Bernstein–Frechet inequalities for the parameter of the first order autoregressive process

Abdelnasser Dahmani; Megdouda Tari

doi:10.1016/j.crma.2004.12.016

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Bernstein–Frechet inequalities for the parameter of the first order autoregressive process

Presented by: Paul Deheuvels

Abdelnasser Dahmani ¹; Megdouda Tari ¹

¹ Laboratory of Applied Mathematics, Department of Mathematics, University of Bejaia, Algeria

Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 309-314.

Abstract
Résumé

The autoregressive process takes an important part in predicting problems leading to decision making. In practice, we use the least squares method to estimate the parameter of the autoregressive process. In the case of the first order autoregressive process, we know that the least squares estimator converges in probability to the unknown parameter θ. In this Note, we show that the least squares estimator converges almost completely to θ and so we construct the inequalities of type Bernstein–Frechet for the coefficient of the first order autoregressive process. Using these inequalities a confidence interval is then obtained.

Les processus linéaires, en général et les processus autoregressifs, en particulier jouent un rôle important dans la prévision qui est fondamentale dans la mesure où elle à la base de l'action. En pratique on utilise la méthode des moindres carrées pour estimer les paramètres d'un processus autoregressif en minimisant la somme des erreurs au carrées. Dans le cas d'un processus autoregressif d'ordre 1, on sait que l'estimateur des moindres carrées converge en probabilités vers le paramètre inconnu θ. Dans cette Note, on montre que cet estimateur converge presque complétement vers θ et nous construisons des inégalités de type Bernstein–Fréchet pour le coefficient du processus autoregressif d'ordre 1. Ces inégalités nous ont permis de construire un intervalle de confiance pour ce coefficient.

Received: 2004-03-12
Accepted: 2004-12-14
Published online: 2005-01-22

DOI: 10.1016/j.crma.2004.12.016

Author's affiliations:

Abdelnasser Dahmani ¹; Megdouda Tari ¹

¹ Laboratory of Applied Mathematics, Department of Mathematics, University of Bejaia, Algeria

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     title = {Bernstein{\textendash}Frechet inequalities for the parameter of the first order autoregressive process},
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     pages = {309--314},
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Abdelnasser Dahmani; Megdouda Tari. Bernstein–Frechet inequalities for the parameter of the first order autoregressive process. Comptes Rendus. Mathématique, Volume 340 (2005) no. 4, pp. 309-314. doi : 10.1016/j.crma.2004.12.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.12.016/

References
Cited by

[1] B.V. Bondarev On an estimator of the unknown drift coefficient parameter for an equation perturbed by Gaussian noise, Theory Probab. Math. Statist., Volume 42 (1991), pp. 1-12

[2] N.H. Chan; C.Z. Wei Limiting distribution of last squares estimates of unstable autoregressive process, Ann. Statist., Volume 16 (1998), pp. 367-401

[3] G.M. Dufour Exact tests and confidence sets in linear regression with autocorrelated errors, Econometrica, Volume 58 (1990), pp. 475-494

[4] L. Galtchouk; V. Konev On uniform asymptotic normality of sequential least squares estimators for the parameters in a stable AR(1), Sequential Anal., Volume 22 (2003), pp. 31-54

[5] C. Gourieroux; A. Monfort Series temporelles et modèles dynamiques, Economica, Paris, 1995

[6] I.I. Guikhman; A.B. Skorokhod Theory of Random Process, vol. 1, Nauka, 1971

[7] T.L. Lai; D. Siegmund Fixed accuracy estimation of an autoregressive parameter, Ann. Statist., Volume 11 (1983), pp. 478-485

[8] B. Nielsen; N. Shephard Likelihood analysis of a first-order autoregressive model with exponential innovations, J. Time Series Anal., Volume 24 (2003) no. 3, pp. 337-344

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