Estimation in autoregressive model with measurement error

Dedecker, Jérôme; Samson, Adeline; Taupin, Marie-Luce

ESAIM: Probability and Statistics, Tome 18 (2014), pp. 277-307.

Consider an autoregressive model with measurement error: we observe $Z_i=X_i+{\epsilon }_i$, where the unobserved $X_i$ is a stationary solution of the autoregressive equation $X_i=g_{{\theta }^0}\left(X_{i-1}\right)+{\xi }_i$. The regression function $g_{{\theta }^0}$ is known up to a finite dimensional parameter ${\theta }^0$ to be estimated. The distributions of ${\xi }_1$and $X_0$ are unknown and $g_{\theta }$ belongs to a large class of parametric regression functions. The distribution of ${\epsilon }_0$ 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 $\widehat{\theta }$ of ${\theta }^0$, for a large class of regression functions and various noise distributions.

Classification : 62J02, 62F12, 62G05, 62G20
Mots clés : autoregressive model, Markov chain, mixing, deconvolution, semi-parametric model

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

[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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

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

[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

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

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

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

[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 | Zbl

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

[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

[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 | Zbl

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

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

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

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

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