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Asymptotics for the Lp -deviation of the variance estimator under diffusion

Published online by Cambridge University Press:  15 September 2004

Paul Doukhan  and
José R. León
Paul Doukhan
Affiliation:
Laboratoire de Statistiques LS-CREST, ENSAE, 3 rue Pierre Larousse, France; [email protected].
José R. León
Affiliation:
Universidad Central de Venezuela, Escuela de Matemática, Facultad de Ciencias, AP. 47197, Los Chaguaramos Caracas 1041-A, Venezuela; [email protected].
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Abstract

We consider a diffusion process X t smoothed with (small)sampling parameter ε. As in Berzin, León and Ortega(2001), we consider a kernel estimate $\widehat{\alpha}_{\varepsilon}$ with window h(ε) of afunction α of its variance. In order to exhibit globaltests of hypothesis, we derive here central limit theorems forthe Lp deviations such as \[ \frac1{\sqrt{h}}\left(\frac{h}\varepsilon\right)^{\frac{p}2}\left(\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p-\mbox{I E}\left\|\widehat{\alpha}_{\varepsilon}-{\alpha}\right\|_p^p\right).\]

Keywords

Variance estimatorkernelLp -deviationcentral limit theorem.
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 8 , August 2004 , pp. 132 - 149
Copyright
© EDP Sciences, SMAI, 2004

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References

Beirlant, J. and Mason, D.M., On the asymptotic normality of the Lp -norm of empirical functional. Math. Methods Statist. 4 (1995) 1–19.
Berzin-Joseph, C., León, J.R. and Ortega, J., Non-linear functionals of the Brownian bridge and some applications. Stoch. Proc. Appl. 92 (2001) 11–30. CrossRef
P. Brugière, Théorème de limite centrale pour un estimateur non paramétrique de la variance d'un processus de diffusion multidimensionnelle. Ann. Inst. Henri Poincaré, Probab. Stat. 29 (1993) 357–389.
P.D. Ditlevsen, S. Ditlevsen and K.K. Andersen, The fast climate fluctuations during the stadial and interstadial climate states. Ann. Glaciology 35 (2002).
Doukhan, P., León, J.R. and Portal, F., Calcul de la vitesse de convergence dans le théorème central limite vis-à-vis des distances de Prohorov, Dudley et Lévy dans le cas de v. a. dépendantes. Probab. Math. Statist. 6 (1985) 19–27.
Genon-Catalot, V., Laredo, C. and Picard, D., Non-parametric estimation of the diffusion coefficient by wavelets methods. Scand. J. Statist. 19 (1992) 317–335.
I.J. Gihman and A.V. Skorohov, Stochastic differential equations. Springer-Verlag, Berlin, New York (1972).
E. Giné, D. Mason and Yu. Zaitsev, The L 1-norm density estimator process. To appear in Ann. Prob.
Gloter, A., Parameter estimation for a discrete sampling of an integrated Ornstein-Uhlenbeck process. Statistics 35 (2000) 225–243. CrossRef
J. Jacod, On continuous conditional martingales and stable convergence in law, sémin. Probab. XXXI, LNM 1655, Springer (1997) 232–246.
P. Major, Multiple Wiener-Itô integrals. Springer-Verlag, New York, Lect. Notes Math. 849 (1981).
Perera, G. and Wschebor, M., Crossings and occupation measures for a class of semimartingales. Ann. Probab. 26 (1998) 253–266.
G. Perera and M. Wschebor, Inference on the variance and smoothing of the paths of diffusions. Ann. Inst. Henri Poincaré, Probab. Stat. 38 (2002) 1009–1022.
Rio, E., About the Lindeberg method for strongly mixing sequences. ESAIM: PS 1 (1995) 35–61. CrossRef
Rosenthal, H.P., On the subspaces of Lp , (p > 2) spanned by sequences of independent random variables. Israël Jour. Math. 8 (1970) 273–303. CrossRef
Shergin, V.V., On the convergence rate in the central limit theorem for m-dependent random variables. Theor. Proba. Appl. 24 (1979) 782–796. CrossRef
Soulier, P., Non-parametric estimation of the diffusion coefficient of a diffusion process. Stoch. Anal. Appl. 16 (1998) 185–200.
G. Terdik, Bilinear Stochastic Models and Related problems of Nonlinear Time Series. Springer-Verlag, New York, Lect. Notes Statist. 142 (1999).