In the regression model with errors in variables, we observe i.i.d. copies of satisfying and involving independent and unobserved random variables plus a regression function , known up to a finite dimensional . The common densities of the ’s and of the ’s are unknown, whereas the distribution of is completely known. We aim at estimating the parameter by using the observations . We propose an estimation procedure based on the least square criterion where is a weight function to be chosen. We propose an estimator and derive an upper bound for its risk that depends on the smoothness of the errors density and on the smoothness properties of . Furthermore, we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of in the case of nonlinear regression functions which are smooth on pieces allowing to gain in the order of the rate of convergence, up to the parametric rate in some cases. We also consider extensions of the estimation procedure, in particular, when a choice of depending on would be more appropriate.
Dans le modèle de régression avec erreurs sur les variables, nous observons v.a. i.i.d. de même loi que satisfaisant aux relations et , où les v.a. sont indépendantes, pas observées, et la fonction de régression est connue à un paramètre de dimension finie près. Les densités de et de sont inconnues tandis que la loi de est entièrement connue. Nous estimons le paramètre à partir des observations . Nous proposons une procédure d’estimation basée sur le critère des moindres carrés , où est une fonction de poids à choisir. Nous définissons l’estimateur et calculons la borne supérieure du risque de cet estimateur, qui dépend de la régularité de la densité des erreurs et de la régularité en de . De plus, nous établissons des conditions suffisantes pour que les estimateurs atteignent la vitesse paramétrique. Nous décrivons des méthodes pratiques pour le choix de dans le cas des fonctions de régression non-linéaires qui sont régulières par morceaux permettant de gagner des ordres de vitesse allant jusqu’à la vitesse paramétrique dans certains cas. Nous considérons également des extensions de cette procédure d’estimation, en particulier au cas où un choix de dépendant de serait plus approprié.
Keywords: asymptotic normality, consistency, deconvolution kernel estimator, errors-in-variables model, M-estimators, ordinary smooth and super-smooth functions, rates of convergence, semi-parametric nonlinear regression
@article{AIHPB_2008__44_3_393_0, author = {Butucea, Cristina and Taupin, Marie-Luce}, title = {New $M$-estimators in semi-parametric regression with errors in variables}, journal = {Annales de l'I.H.P. Probabilit\'es et statistiques}, pages = {393--421}, publisher = {Gauthier-Villars}, volume = {44}, number = {3}, year = {2008}, doi = {10.1214/07-AIHP107}, zbl = {1206.62068}, language = {en}, url = {http://archive.numdam.org/articles/10.1214/07-AIHP107/} }
TY - JOUR AU - Butucea, Cristina AU - Taupin, Marie-Luce TI - New $M$-estimators in semi-parametric regression with errors in variables JO - Annales de l'I.H.P. Probabilités et statistiques PY - 2008 SP - 393 EP - 421 VL - 44 IS - 3 PB - Gauthier-Villars UR - http://archive.numdam.org/articles/10.1214/07-AIHP107/ DO - 10.1214/07-AIHP107 LA - en ID - AIHPB_2008__44_3_393_0 ER -
%0 Journal Article %A Butucea, Cristina %A Taupin, Marie-Luce %T New $M$-estimators in semi-parametric regression with errors in variables %J Annales de l'I.H.P. Probabilités et statistiques %D 2008 %P 393-421 %V 44 %N 3 %I Gauthier-Villars %U http://archive.numdam.org/articles/10.1214/07-AIHP107/ %R 10.1214/07-AIHP107 %G en %F AIHPB_2008__44_3_393_0
Butucea, Cristina; Taupin, Marie-Luce. New $M$-estimators in semi-parametric regression with errors in variables. Annales de l'I.H.P. Probabilités et statistiques, Volume 44 (2008) no. 3, pp. 393-421. doi : 10.1214/07-AIHP107. http://archive.numdam.org/articles/10.1214/07-AIHP107/
[1] A consistent estimator in general functional errors-in-variables models. Metrika 51 (2000) 117-132 (electronic). | MR | Zbl
.[2] Efficient and Adaptative Estimation for Semiparametric Model. Johns Hopkins Univ. Press, Baltimore, MD, 1993. | MR | Zbl
, , and .[3] Efficient estimation in the errors-in-variables model. Ann. Statist. 15 (1987) 513-540. | MR | Zbl
and .[4] Probability and Measure, 3rd edition. Wiley. New York, 1995. | MR | Zbl
[5] Measurement Error in Nonlinear Models. Chapman and Hall, London, 1995. | MR | Zbl
, and .[6] On the polynomial functionnal relationship. J. Roy. Statist. Soc. Ser. B 47 (1985) 510-518. | MR
and .[7] On estimating linear relationships when both variables are subject to errors. J. Roy. Statist. Soc. Ser. B 56 (1994) 167-183. | MR | Zbl
and .[8] Semiparametric estimation in the (auto)-regressive β-mixing model with errors-in-variables. Math. Methods Statist. 10 (2001) 121-160. | MR | Zbl
and .[9] Asymptotic properties in space and time of an estimator in nonlinear functional errors-in-variables models. Random Oper. Stochastic Equations 7 (1999) 389-412. | MR | Zbl
, , , and .[10] Asymptotic properties of estimators in nonlinear functional errors-in-variables with dependent error terms. J. Math. Sci. (New York) 92 (1998) 3890-3895. | MR | Zbl
and .[11] Asimptotika: integraly i ryady. “Nauka”, Moscow, 1987. | MR
.[12] Measurement Error Models. Wiley, New York, 1987. | MR | Zbl
.[13] Improvements of the naive approach to estimation in nonlinear errors-in-variables regression models. Contemp. Math. 112 (1990) 99-114. | MR | Zbl
.[14] Identification and estimation of polynomial errors-in-variables models. J. Econometrics 50 (1991) 273-295. | MR | Zbl
, , and .[15] Nonlinear errors in variables estimation of some engel curves. J. Econometrics 65 (1995) 205-233. | MR | Zbl
, and .[16] A simple estimator for nonlinear error in variable models. J. Econometrics 117 (2003) 1-19. | MR | Zbl
and .[17] Consistent estimation for some nonlinear errors-in-variables models. J. Econometrics 41 (1989) 159-185. | MR | Zbl
.[18] Estimation of nonlinear errors-in-variables models: an approximate solution. Statist. Papers 38 (1997) 1-25. | MR | Zbl
, and .[19] Estimation of structural nonlinear errors-in-variables models by simulated least-squares method. Internat. Econom. Rev. 41 (2000) 523-542. | MR
and .[20] Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist. 27 (1956) 887-906. | MR | Zbl
and .[21] Comparing different estimators in a nonlinear measurement error model. I. Math. Methods Statist. 14 (2005) 53-79. | MR
and .[22] Comparing different estimators in a nonlinear measurement error model. II. Math. Methods Statist. 14 (2005) 203-223. | MR
and .[23] Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 (1998) 123-156. | MR | Zbl
and .[24] Estimation of nonlinear errors-in-variables models: a simulated minimum distance estimator. Statist. Probab. Lett. 47 (2000) 243-248. | MR | Zbl
.[25] Robust and consistent estimation of nonlinear errors-in-variables models. J. Econometrics 110 (2002) 1-26. | MR | Zbl
.[26] Likelihood inference in the errors-in-variables model. J. Multivariate Anal. 59 (1996) 81-108. | MR | Zbl
and .[27] Limit Theorems of Probability Theory. Oxford Science Publications, New York, 1995. | MR | Zbl
.[28] Identifiability of a linear relation between variables which are subject to error. Econometrica. 18 (1950) 375-389. | MR | Zbl
.[29] Semi-parametric estimation in the nonlinear structural errors-in-variables model. Ann. Statist. 29 (2001) 66-93. | MR | Zbl
.[30] Semiparametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 331-457. Lecture Notes in Math. 1781. Berlin, Springer, 2002. | MR | Zbl
.[31] Estimating a real parameter in a class of semiparametric models. Ann. Statist. 16 (1988) 1450-1474. | MR | Zbl
.[32] Efficient estimation in semi-parametric mixture models. Ann. Statist. 24 (1996) 862-878. | MR | Zbl
.[33] Estimation of nonlinear errors-in variables models. Ann. Statist. 10 (1982) 539-548. | MR | Zbl
and .[34] Estimation of the quadratic errors-in-variables model. Biometrika 69 (1982) 175-182. | Zbl
and .Cited by Sources: