Statistics/Probability Theory
A locally asymptotically powerful test for nonlinear autoregressive models
[Test localement asymptotiquemt puissant pour des modèles autoregressifs nonlinéaires]
Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 671-676.

Dans cette Note, nous proposons un test localement asymptotiquement puissant pour traiter simultanément des hypothèses portant sur les fonctions moyenne et variance conditionnelles. Ceci est fait sous des conditions de stationnarité et d'érgodicité sur une classe de modèles semi-paramétriques non-linéaires que nous considérons et lorsque la loi des innovations n'est pas nécessairement spécifiée. Basé sur une version modifiée de la méthode de Le Cam, dûe à Hwang et Basawa (2001), nous établissons la normalité locale asymptotique relative aux modèles étudiés. Le résultat principal montre que la statistique du test, construite en substituant aux résidus et aux paramètres des estimateurs consistants, est asymptotiquement normale. La puissance asymptotique du test proposé est calculée et des simulations ont été effectuées pour évaluer sa performance.

We propose a locally asymptotically powerful test to simultaneously examine hypotheses relative to the parametric form of the conditional mean and the conditional variance functions in a class of nonlinear semi-parametric time series models without a specified error law. On the basis of a modified version of the Le Cam method of Hwang and Basawa (2001), we establish the local asymptotic normality relative to the model. The main result shows that the test statistic built by substituting consistent estimated residuals and parameters for the theoretical ones is asymptotically normal. Its asymptotic power is computed and the result is illustrated by some simulations.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.02.017
Chebana, Fateh 1 ; Laïb, Naâmane 2

1 Université du Québec, INRS-ETE, Québec G1K 9A9, Canada
2 L.S.T.A, Université Paris VI, 175, rue du Chevaleret, 75013 Paris, France
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Chebana, Fateh; Laïb, Naâmane. A locally asymptotically powerful test for nonlinear autoregressive models. Comptes Rendus. Mathématique, Tome 346 (2008) no. 11-12, pp. 671-676. doi : 10.1016/j.crma.2008.02.017. http://www.numdam.org/articles/10.1016/j.crma.2008.02.017/

[1] Diebolt, J.; Laïb, N. A weak invariance principle for cumulated functionals of the regressogram estimator with dependent data, J. Nonparametr. Statist., Volume 4 (1994) no. 2, pp. 149-163

[2] Diebolt, J.; Laïb, N.; Ngatchou-Wandji, J. Limiting distribution of weighted processes of residuals. Application to parametric nonlinear autoregressive models, C. R. Acad. Sci. Paris Sér. I Math., Volume 325 (1997) no. 5, pp. 535-540

[3] Doukhan, P. Mixing, Lecture Notes in Statistics, vol. 85, Springer-Verlag, New York, 1994

[4] J. Gao, M. King, Model specification testing in nonparametric and semiparametric time series econometric models, North American Winter Meetings, vol. 225, Econometric Society, 2004

[5] Hall, W.J.; Mathiason, D.J. On large-sample estimation and testing in parametric models, Internat. Statist. Rev., Volume 58 (1990), pp. 77-97

[6] Härdle, W.; Mammen, E. Comparing nonparametric versus parametric regression fits, Ann. Statist., Volume 21 (1993) no. 4, pp. 1926-1947

[7] Hwang, S.Y.; Basawa, I.V. Nonlinear time series contiguous to AR(1) processes and a related efficient test for linearity, Statist. Probab. Lett., Volume 52 (2001) no. 4, pp. 381-390

[8] Laïb, N. Nonparametric testing for correlation models with dependent data, J. Nonparametr. Statist., Volume 12 (1999) no. 1, pp. 53-82

[9] Laïb, N. Non-parametric testing of conditional variance functions in time series, Aust. N. Z. J. Stat., Volume 45 (2003) no. 4, pp. 461-475

[10] McKeague, I.W.; Zhang, M. Identification of nonlinear time series from first order cumulative characteristics, Ann. Statist., Volume 22 (1994) no. 1, pp. 495-514

[11] Ngatchou-Wandji, J. Checking nonlinear heteroscedastic time series models, J. Statist. Planning and Inference, Volume 133 (2005) no. 1, pp. 33-68

[12] J. Ngatchou-Wandji, N. Laïb, Local power of a cramér-von Mises type test for parametric autoregressive models of order one, Comp. Math. Appl., , 2008 | DOI

[13] Stute, W. Nonparametric model checks for regression, Ann. Statist., Volume 25 (1997) no. 2, pp. 613-641

[14] Taniguchi, M.; Kakizawa, Y. Asymptotic Theory of Statistical Inference for Time Series, Springer-Verlag, New York, 2000

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