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A note on asymptotic linearity of $$M$$-statistics in nonlinear models. (English) Zbl 0882.62053
Summary: For a smooth nonlinear regression model the conditions for the uniform second order asymptotic linearity of the $$M$$-statistics in the regression parameters are given. The existence of the $$\sqrt n$$-consistent estimator of the regression parameters and the role of the rescaling residuals in the $$M$$-estimation are briefly discussed.

##### MSC:
 62J02 General nonlinear regression 62F12 Asymptotic properties of parametric estimators
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##### References:
 [1] D. M. Bates, D. G. Watts: Nonlinear Regression Analysis and Its Applications. J. Wiley & Sons, New York 1988. · Zbl 0728.62062 [2] P. J. Bickel, M. J. Wichura: Convergence criteria for multiparameter stochastic processes and some applications. Ann. Math. Statist. 42 (1971), 1656-1670. · Zbl 0265.60011 · doi:10.1214/aoms/1177693164 [3] M. Csörgö, P. Révész: Strong Approximation in Probability and Statistics. Akademia Kiadó, Budapest 1981. · Zbl 0539.60029 [4] P. L. Davis: Aspects of robust linear regression. Ann. Statist. 21 (1993), 1843-1899. · Zbl 0797.62026 · doi:10.1214/aos/1176349401 [5] F. R. Hampel E. M. Ronchetti P. J. Rousseeuw, W. A. Stahel: Robust Statistics – The Approach Based on Influence Functions. J. Wiley & Sons, New York 1986. · Zbl 0593.62027 [6] P. J. Huber: Robust estimation of a location parameter. Ann. Math. Statist. 35 (1964), 73-101. · Zbl 0136.39805 · doi:10.1214/aoms/1177703732 [7] J. Jurečková: Consistency of $$M$$-estimators in linear model generated by non-monotone and discontinuous $$\psi$$-functions. Probab. Math. Statist. 10 (1988), 1-10. · Zbl 0679.62020 [8] J. Jurečková, B. Procházka: Regression quantiles and trimmed least squares estimator in nonlinear regression model. Nonparametric Statist. 3 (1994), 201-222. · Zbl 1384.62213 [9] J. Jurečková, P. K. Sen: Uniform second order asymptotic linearity of $$M$$-statistics in linear models. Statist. Decisions 7 (1989), 263-276. · Zbl 0676.62056 [10] J. Jurečková, A. H. Welsh: Asymptotic relations between $$L$$- and $$M$$-estimators in the linear model. Ann. Inst. Statist. Math. 42 (1990), 671-698. · Zbl 0732.62027 · doi:10.1007/BF02481144 [11] F. Liese, I. Vajda: Consistency of $$M$$-estimators in general models. J. Multivariate Anal. 50 (1994), 93-114. · Zbl 0872.62071 · doi:10.1006/jmva.1994.1036 [12] A. Marazzi: Algorithms, Routines and S Functions for Robust Statistics. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, California 1992. [13] J. M. Ortega, W. C. Rheinboldt: Iterative Solution of Nonlinear Equations in Several Variables. Academic Press, New York and London 1970. · Zbl 0241.65046 [14] S. Portnoy: Tightness of the sequence of empiric c.d.f. processes defined from regression fractiles. Robust and Nonlinear Time-Series Analysis (J. Franke, W. Hardle, D. Martin, Springer-Verlag, New York, 1983, pp. 231-246. [15] C. R. Rao, L. C. Zhao: On the consistency of $$M$$-estimate in linear model obtained through an estimating equation. Statist. Probab. Lett. 14 (1992), 79-84. · Zbl 0755.62054 · doi:10.1016/0167-7152(92)90214-P [16] A. Rubio L. Aguilar, J. Á. Víšek: Testing for difference between models. Comput. Statist. 8 (1992), 57-70. · Zbl 0775.62176 [17] A. Rubio F. Quintana, J. Á. Víšek: Test for differences of $$M$$-estimates between nonlinear regression models. Probab. Math. Statist. 14 (1993), 2, 189-206. · Zbl 0818.62047 [18] J. Á. Víšek: Stability of regression models estimates with respect to subsamples. Computat. Statist. 7 (1992), 183-203. · Zbl 0775.62182 [19] J. Á. Víšek: Problems connected with selection of robust procedure. Proceedings of PROBASTAT’91 (A. Pázman and J. Volaufová, Printing House of the Technical University of Liptovský Mikuláš 1992, pp. 189-203. [20] J. Á. Víšek: On the role of contamination level and the least favorable behaviour of gross-error sensitivity. Probab. Math. Statist. 14 (1993), 2, 173-187. · Zbl 0818.62033 [21] J. Á. Víšek: A cautionary note on the method of Least Median of Squares reconsidered. Transactions of the Twelfth Prague Conference on Information Theory, Statistical Decision Functions and Random Processes (J. Á. Víšek and P. Lachout, Prague 1994, pp. 254-259. [22] J. Á. Víšek: Sensitivity analysis of $$M$$-estimates. Ann. Inst. Statist. Math., to appear. · Zbl 0925.62131
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