Advertisement

Local Optimality Criteria Based on Asymptotic Normality

  • Luc Pronzato
  • Andrej Pázman
Chapter
Part of the Lecture Notes in Statistics book series (LNS, volume 212)

Abstract

The approach considered in this chapter is probably the most common for designing experiments in nonlinear situations. It consists in optimizing a scalar function of the asymptotic covariance matrix of the estimator and thus relies on asymptotic normality, as considered in  Chaps. 3 and  4. Design based on more accurate characterizations of the precision of the estimation will be considered in  Chap. 6. Additionally to asymptotic normality, the approach also supposes that the asymptotic covariance matrix takes the form of the inverse of an information matrix, as it is the case, for instance, for weighted LS with optimum weights, see  Sect. 3.1.3, or maximum likelihood estimation, see  Sect. 4.2.

Keywords

Design Criterion Asymptotic Normality Information Matrix Directional Derivative Asymptotic Variance 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Alexéev, V., E. Galéev, and V. Tikhomirov (1987). Recueil de Problèmes d’Optimisation. Moscou: MIR.Google Scholar
  2. Atwood, C. (1969). Optimal and efficient designs of experiments. Ann. Math. Statist. 40(5), 1570–1602.MathSciNetzbMATHCrossRefGoogle Scholar
  3. Atwood, C. (1980). Convergent design sequences for sufficiently regular optimality criteria, II singular case. Ann. Statist. 8(4), 894–912.MathSciNetzbMATHCrossRefGoogle Scholar
  4. Avriel, M. (2003). Nonlinear Programming. Analysis and Methods. New York: Dover. [Originally published by Prentice Hall, 1976].Google Scholar
  5. Bloomfield, P. and G. Watson (1975). The inefficiency of least squares. Biometrika 62(1), 121–128.MathSciNetzbMATHCrossRefGoogle Scholar
  6. Box, G. and H. Lucas (1959). Design of experiments in nonlinear situations. Biometrika 46, 77–90.MathSciNetzbMATHGoogle Scholar
  7. Chaloner, K. and I. Verdinelli (1995). Bayesian experimental design: a review. Statist. Sci. 10(3), 273–304.MathSciNetzbMATHCrossRefGoogle Scholar
  8. Cook, D. and V. Fedorov (1995). Constrained optimization of experimental design (invited discussion paper). Statistics 26, 129–178.MathSciNetzbMATHCrossRefGoogle Scholar
  9. Cook, D. and W. Wong (1994). On the equivalence between constrained and compound optimal designs. J. Amer. Statist. Assoc. 89(426), 687–692.MathSciNetzbMATHCrossRefGoogle Scholar
  10. Cook, R., D. Hawkins, and S. Weisberg (1993). Exact iterative computation of the robust multivariate minimum volume ellipsoid estimator. Statist. Probab. Lett. 16, 213–218.MathSciNetCrossRefGoogle Scholar
  11. Cramér, H. (1946). Mathematical Methods of Statistics. Princeton, N.J.: Princeton University Press.zbMATHGoogle Scholar
  12. Davies, L. (1992). The asymptotics of Rousseeuw’s minimum volume ellipsoid estimator. Ann. Statist. 20(4), 1828–1843.MathSciNetzbMATHCrossRefGoogle Scholar
  13. de la Garza, A. (1954). Spacing of information in polynomial regression. Ann. Math. Statist. 25, 123–130.zbMATHCrossRefGoogle Scholar
  14. Dem’yanov, V. and V. Malozemov (1974). Introduction to Minimax. New York: Dover.Google Scholar
  15. Dette, H. (1993). Elfving’s theorem for D-optimality. Ann. Statist. 21(2), 753–766.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Dette, H. and V. Melas (2011). A note on de la Garza phenomenon for locally optimal designs. Ann. Statist. 39(2), 1266–1281.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Dette, H. and W. Studden (1993). Geometry of E-optimality. Ann. Statist. 21(1), 416–433.MathSciNetzbMATHCrossRefGoogle Scholar
  18. Dragalin, V. and V. Fedorov (2006). Adaptive designs for dose-finding based on efficacy-toxicity response. J. Stat. Plann. Inference 136, 1800–1823.MathSciNetzbMATHCrossRefGoogle Scholar
  19. Elfving, G. (1952). Optimum allocation in linear regression. Ann. Math. Statist. 23, 255–262.MathSciNetzbMATHCrossRefGoogle Scholar
  20. Fedorov, V. (1971). The design of experiments in the multiresponse case. Theory Probab. Appl. 16(2), 323–332.zbMATHCrossRefGoogle Scholar
  21. Fedorov, V. (1972). Theory of Optimal Experiments. New York: Academic Press.Google Scholar
  22. Fedorov, V. (1989). Optimal design with bounded density: optimization algorithms of the exchange type. J. Stat. Plann. Inference 22, 1–13.zbMATHCrossRefGoogle Scholar
  23. Fedorov, V. and P. Hackl (1997). Model-Oriented Design of Experiments. Berlin: Springer.zbMATHCrossRefGoogle Scholar
  24. Fedorov, V. and A. Pázman (1968). Design of physical experiments. Fortschritte der Physik 16, 325–355.CrossRefGoogle Scholar
  25. Fellman, J. (1974). On the allocation of linear observations. Comment. Phys. Math. 44, 27–78.MathSciNetGoogle Scholar
  26. Galil, Z. and J. Kiefer (1977). Comparison of simplex designs for quadratic mixture models. Technometrics 19(4), 445–453.MathSciNetzbMATHCrossRefGoogle Scholar
  27. Hadjihassan, S., L. Pronzato, E. Walter, and I. Vuchkov (1997). Robust design for quality improvement by ellipsoidal bounding. In G. Yin and Q. Zhang (Eds.), Proc. 1996 AMS–SIAM Summer Seminar, Math. of Stochastic Manufacturing Systems, Lectures in Applied Math., vol. 33, Providence, Rhode Island, pp. 127–138. American Math. Soc.Google Scholar
  28. Harman, R. (2004a). Lower bounds on efficiency ratios based on ϕ p-optimal designs. In A. Di Bucchianico, H. Läuter, and H. Wynn (Eds.), mODa’7 – Advances in Model–Oriented Design and Analysis, Proc. 7th Int. Workshop, Heeze (Netherlands), Heidelberg, pp. 89–96. Physica Verlag.Google Scholar
  29. Harman, R. (2004b). Minimal efficiency of designs under the class of orthogonally invariant information criteria. Metrika 60, 137–153.MathSciNetzbMATHCrossRefGoogle Scholar
  30. Harman, R. and T. Jurík (2008). Computing c-optimal experimental designs using the simplex method of linear programming. Comput. Statist. Data Anal. 53, 247–254.MathSciNetzbMATHCrossRefGoogle Scholar
  31. Harman, R. and M. Trnovská (2009). Approximate D-optimal designs of experiments on the convex hull of a finite set of information matrices. Math. Slovaca 59(5), 693–704.MathSciNetzbMATHCrossRefGoogle Scholar
  32. Harville, D. (1997). Matrix Algebra from a Statistician’s Perspective. Heidelberg: Springer.zbMATHCrossRefGoogle Scholar
  33. Karlin, S. and W. Studden (1966). Optimal experimental designs. Ann. Math. Statist. 37, 783–815.MathSciNetzbMATHCrossRefGoogle Scholar
  34. Khachiyan, L. (1979). A polynomial algorithm in linear programming. Doklady Akademïa Nauk SSSR 244, 1093–1096. [English transl. Soviet Math. Doklady, 20, 191–194].Google Scholar
  35. Khachiyan, L. (1996). Rounding of polytopes in the real number model of computation. Math. Oper. Res. 21(2), 307–320.MathSciNetzbMATHCrossRefGoogle Scholar
  36. Khachiyan, L. and M. Todd (1993). On the complexity of approximating the maximal inscribed ellipsoid for a polytope. Math. Programming A61(2), 137–159.Google Scholar
  37. Kiefer, J. (1962). Two more criteria equivalent to D-optimality of designs. Ann. Math. Statist. 33(2), 792–796.MathSciNetzbMATHCrossRefGoogle Scholar
  38. Kiefer, J. (1974). General equivalence theory for optimum designs (approximate theory). Ann. Statist. 2(5), 849–879.MathSciNetzbMATHCrossRefGoogle Scholar
  39. Kiefer, J. and J. Wolfowitz (1960). The equivalence of two extremum problems. Canad. J. Math. 12, 363–366.MathSciNetzbMATHCrossRefGoogle Scholar
  40. Liu, S. and H. Neudecker (1997). Kantorovich inequalities and efficiency comparisons for several classes of estimators in linear models. Statistica Neerlandica 51(3), 345–355.MathSciNetzbMATHCrossRefGoogle Scholar
  41. López-Fidalgo, J. and J. Rodríguez-Díaz (1998). Characteristic polynomial criteria in optimal experimental design. In A. Atkinson, L. Pronzato, and H. Wynn (Eds.), Advances in Model–Oriented Data Analysis and Experimental Design, Proc. MODA’5, Marseilles, June 22–26, 1998, pp. 31–38. Heidelberg: Physica Verlag.Google Scholar
  42. López-Fidalgo, J., B. Torsney, and R. Ardanuy (1998). MV-optimisation in weighted linear regression. In A. Atkinson, L. Pronzato, and H. Wynn (Eds.), Advances in Model–Oriented Data Analysis and Experimental Design, Proc. MODA’5, Marseilles, June 22–26, 1998, pp. 39–50. Heidelberg: Physica Verlag.Google Scholar
  43. Magnus, J. and H. Neudecker (1999). Matrix Differential Calculus, with Applications in Statistics and Econometrics. New York: Wiley.zbMATHGoogle Scholar
  44. Mikulecká, J. (1983). On a hybrid experimental design. Kybernetika 19(1), 1–14.MathSciNetzbMATHGoogle Scholar
  45. Müller, C. and A. Pázman (1998). Applications of necessary and sufficient conditions for maximin efficient designs. Metrika 48, 1–19.MathSciNetzbMATHGoogle Scholar
  46. Pázman, A. (1986). Foundations of Optimum Experimental Design. Dordrecht (co-pub. VEDA, Bratislava): Reidel (Kluwer group).Google Scholar
  47. Pázman, A. (2001). Concentration sets, Elfving sets and norms in optimum design. In A. Atkinson, B. Bogacka, and A. Zhigljavsky (Eds.), Optimum Design 2000, Chapter 10, pp. 101–112. Dordrecht: Kluwer.Google Scholar
  48. Pázman, A. (2002a). Optimal design of nonlinear experiments with parameter constraints. Metrika 56, 113–130.MathSciNetCrossRefGoogle Scholar
  49. Pečarić, J., S. Puntanen, and G. Styan (1996). Some further matrix extensions of the Cauchy-Schwarz and Kantorovich inequalities, with some statistical applications. Linear Algebra Appl. 237/238, 455–476.Google Scholar
  50. Pilz, J. (1983). Bayesian Estimation and Experimental Design in Linear Regression Models, Volume 55. Leipzig: Teubner-Texte zur Mathematik. [Also published by Wiley, New York, 1991].Google Scholar
  51. Polak, E. (1987). Optimization. Algorithms and Consistent Approximations. New York: Springer.Google Scholar
  52. Pronzato, L. (2004). A minimax equivalence theorem for optimum bounded design measures. Statist. Probab. Lett. 68, 325–331.MathSciNetzbMATHCrossRefGoogle Scholar
  53. Pronzato, L. (2006). On the sequential construction of optimum bounded designs. J. Stat. Plann. Inference 136, 2783–2804.MathSciNetzbMATHCrossRefGoogle Scholar
  54. Pronzato, L. (2009b). On the regularization of singular c-optimal designs. Math. Slovaca 59(5), 611–626.MathSciNetzbMATHCrossRefGoogle Scholar
  55. Pronzato, L. (2010b). Penalized optimal designs for dose-finding. J. Stat. Plann. Inference 140, 283–296.MathSciNetzbMATHCrossRefGoogle Scholar
  56. Pronzato, L., C.-Y. Huang, and E. Walter (1991). Nonsequential T-optimal design for model discrimination: new algorithms. In A. Pázman and J. Volaufová (Eds.), Proc. ProbaStat’91, Bratislava, pp. 130–136.Google Scholar
  57. Pronzato, L. and E. Walter (1994). Minimal-volume ellipsoids. Internat. J. Adapt. Control Signal Process. 8, 15–30.MathSciNetzbMATHCrossRefGoogle Scholar
  58. Pronzato, L. and E. Walter (1996). Volume-optimal inner and outer ellipsoids. In M. Milanese, J.-P. Norton, H. Piet-Lahanier, and E. Walter (Eds.), Bounding Approaches to System Identification, pp. 119–138. London: Plenum.Google Scholar
  59. Pronzato, L., H. Wynn, and A. Zhigljavsky (2000). Dynamical Search. Boca Raton: Chapman & Hall/CRC.zbMATHGoogle Scholar
  60. Pronzato, L., H. Wynn, and A. Zhigljavsky (2005). Kantorovich-type inequalities for operators via D-optimal design theory. Linear Algebra Appl. 410, 160–169.MathSciNetzbMATHCrossRefGoogle Scholar
  61. Pukelsheim, F. (1993). Optimal Experimental Design. New York: Wiley.Google Scholar
  62. Raynaud, H.-F., L. Pronzato, and E. Walter (2000). Robust identification and control based on ellipsoidal parametric uncertainty descriptions. Eur. J. Control 6(3), 245–257.MathSciNetGoogle Scholar
  63. Rényi, A. (1961). On measures of entropy and information. In Proc. 4th Berkeley Symp. on Math. Statist. and Prob., pp. 547–561.Google Scholar
  64. Rockafellar, R. (1970). Convex Analysis. Princeton: Princeton Univ. Press.zbMATHGoogle Scholar
  65. Sahm, M. and R. Schwabe (2001). A note on optimal bounded designs. In A. Atkinson, B. Bogacka, and A. Zhigljavsky (Eds.), Optimum Design 2000, Chapter 13, pp. 131–140. Dordrecht: Kluwer.Google Scholar
  66. Schwabe, R. (1995). Designing experiments for additive nonlinear models. In C. Kitsos and W. Müller (Eds.), MODA4 – Advances in Model-Oriented Data Analysis, Spetses (Greece), june 1995, pp. 77–85. Heidelberg: Physica Verlag.Google Scholar
  67. Schwabe, R. (1996). Optimum Designs for Multi-Factor Models. New York: Springer.zbMATHCrossRefGoogle Scholar
  68. Schwabe, R. (1997). Maximin efficient designs. Another view at D-optimality. Statist. Probab. Lett. 35, 109–114.Google Scholar
  69. Schwabe, R. and W. Wong (1999). Efficiency bounds for product designs in linear models. Ann. Inst. Statist. Math. 51, 723–730.MathSciNetzbMATHCrossRefGoogle Scholar
  70. Shor, N. and O. Berezovski (1992). New algorithms for constructing optimal circumscribed and inscribed ellipsoids. Optim. Methods Softw. 1, 283–299.CrossRefGoogle Scholar
  71. Sibson, R. (1972). Discussion on a paper by H.P. Wynn. J. Roy. Statist. Soc. B34, 181–183.Google Scholar
  72. Sibson, R. (1974). D A-optimality and duality. Progress in Stat. Colloq. Math. Soc. Janos Bolyai 9, 677–692.MathSciNetGoogle Scholar
  73. Silvey, S. (1980). Optimal Design. London: Chapman & Hall.zbMATHCrossRefGoogle Scholar
  74. Silvey, S. and D. Titterington (1973). A geometric approach to optimal design theory. Biometrika 60(1), 21–32.MathSciNetzbMATHCrossRefGoogle Scholar
  75. Studden, W. (1971). Elfving’s theorem and optimal designs for quadratic loss. Ann. Math. Statist. 42, 1613–1621.MathSciNetzbMATHCrossRefGoogle Scholar
  76. Sun, P. and R. Freund (2004). Computation of minimum-volume covering ellipsoids. Oper. Res. 52(5), 690–706.MathSciNetzbMATHCrossRefGoogle Scholar
  77. Tarasov, S., L. Khachiyan, and I. Erlich (1988). The method of inscribed ellipsoids. Soviet Math. Dokl. 37(1), 226–230.MathSciNetzbMATHGoogle Scholar
  78. Titterington, D. (1975). Optimal design: some geometrical aspects of D-optimality. Biometrika 62(2), 313–320.MathSciNetzbMATHGoogle Scholar
  79. Titterington, D. (1978). Estimation of correlation coefficients by ellipsoidal trimming. J. Roy. Statist. Soc. C27(3), 227–234.Google Scholar
  80. Todd, M. and E. Yildirim (2007). On Khachiyan’s algorithm for the computation of minimum volume enclosing ellipsoids. Discrete Appl. Math. 155, 1731–1744.MathSciNetzbMATHCrossRefGoogle Scholar
  81. Torsney, B. (1986). Moment inequalities via optimal design theory. Linear Algebra Appl. 82, 237–253.MathSciNetzbMATHCrossRefGoogle Scholar
  82. Vandenberghe, L., S. Boyd, and S.-P. Wu (1998). Determinant maximisation with linear matrix inequality constraints. SIAM J. Matrix Anal. Appl. 19(2), 499–533.MathSciNetzbMATHCrossRefGoogle Scholar
  83. Welzl, E. (1991). Smallest enclosing disks (balls and ellipsoids). In New Results and New Trends in Computer Science, Lecture Notes in Comput. Sci., vol. 555, pp. 359–370. Heidelberg: Springer.Google Scholar
  84. Wynn, H. (1977). Optimum designs for finite populations sampling. In S. Gupta and D. Moore (Eds.), Statistical Decision Theory and Related Topics II, pp. 471–478. New York: Academic Press.Google Scholar
  85. Wynn, H. (1982). Optimum submeasures with applications to finite population sampling. In S. Gupta and J. Berger (Eds.), Statistical Decision Theory and Related Topics III. Proc. 3rd Purdue Symp., vol. 2, pp. 485–495. New York: Academic Press.Google Scholar
  86. Yang, M. (2010). On de la Garza phenomenon. Ann. Statist. 38(4), 2499–2524.MathSciNetzbMATHCrossRefGoogle Scholar
  87. Yang, M. and J. Stufken (2009). Support points of locally optimal designs for nonlinear models with two parameters. Ann. Statist. 37(1), 518–541.MathSciNetzbMATHCrossRefGoogle Scholar
  88. Zhang, Y. and L. Gao (2003). On numerical solution of the maximum volume ellipsoid problem. SIAM J. Optim. 14(1), 53–76.MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2013

Authors and Affiliations

  • Luc Pronzato
    • 1
  • Andrej Pázman
    • 2
  1. 1.French National Center for Scientific Research (CNRS)University of NiceSophia AntipolisFrance
  2. 2.Department of Applied Mathematics and StatisticsComenius UniversityBratislavaSlovakia

Personalised recommendations