Siberian Mathematical Journal

, Volume 50, Issue 2, pp 302–315 | Cite as

Asymptotically optimal estimation in the linear regression problem in the case of violation of some classical assumptions

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Abstract

We consider the problem of estimating the unknown parameters of linear regression in the case when the variances of observations depend on the unknown parameters of the model. A two-step method is suggested for constructing asymptotically linear estimators. Some general sufficient conditions for the asymptotic normality of the estimators are found, and an explicit form is established of the best asymptotically linear estimators. The behavior of the estimators is studied in detail in the case when the parameter of the regression model is one-dimensional.

Keywords

linear regression two-step estimation asymptotically normal estimator best asymptotically linear estimator 
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References

  1. 1.
    Borovkov A. A., Mathematical Statistics, Gordon and Breach, Amsterdam (1998).Google Scholar
  2. 2.
    Seber G., Linear Regression Analysis, John Wiley and Sons, New York etc. (1977).MATHGoogle Scholar
  3. 3.
    Kurata H. and Kariya T., “Least upper bound for the covariance matrix of a generalized least squares estimator in regression with applications to a seemingly unrelated regression model and a heteroscedastic model,” Ann. Statist., 24, No. 4, 1547–1559 (1996).MATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Linke Yu. Yu. and Sakhanenko A. I., “Asymptotically normal estimation of a parameter in a linear-fractional regression problem,” Siberian Math. J., 41, No. 1, 125–137 (2000).CrossRefMathSciNetGoogle Scholar
  5. 5.
    Linke Yu. Yu. and Sakhanenko A. I., “Asymptotically normal estimation of a multidimensional parameter in the linear-fractional regression problem,” Siberian Math. J., 42, No. 2, 317–331 (2001).CrossRefMathSciNetGoogle Scholar
  6. 6.
    Linke Yu. Yu. and Sakhanenko A. I., “Asymptotically normal explicit estimation of parameters in the Michaelis-Menten equation,” Siberian Math. J., 42, No. 3, 517–536 (2001).CrossRefMathSciNetGoogle Scholar
  7. 7.
    Sakhanenko A. I. and Linke Yu. Yu., “Asymptotically optimal estimation in a linear-fractional regression problem with random errors in coefficients,” Siberian Math. J., 47, No. 6, 1128–1153 (2006).CrossRefMathSciNetGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2009

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Yugra State UniversityKhanty-MansiĭskRussia

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