornstein–Uhlenbeck-based models and some of their uses in financial economics. J. R. Stat. Soc. Ser. B Stat. Methodol. 63 (2001) 167–241. Zbl0983.60028MR1841412
  2. P. D. Ditlevsen, S. Ditlevsen and K. K. Andersen. The fast climate fluctuations during the stadial and interstadial climate states. Ann. Glaciology 35 (2002) 457–462. 
  3. S. Ditlevsen and M. Sørensen. Inference for observations of integrated diffusion processes. Scand. J. Statist. 31 (2004) 417–429. Zbl1062.62157MR2087834
  4. G. Dohnal. On estimating the diffusion coefficient. J. Appl. Probab. 24 (1987) 105–114. Zbl0615.62109MR876173
  5. V. Genon-Catalot and J. Jacod. On the estimation of the diffusion coefficient for multi-dimensional processes. Ann. Inst. H. Poincaré Probab. Statist. 29 (1993) 119–151. Zbl0770.62070MR1204521
  6. V. Genon-Catalot, T. Jeantheau and C. Laredo. Parameter estimation for discretely observed stochastic volatility models. Bernoulli 5 (1999) 855–872. Zbl0942.62096MR1715442
  7. A. Gloter. Discrete sampling of an integrated diffusion process and parameter estimation of the diffusion coefficient. ESAIM Probab. Statist. 4 (2000) 205–227. Zbl1043.62070MR1808927
  8. A. Gloter. Parameter estimation for a discrete sampling of an integrated Ornstein–Uhlenbeck process. Statistics 35 (2001) 225–243. Zbl0980.62072MR1925514
  9. A. Gloter and J. Jacod. Diffusions with measurement errors. I. Local asymptotic normality. ESAIM Probab. Statist. 5 (2001) 225–242. Zbl1008.60089MR1875672
  10. E. Gobet. Local asymptotic mixed normality property for elliptic diffusion: a Malliavin calculus approach. Bernoulli 7 (2001) 899–912. Zbl1003.60057MR1873834
  11. E. Gobet. LAN property for ergodic diffusions with discrete observations. Ann. Inst. H. Poincaré Probab. Statist. 38 (2002) 711–737. Zbl1018.60076MR1931584
  12. F. Hirsch and S. Song. Criteria of positivity for the density of a Wiener functional. Bull. Sci. Math. 121 (1997) 261–273. Zbl0882.60082MR1456283
  13. F. Hirsch and S. Song. Properties of the set of positivity for the density of a regular Wiener functional. Bull. Sci. Math. 122 (1998) 1–15. Zbl0897.60060MR1617582
  14. I. A. Ibragimov and R. Z. Has’minskii. Statistical Estimation. Asymptotic Theory. Springer-Verlag, New York–Berlin, 1981. (Translated from the Russian by Samuel Kotz.) Zbl0467.62026MR620321
  15. J. Jacod. On continuous conditional Gaussian martingales and stable convergence in law. In Séminaire de probabilité XXXI, 232–246. Lecture Notes in Math. 1655. Springer, Berlin, 1997. Zbl0884.60038MR1478732
  16. P. Jeganathan. On the asymptotic theory of estimation when the limit of the log-likelihood ratios is mixed normal. Sankhyā Ser. A 44 (1982) 173–212. Zbl0584.62042MR688800
  17. P. Jeganathan. Some asymptotic properties of risk functions when the limit of the experiment is mixed normal. Sankhyā Ser. A 45 (1983) 66–87. Zbl0574.62035MR749355
  18. A. Kohatsu-Higa. Lower bounds for densities of uniformly elliptic random variables on Wiener space. Probab. Theory Related Fields 126 (2003) 421–457. Zbl1022.60056MR1992500
  19. P. Krée and C. Soize. Mathematics of Random Phenomena. Random Vibrations of Mechanical Structures. D. Reidel Publishing Co., Reidel, Dordrecht, 1986. Zbl0628.73099MR873731
  20. H. Kunita. Stochastic differential equations and stochastic flows of diffeomorphisms. École d’été de probabilités de Saint-Flour, XII – 1982, 143–303. Lecture Notes in Math. 1097. Springer, Berlin, 1984. Zbl0554.60066MR876080
  21. L. Le Cam and G. Lo Yang. Asymptotics in Statistics. Some Basic Concepts, 2nd edition. Springer-Verlag, New York, 2000. Zbl0952.62002MR1784901
  22. D. Nualart. The Malliavin Calculus and Related Topics. Probability and Its Application. Springer-Verlag, New-York, 1995. Zbl0837.60050MR1344217
  23. D. Nualart. Analysis on Wiener space and anticipating stochastic calculus. Lectures on Probability Theory and Statistics (Saint-Flour, 1995), 123–227. Lecture Notes in Math. 1690. Springer, Berlin, 1998. Zbl0915.60062MR1668111
  24. B. L. S. Prakasa Rao. Statistical Inference for Diffusion Type Processes. Edward Arnold, London; Oxford University Press, New York, 1999. Zbl0952.62077MR1717690
  25. D. Revuz and M. Yor. Continuous Martingales and Brownian Motion, 3rd edition. Springer-Verlag, Berlin, 1999. Zbl0917.60006MR1725357

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.