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Estimation in autoregressive model with measurementerror

Published online by Cambridge University Press:  03 October 2014

Jérôme Dedecker ,
Adeline Samson  and
Marie-Luce Taupin
Jérôme Dedecker
Affiliation:
Laboratoire MAP5 UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris Cité, Paris cedex 6, France
Adeline Samson
Affiliation:
Laboratoire MAP5 UMR CNRS 8145, Université Paris Descartes, Sorbonne Paris Cité, Paris cedex 6, France
Marie-Luce Taupin
Affiliation:
Laboratoire Statistique et Génome, UMR CNRS 8071-USC INRA, Université d’Évry Val d’Essonne, Évry, France. [email protected]
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Abstract

Consider an autoregressive model with measurement error: we observe Zi =Xi +εi, where theunobserved Xi is a stationarysolution of the autoregressive equation Xi =gθ0(Xi− 1) + ξi. Theregression function gθ0 isknown up to a finite dimensional parameter θ0 to be estimated. The distributions ofξ1 and X0 are unknownand gθ belongs to a largeclass of parametric regression functions. The distribution of ε0 is completelyknown. We propose an estimation procedure with a new criterion computed as the Fouriertransform of a weighted least square contrast. This procedure provides an asymptoticallynormal estimator \hbox{$\hat \theta$}θ̂ of θ0, for a large class of regressionfunctions and various noise distributions.

Keywords

Autoregressive modelMarkov chainmixingdeconvolutionsemi–parametric model
Type
Research Article
Information
ESAIM: Probability and Statistics , Volume 18 , 2014 , pp. 277 - 307
Copyright
© EDP Sciences, SMAI 2014

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References

Anderson, B.D.O. and Deistler, M., Identifiability in dynamic errors-in-variables models. J. Time Ser. Anal. 5 (1984) 1–13. Google Scholar
AngoNze, P., Critères d’ergodicité géométrique ou arithmétique de modèles linéaires perturbés à représentation markovienne. C. R. Acad. Sci. Paris Sér. I Math. 326 (1998) 371–376. Google Scholar
Bickel, P.J., Ritov, Y. and Rydén, T., Asymptotic normality of the maximum–likelihood estimator for general hidden Markov models. Ann. Statist. 26 (1998) 1614–1635. Google Scholar
Bradley, R.C., Basic properties of strong mixing conditions, in Dependence in probability and statistics (Oberwolfach 1985). Boston, MA: Birkhäuser Boston, Progr. Probab. Statist. 11 (1986) 165–192. Google Scholar
P.J. Brockwell and R.A. Davis, Time series: theory and methods (Second ed.). Springer Ser. Statistics. New York: Springer-Verlag (1991).
Butucea, C. and Taupin, M.-L., New M-estimators in semiparametric regression with errors in variables. Ann. Inst. Henri Poincaré, Probab. Stat. 44 (2008) 393–421. Google Scholar
Chanda, K.C., Large sample analysis of autoregressive moving-average models with errors in variables. J. Time Ser. Anal. 16 (1995) 1–15. Google Scholar
Chanda, K.C., Asymptotic properties of estimators for autoregressive models with errors in variables. Ann. Statist. 24 (1996) 423–430. Google Scholar
Comte, F. and Taupin, M.-L., Semiparametric estimation in the (auto)-regressive β–mixing model with errors-in-variables. Math. Methods Statist. 10 (2001) 121–160. Google Scholar
Costa, M. and Alpuim, T., Parameter estimation of state space models for univariate observations. J. Statist. Plann. Inference 140 (2010) 1889–1902. Google Scholar
J. Dedecker F. Merlevède and M. Peligrad, A quenched weak invariance principle. Technical report, to appear in Ann. Inst. Henri Poincaré Probab. Statist. (2012). http://fr.arxiv.org/abs/math.ST/arxiv:1204.4554
Dedecker, J. and Prieur, C., New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203–236. Google Scholar
Dedecker, J. and Rio, E., On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Probab. Statist. 36 (2000) 1–34. Google Scholar
Douc, R. and Matias, C., Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381–420. Google Scholar
Douc, R., Moulines, E., Olsson, J. and van Handel, R., Consistency of the maximum likelihood estimator for general hidden markov models. Ann. Statist. 39 (2011) 474–513. Google Scholar
Douc, R., Moulines, É. and Rydén, T., Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 (2004) 2254–2304. Google Scholar
Fuh, C.-D., Efficient likelihood estimation in state space models. Ann. Statist. 34 (2006) 2026–2068. Google Scholar
Genon−Catalot, V. and Laredo, C., Leroux’s method for general hidden Markov models. Stochastic Process. Appl. 116 (2006) 222–243. Google Scholar
Hannan, E.J., The asymptotic theory of linear time−series models. J. Appl. Probab. 10 (1973) 130–145.Google Scholar
Jensen, J.L. and Petersen, N.V., Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 (1999) 514–535. Google Scholar
Leroux, B.G., Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 (1992) 127–143. Google Scholar
Mokkadem, A., Le modèle non linéaire AR(1) général. Ergodicité et ergodicité géométrique. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 889–892. Google Scholar
Na, S., Lee, S. and Park, H., Sequential empirical process in autoregressive models with measurement errors. J. Statist. Plann. Inference 136 (2006) 4204–4216. Google Scholar
Nowak, E., Global identification of the dynamic shock-error model. J. Econom. 27 (1985) 211–219. Google Scholar
Rio, E., Covariance inequalities for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993) 587–597. Google Scholar
Rosenblatt, M., A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43–47. Google Scholar
Staudenmayer, J. and Buonaccorsi, J.P., Measurement error in linear autoregressive models. J. Amer. Statist. Assoc. 100 (2005) 841–852. Google Scholar
A. Trapletti and K. Hornik, tseries: Time Series Analysis and Computational Finance. R package version 0.10-25 (2011).