Table of Contents Author Guidelines Submit a Manuscript
Journal of Probability and Statistics
Volume 2011, Article ID 182049, 15 pages
http://dx.doi.org/10.1155/2011/182049
Research Article

Determinant Efficiencies in Ill-Conditioned Models

Department of Statistics, Virginia Polytechnic Institute, Blacksburg, VA 24061, USA

Received 18 May 2011; Accepted 1 August 2011

Academic Editor: Michael Lavine

Copyright © 2011 D. R. Jensen. 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.

Linked References

  1. J. C. Kiefer, “Optimum experimental designs,” Journal of the Royal Statistical Society. Series B, vol. 21, pp. 272–319, 1959. View at Google Scholar · View at Zentralblatt MATH
  2. J. C. Kiefer, “Optimum designs in regression problems. II,” Annals of Mathematical Statistics, vol. 32, pp. 298–325, 1961. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  3. J. C. Kiefer and J. Wolfowitz, “Optimum designs in regression problems,” Annals of Mathematical Statistics, vol. 30, pp. 271–294, 1959. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  4. A. Wald, “On the efficient design of statistical investigations,” Annals of Mathematical Statistics, vol. 14, pp. 134–140, 1943. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. M. J. Box and N. R. Draper, “Factorial designs, the |XX| criterion, and some related matters,” Technometrics, vol. 13, pp. 731–742, 1971. View at Google Scholar
  6. J. M. Lucas, “Which response surface design is best,” Technometrics, vol. 18, pp. 411–417, 1976. View at Google Scholar
  7. R. H. Myers, A. I. Khuri, and W. H. Carter, Jr., “Response surface methodology: 1966–1988,” Technometrics, vol. 31, no. 2, pp. 137–157, 1989. View at Google Scholar · View at Zentralblatt MATH
  8. R. Sibson, “DA-optimality and duality,” in Progress in Statistics (European Meeting Statisticians, Budapest, 1972), Vol. II, J. Gani, K. Sarkadi, and I. Vincze, Eds., North-Holland, Amsterdam, The Netherlands, 1974. View at Google Scholar · View at Zentralblatt MATH
  9. S. D. Silvey, Optimal Design, Chapman & Hall, London, UK, 1980.
  10. R. C. St. John and N. R. Draper, “D-optimality for regression designs: a review,” Technometrics, vol. 17, pp. 15–23, 1975. View at Google Scholar · View at Zentralblatt MATH
  11. H. Dette, “A generalization of D- and D1-optimal designs in polynomial regression,” The Annals of Statistics, vol. 18, no. 4, pp. 1784–1804, 1990. View at Publisher · View at Google Scholar
  12. H. Dette and T. Franke, “Robust designs for polynomial regression by maximizing a minimum of D- and D1-efficiencies,” The Annals of Statistics, vol. 29, no. 4, pp. 1024–1049, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  13. S. Huda, “On some Ds-optimal designs in spherical regions,” Communications in Statistics, vol. 20, no. 9, pp. 2965–2985, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  14. S. Huda and A. A. Al-Shiha, “On D-optimal designs for estimating slope,” Sankhyá. Series A, vol. 61, no. 3, pp. 488–495, 1999. View at Google Scholar
  15. C. M.-S. Lee, “D-optimal designs for polynomial regression when lower degree parameters are more important,” Utilitas Mathematica, vol. 34, pp. 53–63, 1988. View at Google Scholar · View at Zentralblatt MATH
  16. Y. B. Lim and W. J. Studden, “Efficient Ds-optimal designs for multivariate polynomial regression on the q-cube,” The Annals of Statistics, vol. 16, no. 3, pp. 1225–1240, 1988. View at Publisher · View at Google Scholar
  17. W. J. Studden, “Ds-optimal designs for polynomial regression using continued fractions,” The Annals of Statistics, vol. 8, no. 5, pp. 1132–1141, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  18. D. M. Wardrop and R. H. Myers, “Some response surface designs for finding optimal conditions,” Journal of Statistical Planning and Inference, vol. 25, no. 1, pp. 7–28, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. H. P. Wynn, “The sequential generation of D-optimum experimental designs,” Annals of Mathematical Statistics, vol. 41, pp. 1655–1664, 1970. View at Publisher · View at Google Scholar
  20. H. P. Wynn, “Results in the theory and construction of D-optimum experimental designs,” Journal of the Royal Statistical Society. Series B. Methodological, vol. 34, pp. 133–147, 1972. View at Google Scholar
  21. H. Dette, “On a mixture of the D- and D1-optimality criterion in polynomial regression,” Journal of Statistical Planning and Inference, vol. 35, no. 2, pp. 233–249, 1993. View at Publisher · View at Google Scholar · View at MathSciNet
  22. S. Huda, “On Ds-efficiency of some D-optimal designs,” Calcutta Statistical Association Bulletin, vol. 34, no. 133-134, pp. 119–122, 1985. View at Google Scholar
  23. S. Huda and I. H. Khan, “On efficiency of some D-optimal designs in subset estimation,” Journal of the Indian Society of Agricultural Statistics, vol. 47, no. 2, pp. 151–164, 1995. View at Google Scholar
  24. P. F. Schwartz, C. Gennings, L. K. Teuschler, and M. W. Farris, “Optimizing the precision of toxicity threshold estimation using a two-stage experimental design,” Journal of Agricultural, Biological, and Environmental Statistics, vol. 6, pp. 409–428, 2001. View at Google Scholar
  25. S. L. Meadows-Shropshire, C. Gennings, W. H. Carter, and J. E. Simmons, “Analysis of mixtures of drugs/chemicals along a fixed-ratio ray without single-chemical data to support an additivity model,” Journal of Agricultural, Biological, and Environmental Statistics, vol. 9, pp. 500–514, 2004. View at Google Scholar
  26. D. A. Belsley, “Centering, the constant, first-differencing, and assessing conditioning,” in Model Reliability, D. A. Belsley and E. Kuh, Eds., pp. 117–153, MIT Press, Cambridge, Mass, USA, 1986. View at Google Scholar
  27. E. R. Mansfield and B. P. Helms, “Detecting multicollinearity,” American Statistician, vol. 36, pp. 158–160, 1982. View at Google Scholar
  28. S. D. Simon and J. P. Lesage, “The impact of collinearity involving the intercept term on the numerical accuracy of regression,” Computer Science in Economics and Management, vol. 1, no. 2, pp. 137–152, 1988. View at Publisher · View at Google Scholar
  29. D. A. Belsley, “Comment: well-conditioned collinearity indices,” Statistical Science, vol. 2, no. 1, pp. 86–91, 1987. View at Google Scholar · View at Zentralblatt MATH
  30. H. Hotelling, “Relations between two sets of variates,” Biometrika, vol. 28, pp. 321–377, 1936. View at Google Scholar
  31. Å. Björck and G. H. Golub, “Numerical methods for computing angles between linear subspaces,” Mathematics of Computation, vol. 27, pp. 579–594, 1973. View at Google Scholar · View at Zentralblatt MATH
  32. M. L. Eaton, Multivariate Statistics: A Vector Space Approach, John Wiley & Sons, New York, NY, USA, 1983.
  33. G. W. Stewart, “Collinearity and least squares regression,” Statistical Science, vol. 2, no. 1, pp. 68–100, 1987. View at Google Scholar · View at Zentralblatt MATH
  34. R. H. Myers, Classical and Modern Regression with Applications, PWS–KENT, Boston, Mass, USA, 2nd edition, 1990.
  35. R. M. O'Brien, “A caution regarding rules of thumb for variance inflation factors,” Quality and Quantity, vol. 41, no. 5, pp. 673–690, 2007. View at Publisher · View at Google Scholar
  36. S. S. Wilks, Mathematical Statistics, John Wiley & Sons, New York, NY, USA, 1962.
  37. H. Cramér, Mathematical Methods of Statistics, Princeton Mathematical Series, vol. 9, Princeton University Press, Princeton, NJ, USA, 1946.
  38. K. G. Roquemore, “Hybrid designs for quadratic response surfaces,” Technometrics, vol. 18, pp. 419–423, 1979. View at Google Scholar
  39. H. O. Hartley, “Smallest composite designs for quadratic response surfaces,” Biometrics. Journal of the Biometric Society, vol. 15, pp. 611–624, 1959. View at Google Scholar · View at Zentralblatt MATH
  40. G. E. P. Box and D. W. Behnken, “Some new three level designs for the study of quantitative variables,” Technometrics, vol. 2, pp. 455–475, 1960. View at Google Scholar
  41. G. E. P. Box and K. B. Wilson, “On the experimental attainment of optimum conditions,” Journal of the Royal Statistical Society. Series B, vol. 13, pp. 1–45, 1951. View at Google Scholar · View at Zentralblatt MATH
  42. W. Notz, “Minimal point second order designs,” Journal of Statistical Planning Inference, vol. 6, no. 1, pp. 47–58, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  43. A. T. Hoke, “Economical second-order designs based on irregular fractions of the 3n factorial,” Technometrics, vol. 16, pp. 375–384, 1974. View at Google Scholar · View at Zentralblatt MATH
  44. M. J. Box and N. R. Draper, “On minimum-point second-order designs,” Technometrics, vol. 16, pp. 613–616, 1974. View at Google Scholar · View at Zentralblatt MATH
  45. D. R. Jensen, “Efficiencies of some small second-order designs,” Statistics & Probability Letters, vol. 21, no. 4, pp. 255–261, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet