Estimation in autoregressive model with measurement error
ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 277-307.

Consider an autoregressive model with measurement error: we observe Z i = X i + ε i , where the unobserved X i is a stationary solution of the autoregressive equation X i = g θ 0 ( X i - 1 ) + ξ i . The regression function g θ 0 is known up to a finite dimensional parameter θ 0 to be estimated. The distributions of ξ 1 and X 0 are unknown and g θ belongs to a large class of parametric regression functions. The distribution of ε 0 is completely known. We propose an estimation procedure with a new criterion computed as the Fourier transform of a weighted least square contrast. This procedure provides an asymptotically normal estimator θ ^ of θ 0 , for a large class of regression functions and various noise distributions.

DOI : https://doi.org/10.1051/ps/2013037
Classification : 62J02,  62F12,  62G05,  62G20
Mots clés : autoregressive model, Markov chain, mixing, deconvolution, semi-parametric model
@article{PS_2014__18__277_0,
     author = {Dedecker, J\'er\^ome and Samson, Adeline and Taupin, Marie-Luce},
     title = {Estimation in autoregressive model with measurement error},
     journal = {ESAIM: Probability and Statistics},
     pages = {277--307},
     publisher = {EDP-Sciences},
     volume = {18},
     year = {2014},
     doi = {10.1051/ps/2013037},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/ps/2013037/}
}
Dedecker, Jérôme; Samson, Adeline; Taupin, Marie-Luce. Estimation in autoregressive model with measurement error. ESAIM: Probability and Statistics, Tome 18 (2014) , pp. 277-307. doi : 10.1051/ps/2013037. http://www.numdam.org/articles/10.1051/ps/2013037/

[1] B.D.O. Anderson and M. Deistler, Identifiability in dynamic errors-in-variables models. J. Time Ser. Anal. 5 (1984) 1-13. | MR 747410 | Zbl 0536.93064

[2] P. Angonze, 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. | Zbl 0918.60052

[3] P.J. Bickel, Y. Ritov and T. Rydén, Asymptotic normality of the maximum-likelihood estimator for general hidden Markov models. Ann. Statist. 26 (1998) 1614-1635. | MR 1647705 | Zbl 0932.62097

[4] R.C. Bradley, 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. | MR 899990 | Zbl 0603.60034

[5] P.J. Brockwell and R.A. Davis, Time series: theory and methods (Second ed.). Springer Ser. Statistics. New York: Springer-Verlag (1991). | MR 1093459 | Zbl 0604.62083

[6] C. Butucea and M.-L. Taupin, New M-estimators in semiparametric regression with errors in variables. Ann. Inst. Henri Poincaré, Probab. Stat. 44 (2008) 393-421. | Numdam | MR 2451051 | Zbl 1206.62068

[7] K.C. Chanda, Large sample analysis of autoregressive moving-average models with errors in variables. J. Time Ser. Anal. 16 (1995) 1-15. | MR 1323615 | Zbl 0814.62052

[8] K.C. Chanda, Asymptotic properties of estimators for autoregressive models with errors in variables. Ann. Statist. 24 (1996) 423-430. | MR 1389899 | Zbl 0853.62070

[9] F. Comte and M.-L. Taupin, Semiparametric estimation in the (auto)-regressive β-mixing model with errors-in-variables. Math. Methods Statist. 10 (2001) 121-160. | MR 1851745 | Zbl 1005.62036

[10] M. Costa and T. Alpuim, Parameter estimation of state space models for univariate observations. J. Statist. Plann. Inference 140 (2010) 1889-1902. | MR 2606726 | Zbl 1185.62165

[11] 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 | Numdam | MR 3224292

[12] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203-236. | MR 2199291 | Zbl 1061.62058

[13] J. Dedecker and E. Rio, On the functional central limit theorem for stationary processes. Ann. Inst. Henri Poincaré Probab. Statist. 36 (2000) 1-34. | Numdam | MR 1743095 | Zbl 0949.60049

[14] R. Douc and C. Matias, Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli 7 (2001) 381-420. | MR 1836737 | Zbl 0987.62018

[15] R. Douc, E. Moulines, J. Olsson and R. Van Handel, Consistency of the maximum likelihood estimator for general hidden markov models. Ann. Statist. 39 (2011) 474-513. | MR 2797854 | Zbl 1209.62194

[16] R. Douc, É. Moulines and T. Rydén, Asymptotic properties of the maximum likelihood estimator in autoregressive models with Markov regime. Ann. Statist. 32 (2004) 2254-2304. | MR 2102510 | Zbl 1056.62028

[17] C.-D. Fuh, Efficient likelihood estimation in state space models. Ann. Statist. 34 (2006) 2026-2068. | MR 2283726 | Zbl 1246.62185

[18] V. Genon−Catalot and C. Laredo, Leroux's method for general hidden Markov models. Stochastic Process. Appl. 116 (2006) 222-243. | MR 2197975 | Zbl 1099.60022

[19] E.J. Hannan, The asymptotic theory of linear time−series models. J. Appl. Probab. 10 (1973) 130-145. | MR 365960 | Zbl 0261.62073

[20] J.L. Jensen and N.V. Petersen, Asymptotic normality of the maximum likelihood estimator in state space models. Ann. Statist. 27 (1999) 514-535. | MR 1714719 | Zbl 0952.62023

[21] B.G. Leroux, Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl. 40 (1992) 127-143. | MR 1145463 | Zbl 0738.62081

[22] A. Mokkadem, Le modèle non linéaire AR(1) général. Ergodicité et ergodicité géométrique. C. R. Acad. Sci. Paris Sér. I Math. 301 (1985) 889-892. | Zbl 0588.60070

[23] S. Na, S. Lee and H. Park, Sequential empirical process in autoregressive models with measurement errors. J. Statist. Plann. Inference 136 (2006) 4204-4216. | MR 2323411 | Zbl 1098.62116

[24] E. Nowak, Global identification of the dynamic shock-error model. J. Econom. 27 (1985) 211-219. | MR 786615 | Zbl 0556.62089

[25] E. Rio, Covariance inequalities for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Statist. 29 (1993) 587-597. | Numdam | MR 1251142 | Zbl 0798.60027

[26] M. Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA 42 (1956) 43-47. | MR 74711 | Zbl 0070.13804

[27] J. Staudenmayer and J.P. Buonaccorsi, Measurement error in linear autoregressive models. J. Amer. Statist. Assoc. 100 (2005) 841-852. | MR 2201013 | Zbl 1117.62430

[28] A. Trapletti and K. Hornik, tseries: Time Series Analysis and Computational Finance. R package version 0.10-25 (2011).