Estimating the Minimal Value of a Function in Global Random Search: Comparison of Estimation Procedures

  • Emily Hamilton
  • Vippal Savani
  • Anatoly Zhigljavsky
Part of the Optimization and Its Applications book series (SOIA, volume 4)


In a variety of global random search methods, the minimum of a function is estimated using either one of linear estimators or the the maximum likelihood estimator. The asymptotic mean square errors (MSE) of several linear estimators asymptotically coincide with the asymptotic MSE of the maximum likelihood estimator. In this chapter we consider the non-asymptotic behaviour of different estimators. In particular, we demonstrate that the MSE of the best linear estimator is superior to the MSE of the the maximum likelihood estimator.


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  1. [BGTS04]
    Beirlant, J., Goegebeur, Y., Teugels, J., Segers, J.: Statistics of Extremes. John Wiley & Sons Ltd., Chichester (2004)zbMATHCrossRefGoogle Scholar
  2. [CM89]
    Csörgő, S., Mason, D.M.: Simple estimators of the endpoint of a distribution. In: Extreme Value Theory (Oberwolfach, 1987). Vol. 51 of Lecture Notes in Statist., pp. 132–147. Springer, New York (1989)Google Scholar
  3. [Coo79]
    Cooke, P.: Statistical inference for bounds of random variables. Biometrika, 66(2), 367–374 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  4. [Coo80]
    Cooke, P.: Optimal linear estimation of bounds of random variables. Biometrika, 67(1), 257–258 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  5. [dH81]
    de Haan, L.: Estimation of the minimum of a function using order statistics. J. Amer. Statist. Assoc., 76(374), 467–469 (1981)zbMATHCrossRefMathSciNetGoogle Scholar
  6. [DN03]
    David, H.A., Nagaraja, H.N.: Order Statistics. Wiley Series in Probability and Statistics. 3rd ed. John Wiley & Sons, N.Y. (2003)Google Scholar
  7. [EKM03]
    Embrechts, P., Klüppelberg, C, Mikosch, T.: Modelling extremal events for insurance and finance. Springer-Verlag, Berlin (2003)Google Scholar
  8. [Gal87]
    Galambos, J.: The asymptotic theory of extreme order statistics. 2nd ed. Robert E. Krieger Publishing Co. Inc., Melbourne, FL (1987)zbMATHGoogle Scholar
  9. [Hal82]
    Hall, P.: On estimating the endpoint of a distribution. Ann. Statist., 10(2), 556–568 (1982)zbMATHMathSciNetGoogle Scholar
  10. [KN00]
    Kotz, S., Nadarajah, S.: Extreme Value Distributions. Imperial College Press, London (2000)zbMATHGoogle Scholar
  11. [Nev01]
    Nevzorov, V.B.: Records: Mathematical Theory. Vol. 194 of Translations of Mathematical Monographs. American Mathematical Society, Providence, RI (2001) Translated from the Russian manuscript by D.M. ChibisovGoogle Scholar
  12. [Zhi79]
    Zhigljavsky, A.: PhD: Monte-Carlo methods in global optimisation. University of St. Petersburg, St. Petersburg (1979)Google Scholar
  13. [Zhi91]
    Zhigljavsky, A.: Theory of global random search. Kluwer Academic Publishers, Dordrecht (1991)Google Scholar
  14. [ZZ06]
    Zhigljavsky, A., Zilinskas, A.: Stochastic Global Optimization. Springer, N.Y. et al (2006) to appearGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Emily Hamilton
    • 1
  • Vippal Savani
    • 1
  • Anatoly Zhigljavsky
    • 1
  1. 1.School of MathematicsCardiff UniversityUK

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