Wang, Liqun Estimation of nonlinear models with Berkson measurement errors. (English) Zbl 1068.62072 Ann. Stat. 32, No. 6, 2559-2579 (2004). Summary: This paper is concerned with general nonlinear regression models where the predictor variables are subject to Berkson-type measurement errors [J. Berkson, J. Am. Stat. Assoc. 45, 164–180 (1950; Zbl 0040.22404)]. The measurement errors are assumed to have a general parametric distribution, which is not necessarily normal. In addition, the distribution of the random error in the regression equation is nonparametric. A minimum distance estimator is proposed, which is based on the first two conditional moments of the response variable given the observed predictor variables. To overcome the possible computational difficulty of minimizing an objective function which involves multiple integrals, a simulation-based estimator is constructed. Consistency and asymptotic normality for both estimators are derived under fairly general regularity conditions. Cited in 27 Documents MSC: 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators 62G08 Nonparametric regression and quantile regression 65C60 Computational problems in statistics (MSC2010) 65C05 Monte Carlo methods 62G20 Asymptotic properties of nonparametric inference Keywords:semiparametric model; errors-in-variables; method of moments; weighted least squares; minimum distance estimator; simulation-based estimator; consistency; asymptotic normality Citations:Zbl 0040.22404 Software:nlmdl PDFBibTeX XMLCite \textit{L. Wang}, Ann. Stat. 32, No. 6, 2559--2579 (2004; Zbl 1068.62072) Full Text: DOI arXiv References: [1] Amemiya, T. (1974). The nonlinear two-stage least-squares estimator. J. Econometrics 2 105–110. · Zbl 0282.62089 · doi:10.1016/0304-4076(74)90033-5 [2] Amemiya, T. (1985). Advanced Econometrics . Basil Blackwell Ltd., Oxford. [3] Berkson, J. (1950). Are there two regressions? J. Amer. Statist. Assoc. 45 164–180. · Zbl 0040.22404 · doi:10.2307/2280676 [4] Carroll, R. J., Ruppert, D. and Stefanski, L. A. (1995). Measurement Error in Nonlinear Models . Chapman and Hall, London. · Zbl 0853.62048 [5] Cheng, C. and Van Ness, J. W. (1999). Statistical Regression with Measurement Error . Arnold, London. · Zbl 0947.62046 [6] Evans, M. and Swartz, T. (2000). Approximating Integrals via Monte Carlo and Deterministic Methods . Oxford Univ. Press. · Zbl 0958.65009 [7] Fuller, W. A. (1987). Measurement Error Models . Wiley, New York. · Zbl 0800.62413 [8] Gallant, A. R. (1987). Nonlinear Statistical Models . Wiley, New York. · Zbl 0611.62071 [9] Huwang, L. and Huang, Y. H. S. (2000). On errors-in-variables in polynomial regression—Berkson case. Statist. Sinica 10 923–936. · Zbl 0952.62062 [10] McFadden, D. (1989). A method of simulated moments for estimation of discrete response models without numerical integration. Econometrica 57 995–1026. · Zbl 0679.62101 · doi:10.2307/1913621 [11] Pakes, A. and Pollard, D. (1989). Simulation and the asymptotics of optimization estimators. Econometrica 57 1027–1057. · Zbl 0698.62031 · doi:10.2307/1913622 [12] Rudemo, M., Ruppert, D. and Streibig, J. C. (1989). Random-effect models in nonlinear regression with applications to bioassay. Biometrics 45 349–362. · Zbl 0715.62253 · doi:10.2307/2531482 [13] Seber, G. A. F. and Wild, C. J. (1989). Nonlinear Regression . Wiley, New York. · Zbl 0721.62062 [14] Wang, L. (2003). Estimation of nonlinear Berkson-type measurement error models. Statist. Sinica 13 1201–1210. · Zbl 1034.62055 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.