Bernoulli

  • Bernoulli
  • Volume 22, Number 2 (2016), 1093-1112.

Behavior of R-estimators under measurement errors

Jana Jurečková, Hira L. Koul, Radim Navrátil, and Jan Picek

Full-text: Open access

Abstract

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.

Article information

Source
Bernoulli, Volume 22, Number 2 (2016), 1093-1112.

Dates
Received: February 2014
Revised: October 2014
First available in Project Euclid: 9 November 2015

Permanent link to this document
https://projecteuclid.org/euclid.bj/1447077770

Digital Object Identifier
doi:10.3150/14-BEJ687

Mathematical Reviews number (MathSciNet)
MR3449809

Zentralblatt MATH identifier
06562306

Keywords
contiguity linear rank statistic linear regression model local asymptotic bias measurement error R-estimate

Citation

Jurečková, Jana; Koul, Hira L.; Navrátil, Radim; Picek, Jan. Behavior of R-estimators under measurement errors. Bernoulli 22 (2016), no. 2, 1093--1112. doi:10.3150/14-BEJ687. https://projecteuclid.org/euclid.bj/1447077770


Export citation

References

  • [1] Adcock, R.J. (1877). Note on the method of least squares. The Analyst 4 183–184.
  • [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.
  • [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.
  • [7] Cheng, C.-L. and Van Ness, J.W. (1999). Statistical Regression with Measurement Error. Kendall’s Library of Statistics 6. London: Arnold.
  • [8] Fan, J. and Truong, Y.K. (1993). Nonparametric regression estimation involving errors-in-variables. Ann. Statist. 21 23–37.
  • [9] Fuller, W.A. (1987). Measurement Error Models. Wiley Series in Probability and Mathematical Statistics: Probability and Mathematical Statistics. New York: Wiley.
  • [10] Hájek, J. and Šidák, Z. (1967). Theory of Rank Tests. New York: Academic Press.
  • [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.
  • [13] Heiler, S. and Willers, R. (1988). Asymptotic normality of R-estimates in the linear model. Statistics 19 173–184.
  • [14] Hodges, J.L. Jr. and Lehmann, E.L. (1963). Estimates of location based on rank tests. Ann. Math. Statist. 34 598–611.
  • [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.
  • [18] Jurečková, J. (1969). Asymptotic linearity of a rank statistic in regression parameter. Ann. Math. Statist. 40 1889–1900.
  • [19] Jurečková, J. (1971). Nonparametric estimate of regression coefficients. Ann. Math. Statist. 42 1328–1338.
  • [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.
  • [21] Kelly, B.C. (2007). Some aspects of measurement error in linear regression of astronomical data. The Astrophysical Journal 665 1489–1506.
  • [22] Koul, H.L. (2002). Weighted Empirical Processes in Dynamic Nonlinear Models. Lecture Notes in Statistics 166. New York: Springer.
  • [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.
  • [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.
  • [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.
  • [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.
  • [30] Rocke, D.M. and Lorenzato, S. (1995). A two-component model for measurement error in analytical chemistry. Technometrics 37 176–184.
  • [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.
  • [32] Sen, P.K., Jurečková, J. and Picek, J. (2013). Rank tests for corrupted linear models. J. Indian Statist. Assoc. 51 201–229.