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On the Polynomial Measurement Error Model

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Total Least Squares and Errors-in-Variables Modeling

Abstract

This paper discusses point estimation of the coefficients of polynomial measurement error (errors-in-variables) models. This includes functional and structural models. The connection between these models and total least squares (TLS) is also examined. A compendium of existing as well as new results is presented.

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References

  1. R. J. Adcock. Note on the method of least squares. Analyst, 4: 183–184, 1877.

    Article  MATH  Google Scholar 

  2. Y. Amemiya. Instrumental variable estimator for the nonlinear errors-invariables model. Journal of Econometrics, 28: 273–289, 1985.

    Article  MathSciNet  Google Scholar 

  3. S. Baran. A consistent estimator in general functional errors-in-variables models. Metrika, 51:117 — 132, 2000.

    Google Scholar 

  4. P. A. Bekker. Comment on identification in the linear errors in variables model. Econometrica, 54: 215–217, 1986.

    Article  MathSciNet  MATH  Google Scholar 

  5. H. I. Britt and R. H. Luecke. The estimation of parameters in nonlinear implicit models. Technometrics, 15: 233–247, 1973.

    Article  MATH  Google Scholar 

  6. J. P. Buonaccorsi. A modified estimating equation approach to correcting for measurement error in regression. Biometrika, 83: 433–440, 1996.

    Article  MathSciNet  MATH  Google Scholar 

  7. R. J. Carroll, D. Ruppert, and L. A. Stefanski. Nonlinear Measurement Error Models. Chapman and Hall, New York, 1995.

    MATH  Google Scholar 

  8. L. K. Chan and T. K. Mak. On the polynomial functional relationship. Journal of the Royal Statistical Society, Series B, 47: 510–518, 1985.

    MathSciNet  Google Scholar 

  9. C.-L. Cheng and H. Schneeweiss. Note on two estimators for the polynomial regression with errors in the variables. In Statistical Modeling: Proceedings of the 13th International Workshop on Statistical Modeling,pages 141–147, New Orleans, Lousiana, 1998. B. Marx and H. Friedl.

    Google Scholar 

  10. C.-L. Cheng and H. Schneeweiss. Polynomial regression with errors in the variables. Journal of the Royal Statistical Society, Series B, 60: 189–199, 1998.

    MathSciNet  MATH  Google Scholar 

  11. C.-L. Cheng, H. Schneeweiss, and M. Thamerus. A small sample estimator for a polynomial regression with errors in the variables. Journal of the Royal Statistical Society, Series B, 62: 699–709, 2000.

    MathSciNet  MATH  Google Scholar 

  12. C.-L. Cheng and J. W. Van Ness. On estimating linear relationships when both variables are subject to errors. Journal of the Royal Statistical Society, Series B, 57: 167–183, 1994.

    Google Scholar 

  13. C.-L. Cheng and Van Ness J. W. Statistical Regression with Measurement Error. Arnold, 1999.

    Google Scholar 

  14. A. Chesher. Polynomial regression with covariate measurement error. Discussion Papaer No. 98/448, University of Bristol, Department of Economics, 1998.

    Google Scholar 

  15. I. Fazekas, S. Baran, A. Kukush, and J. Lauridsen. On inconsistency of the least squares estimator in nonlinear functional relations. Research Report NI, Odense University, Denmark, Department of Statistic and Demography, 1998.

    Google Scholar 

  16. I. Fazekas, S. Baran, A. Kukush, and J. Lauridsen. Asymptotic properties in space and time of an estimator in nonlinear functional errors-in-variables models. Random Operators and Stochastic Equations, 7: 389–412, 1999.

    MathSciNet  MATH  Google Scholar 

  17. W. A. Fuller. Measurement Error Models. John Wiley and Sons, New York, 1987.

    Book  MATH  Google Scholar 

  18. L. J. Gleser and J. T. Hwang. The nonexistence of 100(1 — a)% confidence sets of finite expected diameter in errors-in-variables and related models. The Annals of Statistics, 15: 1351–1362, 1987.

    Article  MathSciNet  MATH  Google Scholar 

  19. G. H. Golub and C. F. Van Loan. An analysis of total least squares problem. SIAM J. Numer. Anal., 17: 883–893, 1980.

    Article  MathSciNet  MATH  Google Scholar 

  20. Z. Griliches and V. Ringstad. Error-in-the-variables bias in nonlinear contexts. Econometrica, 38: 368–370, 1970.

    Article  Google Scholar 

  21. J. A. Hausman and W. K. Newey. Identification and estimation of polynomical errors-in-variables models. Journal of Econometrics, 50: 273–295, 1991.

    Article  MathSciNet  MATH  Google Scholar 

  22. J. A. Hausman, W. K. Newey, and J. L. Powell. Nonlinear errors in variables estimation of some engel curves. Journal of Econometrics, 50: 205–233, 1995.

    Article  MathSciNet  Google Scholar 

  23. Y. H. Huang and L. C. Huwang. On the polynomial structural relationship. Canadian Journal of Statistics, page to appear, 2001.

    Google Scholar 

  24. P. J. Huber. The behavior of maximum likelihood estimates under nonstandard condition. In Proceedings Fifth Berkeley Symposium on Mathematical Statistics and Probability, volume 1, pages 221–233, University of California Press, Berkeley, 1967. L. LeCam and J. Neyman

    Google Scholar 

  25. Y. H. Huwang, L. C. amd Huang. On errors-in-variables in polynomial regression-berkson case. Statistica Sinica, 10: 923–936, 2000.

    MathSciNet  MATH  Google Scholar 

  26. S. J. Iturria, R. J. Carroll, and D. Firth. Polynomial regression and estimating functions in the presence of multiplicative measurement error. Journal of the Royal Statistical Society, Series B, 61 (3): 547–561, 1999.

    Article  MathSciNet  MATH  Google Scholar 

  27. M. G. Kendall and A. Stuart. The Advanced Theory of Statistics, volume 2. Hafner, New York, 4th edition, 1979.

    Google Scholar 

  28. J. T. Kuha and J. Temple. Covariate measurement error in quadratic regression. Discussion Paper 1999-W2, Nuffield College Oxford, 1999.

    Google Scholar 

  29. A. Kukush and H. Schneeweiss. A comparison of asymptotic covariance matrices of adjusted least squares and structural least squares in error ridden polynomial regression. Discussion Paper 218, Sonderforschungsbereich 386, University of Munich, 2000.

    Google Scholar 

  30. A. Kukush, H. Schneeweiss, and R. Wolf. Comparison of three estimators in a polynomial regression with measurement errors. Discussion Paper 233, Sonderforschungsbereich 386, University of Munich, 2001.

    Google Scholar 

  31. D. V. Lindley. Regression lines and the linear functional relationship. Journal of the Royal Statistical Society, Supplement, 9: 218–244, 1947.

    MathSciNet  MATH  Google Scholar 

  32. A. Madansky. The fitting of straight lines when both variables are subject to error. Journal of the American Statistical Association, 54: 173–205, 1959.

    Article  MathSciNet  MATH  Google Scholar 

  33. L. Moberg and R. Sundberg. Maximum likelihood estimation of a linear functional relationship when one of the departure variances is known. Scandinavian Journal of Statistics, 5: 61–64, 1978.

    MathSciNet  MATH  Google Scholar 

  34. M. S. Moon and R. F. Gunst. Polynomial measurement error modeling. Computational Statistics and Data Analysis, 19: 1–21, 1995.

    Article  MathSciNet  MATH  Google Scholar 

  35. P. A. P. Moran. Estimating structural and functional relationships. Journal of Multivariate Analysis, 1: 232–255, 1971.

    Article  MathSciNet  MATH  Google Scholar 

  36. T. Nakamura. Corrected score functions for errors-in-variables models: Methodology and application to generalized linear models. Biometrika, 77: 127–137, 1990.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. Neyman and E. L. Scott. Consistent estimates based on partially consistent observations. Econometrica, 16: 1–32, 1948.

    Article  MathSciNet  Google Scholar 

  38. M. O’Neill, L. G. Sinclair, and F. J. Smith. Polynomial curve fitting when abscissas and ordinates are both subject to error. Computer Journal, 12: 52–56, 1969.

    Article  MathSciNet  MATH  Google Scholar 

  39. O. Reiersol. Identifiability of a linear relation between variables which are subject to error. Econometrica, 18: 375–389, 1950.

    Article  MathSciNet  Google Scholar 

  40. H. Schneeweiß and H. J. Mittag. Lineare Modelle mit fehlerbehafteten Daten. Physica, Heidelberg, 1986.

    Book  Google Scholar 

  41. H. Schneeweiss and T. Nittner. Estimating a polynomial regression with measurement errors in the structural and in the functional case - a comparison. Discussion Paper 197, Sonderforschungsbereich 386, University of Munich, 2000.

    Google Scholar 

  42. P. Sprent. A generalized least-squares approach to linear functional relationships. Journal of the Royal Statistical Society, Series B, 28: 278–297, 1966.

    MathSciNet  MATH  Google Scholar 

  43. Leonard A. Stefanski. Unbiased estimation of a nonlinear function of a normal mean with application to measurement error models. Communications in Statistics, Part A–Theory and Methods, 18: 4335–4358, 1989.

    Article  MathSciNet  MATH  Google Scholar 

  44. M. Thamerus. Different nonlinear regression models with incorrectly observed covariates. In Econometrics in Theory and Practice, Festschrift for Hans Schneeweiss., pages 31–44. Galata, R. and Kuechenhoff, H., 1998.

    Chapter  Google Scholar 

  45. S. Van Huffel and J. Vandewalle. The Total Least Squares Problem: Computational Aspects and Analysis. SIAM, Philadelphia, 1991.

    Google Scholar 

  46. K. Van Montfort. Estimating in Structural Models with Non-normal Distributed Variables: Some Alternative Approaches. Leiden: DSWO, 1989.

    Google Scholar 

  47. M. M. Wall and Y. Amemiya. Estimation for polynomial structural equation models. Journal of the American Statistical Association, 95: 929–940, 2000.

    Article  MathSciNet  MATH  Google Scholar 

  48. K. M. Wolter and W. A. Fuller. Estimation of nonlinear errors-in-variables models. The Annals of Statistics, 10: 539–548, 1982.

    Article  MathSciNet  MATH  Google Scholar 

  49. K. M. Wolter and W. A. Fuller. Estimation of the quadratic errors-in-variables model. Biometrika, 69: 175–182, 1982.

    MATH  Google Scholar 

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Author information

Authors and Affiliations

  1. Institute of Statistical Science, Academia Sinica, Taipei, Taiwan, R.O.C

    Chi-Lun Cheng

  2. Department of Statistics, University of Munich, Munich, Germany

    Hans Schneeweiss

Authors
  1. Chi-Lun Cheng
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  2. Hans Schneeweiss
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Editor information

Editors and Affiliations

  1. Department of Electrical Engineering, Katholieke Universiteit, Leuven-Heverlee, Belgium

    Sabine Van Huffel  & Philippe Lemmerling  & 

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© 2002 Springer Science+Business Media Dordrecht

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Cheng, CL., Schneeweiss, H. (2002). On the Polynomial Measurement Error Model. In: Van Huffel, S., Lemmerling, P. (eds) Total Least Squares and Errors-in-Variables Modeling. Springer, Dordrecht. https://doi.org/10.1007/978-94-017-3552-0_12

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  • DOI: https://doi.org/10.1007/978-94-017-3552-0_12

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-90-481-5957-4

  • Online ISBN: 978-94-017-3552-0

  • eBook Packages: Springer Book Archive

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