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# Traitement statistique des processus alpha-stables: mesures de dépendance et identification des ar stables. Test séquentiels tronqués

Abstract : In this work, we thoroughly study the $\al$-stable distributions (laws with infinite variance). In the first chapter, we recall the various properties of the univariate $\al$-stable distribution (stability, calculus of moments, simulation). Then we introduce the symmetric $\al$-stable (\SaS) multivariate distributions. After having stressed the importance of spectral measure with respect to the notion of independence for the \SaS\ vectors, we concentrate on the measures of dependence. In the second chapter, noting that the coefficient of covariation, widely used by statisticians, admits some limit, we build a new measure of dependence, called symmetric coefficient of covariation. This last one allows us discovering some unexpected things. Indeed, contrary to the Gaussian vectors, for certain \SaS\ vectors one can obtain both a positive dependence and a negative dependence. After having concluded the chapter by the study of the asymptotic law of the estimator of the coefficient of covariation, in the third chapter, we address the autoregressive processes with stable innovations. We present various methods of identification of the order of a AR process: partial autocorrelations (Brockwell and Davis) and asymptotically invariant rank-based quadratic statistics (Garel and Hallin). Many simulations, carried out in Matlab and Fortran, enable us to compare these methods and to note the importance of the role played by the rank-statistics in this field. To finish, a sequential test problem, developed at the occasion of an industrial contract, gives us the opportunity to introduce the concept of confidence level after decision.
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Contributor : Ludovic d'Estampes Connect in order to contact the contributor
Submitted on : Thursday, March 4, 2004 - 8:54:11 PM
Last modification on : Thursday, October 21, 2021 - 3:06:48 AM
Long-term archiving on: : Friday, April 2, 2010 - 8:11:45 PM

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• HAL Id : tel-00005216, version 1

### Citation

Ludovic d'Estampes. Traitement statistique des processus alpha-stables: mesures de dépendance et identification des ar stables. Test séquentiels tronqués. Mathématiques [math]. Institut National Polytechnique de Toulouse - INPT, 2003. Français. ⟨tel-00005216⟩

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