## On nonlinear regression estimator with denoised variables.(English)Zbl 1284.62402

Summary: In this paper, a class of denoised nonlinear regression estimators is suggested for a nonlinear measurement error model where the variables in error are observed together with an auxiliary variable. The programming involved in this denoised nonlinear regression estimation is relatively simple and it can be modified with a little effort from the existing programs for nonlinear regression estimation. We establish the consistency and asymptotic normality of such denoised estimators based on the least squares and M-methods. A simulation study is carried out to illustrate the performance of these estimates. An empirical application of the model to production models in economics further demonstrates the potential of the proposed modeling procedures.

### MSC:

 62J02 General nonlinear regression 65C60 Computational problems in statistics (MSC2010)
Full Text:

### References:

 [1] Arrow, K.J.; Chenery, H.B.; Minhas, B.S.; Solow, R.M., Capital-labor substitution and economic efficiency, Rev. econ. statist., XLIII, 225-251, (1961) [2] Bard, Y., Nonlinear parameters estimation, (1974), Academic Press New York [3] Bates, D.M.; Watts, D.G., Nonlinear regression analysis and its applications, (1988), Wiley New York · Zbl 0728.62062 [4] Bodkin, R.G.; Klein, L.R., Nonlinear estimation of aggregate production functions, Rev. econ. statist., XLIX, 28-34, (1967) [5] Carroll, R.J.; Ruppert, D.; Stefanski, L.A., Measurement error in nonlinear models, (1995), Chapman & Hall · Zbl 0853.62048 [6] Cheng, C.L.; Van Ness, J.W., Statistical regression with measurement error, (1999), Oxford University Press Inc. New York · Zbl 0947.62046 [7] Cai, Z.W; Naik, P.A.; Tsai, C.L., Denoised least squares estimators: an application to estimating advertising effectiveness, Statist. sinica, 10, 1231-1243, (2000) · Zbl 0960.62134 [8] Cui, H.J.; He, X.M.; Zhu, L.X., On regression estimators with denoised variables, Statist. sinica, 12, 1191-1205, (2002) · Zbl 1004.62038 [9] Cui, H.J.; Li, R.C., On parameter estimation for semi-linear error-in-variables model, J. multivariate anal., 64, 1-24, (1998) [10] Cui, H.J.; Kong, E., Empirical likelihood confidence regions for semi-parametric errors-in-variables models, Scand. J. statist., 33, 153-168, (2006) · Zbl 1121.62042 [11] Fuller, W.A., Measurement error models, (1987), Wiley New York · Zbl 0800.62413 [12] Gallant, A.R., Nonlinear regression, The American Statistician, 29, 73-81, (1975) · Zbl 0328.62043 [13] Gasser, T.; Müller, H.G., Kernel estimation of regression function, () [14] Gonin, R.; Money, A.H., Nonlinear $$L^p$$-norm estimation, (1989), Marcel Dekker New York · Zbl 0701.62073 [15] He, X.; Shao, Q.M., A general bahadur representation of M-estimators and its application to linear regression with nonstochastic designs, Ann. statist., 24, 2608-2630, (1996) · Zbl 0867.62012 [16] Huber, P., Robust statistics, (1981), John Wiley New York · Zbl 0536.62025 [17] Jennrich, R.J., Asymptotic properties of nonlinear least squares estimators, Ann. math. statist., 40, 633-643, (1969) · Zbl 0193.47201 [18] Duffy, J.; Papageorgiou, C., A cross-country empirical investigation of the aggregate production function specification, J. econ. growth, 5, 87-120, (2000) · Zbl 0967.91028 [19] Nocedal, J.; Wright, S.J., Numerical optimization, (1999), Springer · Zbl 0930.65067 [20] Pollard, D.; Radchenko, P., Nonlinear least-squares estimation, J. multivariate anal., 97, 548-562, (2006) · Zbl 1085.62027 [21] Seber, G.A.F.; Wild, C.J., Nonlinear regression, (1989), John Wiley New York · Zbl 0721.62062 [22] Schennach, S.M., Estimation of nonlinear models with measurement error, Econometrica, 72, 33-75, (2004) · Zbl 1151.91726 [23] Wu, C.F., Asymptotic theory of nonlinear least squares estimation, Ann. statist., 3, 501-513, (1981) · Zbl 0475.62050 [24] You, J.H.; Zhou, H., On semi-parametric EV models with serially correlated errors in both regression models and measured covariates, Scand. J. statist., 34, 365-383, (2007) · Zbl 1142.62043 [25] You, J.H.; Zhou, X.; Zhu, L.X., Inference on a regression model with noised variables and serially correlated errors, J. multivariate anal., 100, 1182-1197, (2009) · Zbl 1159.62029 [26] Zhou, Y.; Liang, H., Statistical inference for semiparametric varying-coefficient partially linear models with error-prone linear covariates, Ann. statist., 1, 427-458, (2009) · Zbl 1156.62036
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.