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Estimation of a semiparametric varying-coefficient partially linear errors-in-variables model. (English) Zbl 1085.62043
Summary: This paper studies the estimation of a varying-coefficient partially linear regression model which is a generalization of the partially linear regression model and varying-coefficient regression model of {\it J. Fan} and {\it T. Huang} [Profile likelihood inferences on semiparametric varying-coefficient partially linear models. Manuscript, Univ. North Carolina, Chapel Hill, USA (2002)]. We focus on the case where some covariates are measured with additive errors. The usual profile least squares and local polynomial estimations lead to biased estimators of the parametric and nonparametric components, respectively, when measurement errors are ignored. By correcting the attenuation we propose a modified profile least squares estimator for the parametric component and a local polynomial estimator for the nonparametric component. We show that the former is consistent, asymptotically normal and achieves the rate in the law of the iterated logarithm, and the latter achieves the optimal strong convergence rate of the usual nonparametric regression. In addition, a consistent estimator is also developed for the error variance. These results can be used to make asymptotically valid statistical inferences. Some simulation studies are conducted to illustrate the finite sample performance of the proposed estimators.

##### MSC:
 62G08 Nonparametric regression 62G20 Nonparametric asymptotic efficiency 62H12 Multivariate estimation
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##### References:
 [1] Brumback, B.; Rice, J. A.: Smoothing spline models for the analysis of nested and crossed samples of curves with discussion. J. amer. Statist. assoc. 93, 961-994 (1998) · Zbl 1064.62515 [2] Carroll, R. J.; Ruppert, D.; Stefanski, L. A.: Measurement error in nonlinear models. (1995) · Zbl 0853.62048 [3] Carroll, R. J.; Ruppert, D.; Welesh, A. H.: Nonparametric estimation via local estimating equations. J. amer. Statist. assoc. 93, 214-227 (1998) · Zbl 0910.62033 [4] Chen, H.: Convergence rates for parametric components in a partially linear model. Ann. statist. 16, 136-146 (1988) · Zbl 0637.62067 [5] Chen, H.; Shiau, J.: A two-stage spline smoothing method for partially linear models. J. statist. Plann. inference 27, 187-202 (1991) · Zbl 0741.62039 [6] Chen, H.; Shiau, J.: Data-driven efficient estimation for a partially linear model. Ann. statist. 22, 211-237 (1994) · Zbl 0806.62029 [7] Cheng, C. L.; Van Ness, J.: Statistical regression with measurement error. (1999) · Zbl 0947.62046 [8] Chow, Y. S.; Teicher, H.: Probability theory. (1980) [9] Cui, H.; Li, R.: On parameter estimation for semi-linear errors-in-variables models. J. multi. Anal. 64, 1-24 (1998) · Zbl 0909.62068 [10] Donald, G.; Newey, K.: Series estimation of semilinear models. J. multi. Anal. 50, 30-40 (1994) · Zbl 0798.62074 [11] Engle, R. F.; Granger, W. J.; Rice, J.; Weiss, A.: Semiparametric estimates of the relation between weather and electricity sales. J. amer. Statist. assoc. 80, 310-319 (1986) [12] Eubank, R.; Speckman, P.: Trigonometric series regression estimators with an application to partially linear models. J. multi. Anal. 32, 70-83 (1990) · Zbl 0709.62041 [13] Fan, J.; Huang, T.: Profile likelihood inferences on semiparametric varying-coefficient partially linear models, manuscript. (2002) · Zbl 1098.62077 [14] Fan, J.; Zhang, W.: Statistical estimation in varying coefficient models. Ann. statist. 27, 1491-1518 (1999) · Zbl 0977.62039 [15] Fan, J.; Zhang, C.; Zhang, J.: Generalized likelihood ratio statistics and wilks phenomenon. Ann. statist. 29, 153-193 (2001) · Zbl 1029.62042 [16] Fuller, W. A.: Measurement error models. (1987) [17] Gao, J.: The laws of the iterated logarithm of some estimates in a partly linear models. Statist. probab. Lett. 25, 153-162 (1995) · Zbl 0837.62041 [18] Hamilton, A.; Truong, K.: Local linear estimation in partly linear models. J. multi. Anal. 60, 1-19 (1997) · Zbl 0883.62041 [19] Härdle, W.; Liang, H.; Gao, J.: Partially linear models. (2000) · Zbl 0968.62006 [20] Hastie, T. J.; Tibshirani, R.: Varying-coefficient models. J. roy. Statist. soc. B 55, 757-796 (1993) · Zbl 0796.62060 [21] Hoover, D. R.; Rice, J. A.; Wu, C. O.; Yang, L. P.: Nonparametric smoothing estimates of time-varying coeffiocient models with longitudinal data. Biometrika 85, 809-822 (1998) · Zbl 0921.62045 [22] Huang, J. Z.; Wu, C. O.; Zhou, L.: Varying-coefficient model and biasis function approximations for the analysis of repeated measurements. Biometrika 89, 809-822 (2002) [23] Hwang, J. T.: The multiplicative errors-in-variables models with applications to the recent data released by the U.S. Department of energy. J. amer. Statist. assoc. 81, 680-688 (1986) · Zbl 0621.62072 [24] Iturria, S. J.; Carroll, R.; Firth, D.: Polynomial regression and estimating functions in the presence of multiplicative measurement error. J. roy. Statist. soc. B 61, 547-561 (1999) · Zbl 0924.62071 [25] Lai, T. L.; Robbins, H.; Wei, C. Z.: Strong consistency of least squares estimates in multiple regression II. J. multi. Anal. 9, 343-361 (1979) · Zbl 0416.62051 [26] Li, Q.; Huang, C. J.; Li, D.; Fu, T. T.: Semiparametric smooth coefficient models. J. bus. Econ. statist. 3, 412-422 (2002) [27] Liang, H.: Asymptotic normality of parametric part in partially linear models with measurement error in the nonparametric part. J. statist. Plann. inference 86, 51-62 (2000) · Zbl 0952.62036 [28] Liang, H.; Härdle, W.; Carroll, R. J.: Estimation in a semiparametric partially linear errors-in-variables model. Ann. statist. 27, 1519-1535 (1999) · Zbl 0977.62036 [29] Rice, J.: Convergence rates for partially splined models. Statist. probab. Lett. 4, 203-208 (1986) · Zbl 0628.62077 [30] Robinson, P.: Root-n-consistent semiparametric regression. Econometrica 56, 931-954 (1988) · Zbl 0647.62100 [31] Shi, P.; Li, G.: A note of the convergence rates of M-estimates for partly linear model. Statistics 26, 27-47 (1995) · Zbl 0812.62046 [32] Speckman, P.: Kernel smoothing in partial linear models. J. roy. Statist. soc. Ser. B 50, 413-436 (1988) · Zbl 0671.62045 [33] Stout, W. F.: Almost sure convergence. (1974) · Zbl 0321.60022 [34] Tosteson, T.; Stefanski, L. A.; Schafer, D. W.: A measurement error model for binary and ordinal regression. Statist. med. 8, 1139-1147 (1989) [35] Wang, N.; Carroll, R. J.; Liang, K. Y.: Quasilikelihood and variance functions in measurement error models with replicates. Biometrics 52, 423-432 (1996) · Zbl 0875.62333 [36] Xia, Y.; Li, W. K.: On the estimation and testing of functional-coefficient linear models. Statist. sinica 9, 737-757 (1999) · Zbl 0958.62040 [37] J. You, X. Zhou, Corrected local polynomial estimation in varying-coefficient models with measurement errors, Canad. J. Statist. 2004, submitted for publication. [38] Zhang, W.; Lee, S. Y.; Song, X.: Local polynomial Fitting in semivarying coefficient models. J. multi. Anal. 82, 166-188 (2002) · Zbl 0995.62038 [39] Zhou, X.; You, J.: Wavelet estimation in varying-coefficient partially linear regression models. Statist. probab. Lett. 68, 91-104 (2004) · Zbl 1058.62036 [40] Zhu, L.; Cui, H.: A semiparametric regression model with errors in variables. Scan. J. Statist. 30, 429-442 (2003) · Zbl 1053.62053