New $M$-estimators in semi-parametric regression with errors in variables

Butucea, Cristina; Taupin, Marie-Luce

Annales de l'I.H.P. Probabilités et statistiques, Tome 44 (2008) no. 3, pp. 393-421.

Dans le modèle de régression avec erreurs sur les variables, nous observons $n$ v.a. i.i.d. de même loi que $(Y,Z)$ satisfaisant aux relations $Y=f_{{\theta }^0}\left(X\right)+\xi$ et $Z=X+\epsilon$, où les v.a. $X,\xi ,\epsilon$ sont indépendantes, pas observées, et la fonction de régression $f_{{\theta }^0}$ est connue à un paramètre de dimension finie ${\theta }^0$ près. Les densités de $X$ et de $\xi$ sont inconnues tandis que la loi de $\epsilon$ est entièrement connue. Nous estimons le paramètre ${\theta }^0$ à partir des observations $(Y_1,Z_1),...,(Y_n,Z_n)$. Nous proposons une procédure d’estimation basée sur le critère des moindres carrés ${\tilde{S}}_{{\theta }^0,g}\left(\theta \right)={𝔼}_{{\theta }^0,g}[\left({(Y-f_{\theta }\left(X\right))}^{2}w\left(X\right)\right]$, où $w$ 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 $p_{\epsilon }$ et de la régularité en $x$ de $w\left(x\right)f_{\theta }\left(x\right)$. De plus, nous établissons des conditions suffisantes pour que les estimateurs atteignent la vitesse paramétrique. Nous décrivons des méthodes pratiques pour le choix de $x$ 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 $w_{\theta }$ dépendant de $\theta$ serait plus approprié.

In the regression model with errors in variables, we observe $n$ i.i.d. copies of $(Y,Z)$ satisfying $Y=f_{{\theta }^0}\left(X\right)+\xi$ and $Z=X+\epsilon$ involving independent and unobserved random variables $X,\xi ,\epsilon$ plus a regression function $f_{{\theta }^0}$, known up to a finite dimensional ${\theta }^0$. The common densities of the $X_i$’s and of the ${\xi }_i$’s are unknown, whereas the distribution of $\epsilon$ is completely known. We aim at estimating the parameter ${\theta }^0$ by using the observations $(Y_1,Z_1),...,(Y_n,Z_n)$. We propose an estimation procedure based on the least square criterion ${\tilde{S}}_{{\theta }^0,g}\left(\theta \right)={𝔼}_{{\theta }^0,g}[\left({(Y-f_{\theta }\left(X\right))}^{2}w\left(X\right)\right]$ where $w$ 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 $p_{\epsilon }$ and on the smoothness properties of $w\left(x\right)f_{\theta }\left(x\right)$. Furthermore, we give sufficient conditions that ensure that the parametric rate of convergence is achieved. We provide practical recipes for the choice of $w$ 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 $w_{\theta }$ depending on $\theta$ would be more appropriate.

Classification : 62J02, 62F12, 62G05, 62G20
Mots clés : 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

[1] S. Baran. A consistent estimator in general functional errors-in-variables models. Metrika 51 (2000) 117-132 (electronic). | MR | Zbl

[2] P. J. Bickel, A. J. C. Klaassen, Y. Ritov and J. A. Wellner. Efficient and Adaptative Estimation for Semiparametric Model. Johns Hopkins Univ. Press, Baltimore, MD, 1993. | MR | Zbl

[3] Bickel, P. J. and A. J. C. Ritov. Efficient estimation in the errors-in-variables model. Ann. Statist. 15 (1987) 513-540. | MR | Zbl

[4] Billingsley, P. Probability and Measure, 3rd edition. Wiley. New York, 1995. | MR | Zbl

[5] R. J. Carroll, D. Ruppert and L. A. Stefanski. Measurement Error in Nonlinear Models. Chapman and Hall, London, 1995. | MR | Zbl

[6] L. K. Chan and T. K. Mak. On the polynomial functionnal relationship. J. Roy. Statist. Soc. Ser. B 47 (1985) 510-518. | MR

[7] C. H. Cheng and J. W. Van Ness. On estimating linear relationships when both variables are subject to errors. J. Roy. Statist. Soc. Ser. B 56 (1994) 167-183. | MR | Zbl

[8] 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

[9] I. Fazekas, S. Baran, A. Kukush, and J. Lauridsen. 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

[10] I. Fazekas and A. G. Kukush. 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

[11] M. V. Fedoryuk. Asimptotika: integraly i ryady. “Nauka”, Moscow, 1987. | MR

[12] W. A. Fuller. Measurement Error Models. Wiley, New York, 1987. | MR | Zbl

[13] L. J. Gleser. Improvements of the naive approach to estimation in nonlinear errors-in-variables regression models. Contemp. Math. 112 (1990) 99-114. | MR | Zbl

[14] J. A. Hausman, W. K. Newey, I. Ichimura and J. L. Powell. Identification and estimation of polynomial errors-in-variables models. J. Econometrics 50 (1991) 273-295. | MR | Zbl

[15] J. A. Hausman, W. K. Newey and J. L. Powell. Nonlinear errors in variables estimation of some engel curves. J. Econometrics 65 (1995) 205-233. | MR | Zbl

[16] H. Hong and E. Tamer. A simple estimator for nonlinear error in variable models. J. Econometrics 117 (2003) 1-19. | MR | Zbl

[17] C. Hsiao. Consistent estimation for some nonlinear errors-in-variables models. J. Econometrics 41 (1989) 159-185. | MR | Zbl

[18] C. Hsiao, L. Wang and Q. Wang. Estimation of nonlinear errors-in-variables models: an approximate solution. Statist. Papers 38 (1997) 1-25. | MR | Zbl

[19] C. Hsiao and Q. K. Wang. Estimation of structural nonlinear errors-in-variables models by simulated least-squares method. Internat. Econom. Rev. 41 (2000) 523-542. | MR

[20] J. Kiefer and J. Wolfowitz. Consistency of the maximum likelihood estimator in the presence of infinitely many nuisance parameters. Ann. Math. Statist. 27 (1956) 887-906. | MR | Zbl

[21] A. Kukush and H. Schneeweiss. Comparing different estimators in a nonlinear measurement error model. I. Math. Methods Statist. 14 (2005) 53-79. | MR

[22] A. Kukush and H. Schneeweiss. Comparing different estimators in a nonlinear measurement error model. II. Math. Methods Statist. 14 (2005) 203-223. | MR

[23] O. V. Lepski and B. Y. Levit. Adaptive minimax estimation of infinitely differentiable functions. Math. Methods Statist. 7 (1998) 123-156. | MR | Zbl

[24] T. Li. Estimation of nonlinear errors-in-variables models: a simulated minimum distance estimator. Statist. Probab. Lett. 47 (2000) 243-248. | MR | Zbl

[25] T. Li. Robust and consistent estimation of nonlinear errors-in-variables models. J. Econometrics 110 (2002) 1-26. | MR | Zbl

[26] S. A. Murphy and A. W. Van Der Vaart. Likelihood inference in the errors-in-variables model. J. Multivariate Anal. 59 (1996) 81-108. | MR | Zbl

[27] V. V. Petrov. Limit Theorems of Probability Theory. Oxford Science Publications, New York, 1995. | MR | Zbl

[28] O. Reiersøl. Identifiability of a linear relation between variables which are subject to error. Econometrica. 18 (1950) 375-389. | MR | Zbl

[29] M.-L. Taupin. Semi-parametric estimation in the nonlinear structural errors-in-variables model. Ann. Statist. 29 (2001) 66-93. | MR | Zbl

[30] A. Van Der Vaart. Semiparametric statistics. Lectures on Probability Theory and Statistics (Saint-Flour, 1999) 331-457. Lecture Notes in Math. 1781. Berlin, Springer, 2002. | MR | Zbl

[31] A. W. Van Der Vaart. Estimating a real parameter in a class of semiparametric models. Ann. Statist. 16 (1988) 1450-1474. | MR | Zbl

[32] A. W. Van Der Vaart. Efficient estimation in semi-parametric mixture models. Ann. Statist. 24 (1996) 862-878. | MR | Zbl

[33] K. M. Wolter and W. A. Fuller. Estimation of nonlinear errors-in variables models. Ann. Statist. 10 (1982) 539-548. | MR | Zbl

[34] K. M. Wolter and W. A. Fuller. Estimation of the quadratic errors-in-variables model. Biometrika 69 (1982) 175-182. | Zbl