Advances in Decision Sciences

Advances in Decision Sciences / 2007 / Article
Special Issue

Statistics and Applied Probability: A Tribute to Jeffrey J. Hunter

View this Special Issue

Review Article | Open Access

Volume 2007 |Article ID 094515 | https://doi.org/10.1155/2007/94515

K. Gustafson, "The Geometry of Statistical Efficiency and Matrix Statistics", Advances in Decision Sciences, vol. 2007, Article ID 094515, 16 pages, 2007. https://doi.org/10.1155/2007/94515

The Geometry of Statistical Efficiency and Matrix Statistics

Academic Editor: Paul Cowpertwait
Received23 Mar 2007
Accepted08 Aug 2007
Published13 Nov 2007

Abstract

We will place certain parts of the theory of statistical efficiency into the author's operator trigonometry (1967), thereby providing new geometrical understanding of statistical efficiency. Important earlier results of Bloomfield and Watson, Durbin and Kendall, Rao and Rao, will be so interpreted. For example, worse case relative least squares efficiency corresponds to and is achieved by the maximal turning antieigenvectors of the covariance matrix. Some little-known historical perspectives will also be exposed. The overall view will be emphasized.

References

  1. K. Gustafson, “On geometry of statistical efficiency,” preprint, 1999. View at: Google Scholar
  2. K. Gustafson, “An unconventional computational linear algebra: operator trigonometry,” in Unconventional Models of Computation, UMC'2K, I. Antoniou, C. Calude, and M. Dinneen, Eds., pp. 48–67, Springer, London, UK, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  3. K. Gustafson, “Operator trigonometry of statistics and econometrics,” Linear Algebra and Its Applications, vol. 354, no. 1–3, pp. 141–158, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  4. P. Bloomfield and G. S. Watson, “The inefficiency of least squares,” Biometrika, vol. 62, no. 1, pp. 121–128, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  5. M. Knott, “On the minimum efficiency of least squares,” Biometrika, vol. 62, no. 1, pp. 129–132, 1975. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  6. C. R. Rao and C. V. Rao, “Stationary values of the product of two Raleigh quotients: homologous canonical correlations,” Sankhyā B, vol. 49, no. 2, pp. 113–125, 1987. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  7. S. G. Wang and S.-C. Chow, Advanced Linear Models: Theory and Applications, vol. 141 of Statistics: Textbooks and Monographs, Marcel Dekker, New York, NY, USA, 1994. View at: Zentralblatt MATH | MathSciNet
  8. K. Gustafson, “A min-max theorem,” Notices of the American Mathematical Society, vol. 15, p. 799, 1968. View at: Google Scholar
  9. K. Gustafson, Lectures on Computational Fluid Dynamics, Mathematical Physics, and Linear Algebra, World Scientific, River Edge, NJ, USA, 1997. View at: Zentralblatt MATH | MathSciNet
  10. K. Gustafson and D. K. M. Rao, Numerical Range: The Field of Values of Linear Operators and Matrices, Universitext, Springer, New York, NY, USA, 1997. View at: MathSciNet
  11. K. Gustafson, “Commentary on topics in the analytic theory of matrices,” in Collected Works of Helmut Wielandt 2, B. Huppert and H. Schneider, Eds., pp. 356–367, DeGruyters, Berlin, Germaney, 1996. View at: Google Scholar
  12. K. Gustafson, “Antieigenvalue inequalities in operator theory,” in Proceedings of the 3rd Symposium on Inequalities, O. Shisha, Ed., pp. 115–119, Academic Press, Los Angeles, Calif, USA, September 1972. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  13. K. Gustafson, “Antieigenvalues,” Linear Algebra and Its Applications, vol. 208-209, pp. 437–454, 1994. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  14. K. Gustafson, “An extended operator trigonometry,” Linear Algebra and Its Applications, vol. 319, no. 1–3, pp. 117–135, 2000. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  15. K. Gustafson, “A note on left multiplication of semigroup generators,” Pacific Journal of Mathematics, vol. 24, no. 3, pp. 463–465, 1968. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  16. R. L. Plackett, “A historical note on the method of least squares,” Biometrika, vol. 36, no. 3-4, pp. 458–460, 1949. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  17. A. C. Aitken, “On least squares and linear combination of observations,” Proceedings of the Royal Society of Edinburgh Section A, vol. 55, pp. 42–48, 1935. View at: Google Scholar | Zentralblatt MATH
  18. J. Durbin and M. G. Kendall, “The geometry of estimation,” Biometrika, vol. 38, no. 1-2, pp. 150–158, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  19. G. S. Watson, “Serial correlation in regression analysis—I,” Biometrika, vol. 42, no. 3-4, pp. 327–341, 1955. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  20. G. S. Watson, “Linear least squares regression,” Annals of Mathematical Statistics, vol. 38, no. 6, pp. 1679–1699, 1967. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  21. J. Durbin and G. S. Watson, “Testing for serial correlation in least squares regression—I,” Biometrika, vol. 37, no. 3-4, pp. 409–428, 1950. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  22. J. Durbin and G. S. Watson, “Testing for serial correlation in least squares regression—II,” Biometrika, vol. 38, no. 1-2, pp. 159–178, 1951. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  23. T. W. Anderson, “On the theory of testing serial correlation,” Skandinavisk Aktuarietidskrift, vol. 31, pp. 88–116, 1948. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  24. R. L. Anderson and T. W. Anderson, “Distribution of the circular serial correlation coefficient for residuals from a fitted Fourier series,” Annals of Mathematical Statistics, vol. 21, no. 1, pp. 59–81, 1950. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  25. U. Grenander, “On the estimation of regression coefficients in the case of an autocorrelated disturbance,” Annals of Mathematical Statistics, vol. 25, no. 2, pp. 252–272, 1954. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  26. U. Grenander and M. Rosenblatt, Statistical Analysis of Stationary Time Series, John Wiley & Sons, New York, NY, USA, 1957. View at: Zentralblatt MATH | MathSciNet
  27. R. A. Fisher, “On the mathematical foundations of theoretical statistics,” Philosophical Transactions of the Royal Society of London A, vol. 222, pp. 309–368, 1922. View at: Publisher Site | Google Scholar
  28. J. von Neumann, “Distribution of the ratio of the mean square successive difference to the variance,” Annals of Mathematical Statistics, vol. 12, no. 4, pp. 367–395, 1941. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  29. J. von Neumann, R. H. Kent, H. R. Bellinson, and B. I. Hart, “The mean square successive difference,” Annals of Mathematical Statistics, vol. 12, pp. 153–162, 1941. View at: Google Scholar | Zentralblatt MATH | MathSciNet
  30. K. Gustafson, Introduction to Partial Differential Equations and Hilbert Space Methods, Dover, Mineola, NY, USA, 3rd edition, 1999. View at: Zentralblatt MATH | MathSciNet
  31. K. Gustafson, “Operator trigonometry of the model problem,” Numerical Linear Algebra with Applications, vol. 5, no. 5, pp. 377–399, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet
  32. H. Cramér, Mathematical Methods of Statistics, vol. 9 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1946. View at: Zentralblatt MATH | MathSciNet
  33. R. A. Fisher, “Theory of statistical estimation,” Proceedings of the Cambridge Philosophical Society, vol. 22, pp. 700–725, 1925. View at: Google Scholar
  34. I. Miller and M. Miller, Freund's Mathematical Statistics, Prentice-Hall, Upper Saddle River, NJ, USA, 6th edition, 1999. View at: Zentralblatt MATH
  35. K. Gustafson, “Parallel computing forty years ago,” Mathematics and Computers in Simulation, vol. 51, no. 1-2, pp. 47–62, 1999. View at: Publisher Site | Google Scholar | MathSciNet
  36. S. Puntanen and G. P. H. Styan, “The equality of the ordinary least squares estimator and the best linear unbiased estimator,” The American Statistician, vol. 43, no. 3, pp. 153–164, 1989. View at: Publisher Site | Google Scholar | MathSciNet
  37. K. Gustafson, “The trigonometry of matrix statistics,” International Statistical Review, vol. 74, no. 2, pp. 187–202, 2006. View at: Google Scholar
  38. K. Gustafson, Noncommutative trigonometry, vol. 167 of Operator Theory: Advances and Applications, Birkhäuser Basel, Birkhäuser Verlag AG, Viaduktstraße 42, 4051 Basel, Switzerland, 2006. View at: Publisher Site

Copyright © 2007 K. Gustafson. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


More related articles

118 Views | 382 Downloads | 3 Citations
 PDF Download Citation Citation
 Order printed copiesOrder

Related articles

We are committed to sharing findings related to COVID-19 as quickly as possible. We will be providing unlimited waivers of publication charges for accepted research articles as well as case reports and case series related to COVID-19. Review articles are excluded from this waiver policy. Sign up here as a reviewer to help fast-track new submissions.