Skip to main content
SpringerLink
Log in

On the asymptotic optimality of orthoregressional estimators

  • Published:
Journal of Applied and Industrial Mathematics Aims and scope Submit manuscript

Abstract

It is shown that the orthoregressional (STLS) parameter estimators in linear algebraic systems (including autonomous difference equations with matrix coefficients) converge to the maximum likelihood estimators and thus become asymptotically best in the limit case of large variances of the random coordinates on the variety of solutions to the system observed with additive random perturbations.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. R. J. Adcock, “Note on the Method of Least Squares,” The Analyst 4, 183–184 (1877).

    Article  MATH  Google Scholar 

  2. R. J. Adcock, “A Problem in Least Squares,” The Analyst 5, 53–54 (1878).

    Article  Google Scholar 

  3. C. H. Kummel, “Reduction ofObserved Equations Which Contain More Than One Observed Quantity,” The Analyst 6, 97–105 (1879).

    Article  Google Scholar 

  4. K. Pearson, “On Lines and Planes of Closest Fit to Systems of Points in Space,” Philosophical Magazine 6 (2), 559–572 (1901).

    Article  MATH  Google Scholar 

  5. K. F. Gauss, Selected Geodesic Works, Vol. 1 (Izd. Geodez. Lit, Moscow, 1957) [in Russian].

    Google Scholar 

  6. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, 1946; Nauka, Moscow, 1975).

    MATH  Google Scholar 

  7. G. H. Golub and C. F. Van Loan, “An Analysis of the Total Least Squares Problem,” SIAM J. Numer. Anal. 17, 883–893 (1980).

    Article  MathSciNet  MATH  Google Scholar 

  8. T. C. Koopmans, Linear Regression Analysis of Economic Time Series (Haarlem, 1937).

    MATH  Google Scholar 

  9. M. J. Levin, “Estimation of a System Pulse Transfer Function in the Presence of Noise,” IEEE Trans. Automat Control AC-9, 229–235 (1964).

    Article  Google Scholar 

  10. I. Markovsky and S. Van Huffel, “Overview of Total Least Squares Methods,” Signal Processing 87, 2283–2302 (2007).

    Article  MATH  Google Scholar 

  11. A. O. Egorshin, “Closed ComputationalMethods for Identification of Linear Objects,” in Optimal and Self-Adjusting Systems (Inst.Mat., Novosibirsk, 1971), pp. 40–53.

    Google Scholar 

  12. A. O. Egorshin, “The Least-Squares Method and’ Fast’ Algorithms in Variational Problems of Identification and Filtration (Method VI),” Avtometriya, No. 1, 30–42 (1988).

    Google Scholar 

  13. B. DeMoor, “Structured Total Least Squares and L2 Approximation Problems,” Linear Algebra Appl. 188–189, 163–207 (1993).

    Article  MathSciNet  Google Scholar 

  14. A. A. Lomov, “Variational Identification Methods for Linear Dynamic Systems and Local Extrema Problem,” Upravlenie Bol’shimi Sistemami, No. 39, 53–94 (2012) [http://ubs.mtas.ru/upload/library/UBS3903.pdf].

    Google Scholar 

  15. A. A. Lomov, “On Consistency of a GeneralizedOrthoregressive Parameter Estimator for a Linear Dynamical System,” Sibirsk. Zh. Industr.Mat. 16 (4), 87–93 (2013) [J. Appl. Indust. Math. 8 (1), 86–91 (2014)].

    MATH  Google Scholar 

  16. M. Kahn, M. S. Mackisack, M. R. Osborne, and G.K. Smyth, “On the Consistency of Prony’sMethod and Related Algorithms,” J. Comput. Graph. Statist. 1, 329–349 (1992).

    MathSciNet  Google Scholar 

  17. C. Heij and W. Scherrer, “Consistency of System Identification by Global Total Least Squares,” Automatica 35, 993–1008 (1999).

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Kukush, I. Markovsky, and S. Van Huffel, “Consistency of the Structured Total Least Squares Estimator in aMultivariate Errors-in-VariablesModel,” J. Statist. Plann. Inference. 133 (2), 315–358 (2005).

    Article  MathSciNet  MATH  Google Scholar 

  19. L. J. Gleser, “Estimation in a Multivariate ‘Errors in Variables’ Regression Model: Large Sample Results,” Ann. Statist. 9 (1), 24–44 (1981).

    Article  MathSciNet  MATH  Google Scholar 

  20. W. A. Fuller, Measurement ErrorModels (Wiley, New York, 1987).

    Google Scholar 

  21. A. A. Lomov, “Orthoregressive Estimates for the Parameters of Systems of Linear Difference Equations,” Sibirsk.Zh. Industr. Mat. 8 (3), 102–119 (2005) [J. Appl. Indust. Math. 1 (1), 59–76 (2005)].

    MathSciNet  MATH  Google Scholar 

  22. A. A. Lomov, “Identification of Linear Dynamic Systems by Short Parts of Transients with Additive Measuring Disturbances,” Izv. Akad. Nauk Teor. Sist. Upravl. No. 3, 20–26 (1997) [J. Comput. Syst. Sci. Int. 36 (3), 346–352 (1997)].

    MATH  Google Scholar 

  23. A. A. Lomov, “Distinguishability Conditions for Stationary Linear Systems,” Differentsial’nye Uravneniya 39 (2), 261–266 (2003) [Differential Equations 39 (2), 283–288 (2003)].

    MathSciNet  MATH  Google Scholar 

  24. A. A. Lomov, “On Identifiability of Stationary Linear Systems with Coefficients That Depend on a Parameter,” Sibirsk.Zh. Industr. Mat. 6 (4), 60–66 (2003).

    MathSciNet  MATH  Google Scholar 

  25. Ph. Lemmerling, N. Mastronardi and S. Van Huffel, “Fast Algorithm for Solving the Hankel/Toeplitz Structured Total Least Squares Problem,” Numer. Algorithms 23, 371–392 (2000).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. A. Borovkov, Mathematical Statistics (Nauka, Novosibirsk, 1997; Gordon and Breach, Amsterdam, 1998).

    MATH  Google Scholar 

  27. R. C. K. Lee, Optimal Estimation, Identification and Control (The M.I.T. Press, Cambridge, MS, 1964; Nauka, Moscow, 1966).

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Sobolev Institute of Mathematics, pr. Akad. Koptyuga 4, Novosibirsk, 630090, Russia

    A. A. Lomov

  2. Novosibirsk State University, ul. Pirogova 2, Novosibirsk, 630090, Russia

    A. A. Lomov

Authors
  1. A. A. Lomov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to A. A. Lomov.

Additional information

Original Russian Text © A.A. Lomov, 2016, published in Sibirskii Zhurnal Industrial’noi Matematiki, 2016, Vol. XIX, No. 4, pp. 51–60.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lomov, A.A. On the asymptotic optimality of orthoregressional estimators. J. Appl. Ind. Math. 10, 511–519 (2016). https://doi.org/10.1134/S1990478916040074

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S1990478916040074

Keywords

Search

Navigation