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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; doukhan@ensae.fr.
José R. León
Affiliation:
Universidad Central de Venezuela, Escuela de Matemática, Facultad de Ciencias, AP. 47197, Los Chaguaramos Caracas 1041-A, Venezuela; jleon@euler.ciens.ucv.ve.
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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.

Information

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

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