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Behavior of R-estimators under measurement errors. (English) Zbl 1388.62207

Summary: As was shown recently, the measurement errors in regressors affect only the power of the rank test, but not its critical region. Noting that, we study the effect of measurement errors on R-estimators in linear model. It is demonstrated that while an R-estimator admits a local asymptotic bias, its bias surprisingly depends only on the precision of measurements and does neither depend on the chosen rank test score-generating function nor on the regression model error distribution. The R-estimators are numerically illustrated and compared with the LSE and \(L_{1}\) estimators in this situation.

MSC:

62J05 Linear regression; mixed models
62G20 Asymptotic properties of nonparametric inference
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[1] Adcock, R.J. (1877). Note on the method of least squares. The Analyst 4 183-184. · JFM 09.0154.04
[2] Akritas, M.G. and Bershady, M.A. (1996). Linear regression for astronomical data with measurement errors and intrinsic scatter. Astrophysical Journal 470 706-728.
[3] Arias, O., Hallock, K.F. and Sosa-Escudero, W. (2001). Individual heterogeneity in the returns to schooling: Instrumental variables quantile regression using twins data. Empirical Economics 26 7-40.
[4] Carroll, R.J., Delaigle, A. and Hall, P. (2007). Non-parametric regression estimation from data contaminated by a mixture of Berkson and classical errors. J. R. Stat. Soc. Ser. B Stat. Methodol. 69 859-878.
[5] Carroll, R.J., Maca, J.D. and Ruppert, D. (1999). Nonparametric regression in the presence of measurement error. Biometrika 86 541-554. · Zbl 0938.62039
[6] Carroll, R.J., Ruppert, D., Stefanski, L.A. and Crainiceanu, C.M. (2006). Measurement Error in Nonlinear Models. A Modern Perspective , 2nd ed. Monographs on Statistics and Applied Probability 105 . Boca Raton, FL: Chapman & Hall/CRC. · Zbl 1119.62063
[7] Cheng, C.-L. and Van Ness, J.W. (1999). Statistical Regression with Measurement Error. Kendall’s Library of Statistics 6 . London: Arnold. · Zbl 0947.62046
[8] Fan, J. and Truong, Y.K. (1993). Nonparametric regression estimation involving errors-in-variables. Ann. Statist. 21 23-37. · Zbl 0791.62042
[9] Fuller, W.A. (1987). Measurement Error Models. Wiley Series in Probability and Mathematical Statistics : Probability and Mathematical Statistics . New York: Wiley. · Zbl 0800.62413
[10] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests . New York: Academic Press. · Zbl 0161.38102
[11] Hausman, J. (2001). Mismeasured variables in econometric analysis: Problems from the right and problems from the left. J. Econ. Perspect. 15 57-67.
[12] He, X. and Liang, H. (2000). Quantile regression estimates for a class of linear and partially linear errors-in-variables models. Statist. Sinica 10 129-140. · Zbl 0970.62043
[13] Heiler, S. and Willers, R. (1988). Asymptotic normality of R-estimates in the linear model. Statistics 19 173-184. · Zbl 0658.62040
[14] Hodges, J.L. Jr. and Lehmann, E.L. (1963). Estimates of location based on rank tests. Ann. Math. Statist. 34 598-611. · Zbl 0203.21105
[15] Hyk, W. and Stojek, Z. (2013). Quantifying uncertainty of determination by standard additions and serial dilutions methods taking into account standard uncertainties in both axes. Anal. Chem. 85 5933-5939.
[16] Hyslop, D.R. and Imbens, G.W. (2001). Bias from classical and other forms of measurement error. J. Bus. Econom. Statist. 19 475-481.
[17] Jaeckel, L.A. (1972). Estimating regression coefficients by minimizing the dispersion of the residuals. Ann. Math. Statist. 43 1449-1458. · Zbl 0277.62049
[18] Jurečková, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40 1889-1900. · Zbl 0188.51003
[19] Jurečková, J. (1971). Nonparametric estimate of regression coefficients. Ann. Math. Statist. 42 1328-1338. · Zbl 0225.62052
[20] Jurečková, J., Picek, J. and Saleh, A.K.Md.E. (2010). Rank tests and regression and rank score tests in measurement error models. Comput. Statist. Data Anal. 54 3108-3120. · Zbl 1284.62420
[21] Kelly, B.C. (2007). Some aspects of measurement error in linear regression of astronomical data. The Astrophysical Journal 665 1489-1506. · Zbl 0438.73057
[22] Koul, H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics 166 . New York: Springer. · Zbl 1007.62047
[23] Marques, T.A. (2004). Predicting and correcting bias caused by measurement error in line transect sampling using multiplicative error models. Biometrics 60 757-763. · Zbl 1274.62836
[24] Müller, I. (1996). Robust methods in the linear calibration model. Ph.D. thesis, Charles Univ. in Prague.
[25] Navrátil, R. (2012). Rank Tests and R-estimates in Location Model with Measurement errors. In Proceedings of Workshop of the Jaroslav Hájek Center and Financial Mathematics in Practice I. Book of Short Papers (J. Zelinka and J. Horová, eds.). Brno: Masaryk Univ.
[26] Navrátil, R. and Saleh, A.K.Md.E. (2011). Rank tests of symmetry and R-estimation of location parameter under measurement errors. Acta Univ. Palack. Olomuc. Fac. Rerum Natur. Math. 50 95-102. · Zbl 1244.62067
[27] Oosterhoff, J. and van Zwet, W.R. (1979). A note on contiguity and Hellinger distance. In Contributions to Statistics. Jaroslav Hájek Memorial Volume (J. Jurečková, ed.) 157-166. Dordrecht: Reidel. · Zbl 0418.62020
[28] Picek, J. (1996). Statistical procedures based on regression rank scores. Ph.D. thesis, Charles Univ. in Prague.
[29] Pollard, D. (1991). Asymptotics for least absolute deviation regression estimators. Econometric Theory 7 186-199. · Zbl 04504753
[30] Rocke, D.M. and Lorenzato, S. (1995). A two-component model for measurement error in analytical chemistry. Technometrics 37 176-184. · Zbl 0822.62103
[31] Saleh, A.K.Md.E., Picek, J. and Kalina, J. (2012). R-estimation of the parameters of a multiple regression model with measurement errors. Metrika 75 311-328. · Zbl 1239.62081
[32] Sen, P.K., Jurečková, J. and Picek, J. (2013). Rank tests for corrupted linear models. J. Indian Statist. Assoc. 51 201-229.
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