Estimation in autoregressive model with measurement error

Jérôme Dedecker; Adeline Samson; Marie-Luce Taupin

ESAIM: Probability and Statistics (2014)

  • Volume: 18, page 277-307
  • ISSN: 1292-8100

Abstract

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Consider an autoregressive model with measurement error: we observe Zi = Xi + εi, where the unobserved Xi is a stationary solution of the autoregressive equation Xi = gθ0(Xi − 1) + ξi. The regression function gθ0 is known up to a finite dimensional parameter θ0 to be estimated. The distributions of ξ1 and X0 are unknown and gθ belongs to a large class of parametric regression functions. The distribution of ε0is 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 ^ θ̂ of θ0, for a large class of regression functions and various noise distributions.

How to cite

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Dedecker, Jérôme, Samson, Adeline, and Taupin, Marie-Luce. "Estimation in autoregressive model with measurement error." ESAIM: Probability and Statistics 18 (2014): 277-307. <http://eudml.org/doc/273642>.

@article{Dedecker2014,
abstract = {Consider an autoregressive model with measurement error: we observe Zi = Xi + εi, where the unobserved Xi is a stationary solution of the autoregressive equation Xi = gθ0(Xi − 1) + ξi. The regression function gθ0 is known up to a finite dimensional parameter θ0 to be estimated. The distributions of ξ1 and X0 are unknown and gθ belongs to a large class of parametric regression functions. The distribution of ε0is 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 $\hat\{\theta \}$θ̂ of θ0, for a large class of regression functions and various noise distributions.},
author = {Dedecker, Jérôme, Samson, Adeline, Taupin, Marie-Luce},
journal = {ESAIM: Probability and Statistics},
keywords = {autoregressive model; Markov chain; mixing; deconvolution; semi–parametric model; semiparametric model},
language = {eng},
pages = {277-307},
publisher = {EDP-Sciences},
title = {Estimation in autoregressive model with measurement error},
url = {http://eudml.org/doc/273642},
volume = {18},
year = {2014},
}

TY - JOUR
AU - Dedecker, Jérôme
AU - Samson, Adeline
AU - Taupin, Marie-Luce
TI - Estimation in autoregressive model with measurement error
JO - ESAIM: Probability and Statistics
PY - 2014
PB - EDP-Sciences
VL - 18
SP - 277
EP - 307
AB - Consider an autoregressive model with measurement error: we observe Zi = Xi + εi, where the unobserved Xi is a stationary solution of the autoregressive equation Xi = gθ0(Xi − 1) + ξi. The regression function gθ0 is known up to a finite dimensional parameter θ0 to be estimated. The distributions of ξ1 and X0 are unknown and gθ belongs to a large class of parametric regression functions. The distribution of ε0is 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 $\hat{\theta }$θ̂ of θ0, for a large class of regression functions and various noise distributions.
LA - eng
KW - autoregressive model; Markov chain; mixing; deconvolution; semi–parametric model; semiparametric model
UR - http://eudml.org/doc/273642
ER -

References

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  1. [1] B.D.O. Anderson and M. Deistler, Identifiability in dynamic errors-in-variables models. J. Time Ser. Anal.5 (1984) 1–13. Zbl0536.93064MR747410
  2. [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. Zbl0918.60052
  3. [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. Zbl0932.62097MR1647705
  4. [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. Zbl0603.60034MR899990
  5. [5] P.J. Brockwell and R.A. Davis, Time series: theory and methods (Second ed.). Springer Ser. Statistics. New York: Springer-Verlag (1991). Zbl0604.62083MR1093459
  6. [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. Zbl1206.62068MR2451051
  7. [7] K.C. Chanda, Large sample analysis of autoregressive moving-average models with errors in variables. J. Time Ser. Anal.16 (1995) 1–15. Zbl0814.62052MR1323615
  8. [8] K.C. Chanda, Asymptotic properties of estimators for autoregressive models with errors in variables. Ann. Statist.24 (1996) 423–430. Zbl0853.62070MR1389899
  9. [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. Zbl1005.62036MR1851745
  10. [10] M. Costa and T. Alpuim, Parameter estimation of state space models for univariate observations. J. Statist. Plann. Inference140 (2010) 1889–1902. Zbl1185.62165MR2606726
  11. [11] J. DedeckerF. 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 Zbl1304.60031MR3224292
  12. [12] J. Dedecker and C. Prieur, New dependence coefficients. Examples and applications to statistics. Probab. Theory Relat. Fields 132 (2005) 203–236. Zbl1061.62058MR2199291
  13. [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. Zbl0949.60049MR1743095
  14. [14] R. Douc and C. Matias, Asymptotics of the maximum likelihood estimator for general hidden Markov models. Bernoulli7 (2001) 381–420. Zbl0987.62018MR1836737
  15. [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. Zbl1209.62194MR2797854
  16. [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. Zbl1056.62028MR2102510
  17. [17] C.-D. Fuh, Efficient likelihood estimation in state space models. Ann. Statist.34 (2006) 2026–2068. Zbl1246.62185MR2283726
  18. [18] V. Genon−Catalot and C. Laredo, Leroux’s method for general hidden Markov models. Stochastic Process. Appl. 116 (2006) 222–243. Zbl1099.60022MR2197975
  19. [19] E.J. Hannan, The asymptotic theory of linear time−series models. J. Appl. Probab. 10 (1973) 130–145. Zbl0261.62073MR365960
  20. [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. Zbl0952.62023MR1714719
  21. [21] B.G. Leroux, Maximum-likelihood estimation for hidden Markov models. Stochastic Process. Appl.40 (1992) 127–143. Zbl0738.62081MR1145463
  22. [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. Zbl0588.60070
  23. [23] S. Na, S. Lee and H. Park, Sequential empirical process in autoregressive models with measurement errors. J. Statist. Plann. Inference136 (2006) 4204–4216. Zbl1098.62116MR2323411
  24. [24] E. Nowak, Global identification of the dynamic shock-error model. J. Econom.27 (1985) 211–219. Zbl0556.62089MR786615
  25. [25] E. Rio, Covariance inequalities for strongly mixing processes. Ann. Inst. Henri Poincaré Probab. Statist.29 (1993) 587–597. Zbl0798.60027MR1251142
  26. [26] M. Rosenblatt, A central limit theorem and a strong mixing condition. Proc. Natl. Acad. Sci. USA42 (1956) 43–47. Zbl0070.13804MR74711
  27. [27] J. Staudenmayer and J.P. Buonaccorsi, Measurement error in linear autoregressive models. J. Amer. Statist. Assoc.100 (2005) 841–852. Zbl1117.62430MR2201013
  28. [28] A. Trapletti and K. Hornik, tseries: Time Series Analysis and Computational Finance. R package version 0.10-25 (2011). 

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