Mathematical Methods of Statistics

, Volume 24, Issue 2, pp 110–121 | Cite as

A characterization of maximin tests for two composite hypotheses

  • A. Gushchin


We consider the problem of testing two composite hypotheses in the minimax setting. To find maximin tests, we propose a new dual optimization problem which has a solution under a mild additional assumption. This allows us to characterize maximin tests in considerable generality. We give a simple example where the null hypothesis and the alternative are strictly separated, however, a maximin test is purely randomized.


dual problem testing composite hypotheses maximin test 

2000 Mathematics Subject Classification

62F03 62G10 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    V. Baumann, “Eine parameterfreie Theorie der ungünstigsten Verteilungen für das Testen von Hypothesen”, Z. Wahrsch. verw. Gebiete 11, 41–60 (1968).MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    J. Cvitanić and I. Karatzas, “Generalized Neyman-Pearson Lemma via Convex Duality”, Bernoulli 7, 79–97 (2001).MathSciNetCrossRefGoogle Scholar
  3. 3.
    J. Cvitanić, W. Schachermayer, and H. Wang, “Utility Maximization in Incomplete Markets with Random Endowment”, Finance Stoch. 5, 259–272 (2001).MathSciNetCrossRefGoogle Scholar
  4. 4.
    F. Delbaen and W. Schachermayer, “AGeneral Version of the Fundamental Theorem of Asset Pricing”, Math. Ann. 300, 463–520 (1994).MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    A. A. Gushchin, “On an Extension of the Notion of f-Divergence”, Theory Probab. Appl. 52, 439–455 (2008).MathSciNetCrossRefGoogle Scholar
  6. 6.
    A. A. Gushchin, “A Characterization of a Minimax Test in the Problem of Testing Two Composite Hypotheses”, Dokl.Math. 87, 345–347 (2013).MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    J. Komlós, “A Generalization of a Problem of Steinhaus”, Acta Math. Acad. Sci. Hungar. 18, 217–229 (1967).MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    O. Krafft and H. Witting, “Optimale Tests und ungünstigste Verteilungen”, Z. Wahrsch. verw. Gebiete 7, 289–302 (1967).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    E. L. Lehmann, “On the Existence of Least Favorable Distributions”, Ann. Math. Statist. 23, 408–416 (1952).CrossRefGoogle Scholar
  10. 10.
    E. L. Lehmann, Testing Statistical Hypotheses (Wiley, New York, 1959).MATHGoogle Scholar
  11. 11.
    E. L. Lehmann and C. Stein, “Most Powerful Tests of Composite Hypotheses. I.Normal Distributions”, Ann. Math. Statist. 19, 495–516 (1948).MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    J. Neyman and E. S. Pearson, “On the Problem of the Most Efficient Tests of Statistical Hypotheses”, Phil. Trans. Roy. Soc., Ser. A 231, 289–337 (1933).CrossRefGoogle Scholar
  13. 13.
    R. R. Phelps, Lectures on Choquet’s Theorem, 2nd ed., in Lecture Notes inMath. (Springer, Berlin, 2001), Vol. 1757.Google Scholar
  14. 14.
    R. T. Rockafellar, “Extension of Fenchel’s Duality Theorem for Convex Functions”, DukeMath. J. 33, 81–89 (1966).MathSciNetCrossRefGoogle Scholar
  15. 15.
    W. Rudin, Functional Analysis (McGraw-Hill, 1991).Google Scholar
  16. 16.
    B. Rudloff and I. Karatzas, “Testing Composite Hypotheses via Convex Duality”, Bernoulli 16, 1224–1239 (2010).MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    H. Strasser, Mathematical Theory of Statistics (De Gruyter, Berlin-New York, 1985).CrossRefGoogle Scholar
  18. 18.
    E. Torgersen, Comparison of Statistical Experiments (Cambridge Univ. Press, Cambridge, 1991).CrossRefMATHGoogle Scholar
  19. 19.
    H. Witting, Mathematische Statistik (Teubner, Stuttgart, 1985), Vol. I.CrossRefMATHGoogle Scholar
  20. 20.
    K. Yosida and E. Hewitt, “Finitely Additive Measures”, Trans. Amer. Math. Soc. 72, 46–66 (1952).MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2015

Authors and Affiliations

  1. 1.Steklov Mathematical InstituteMoscowRussia
  2. 2.International Laboratory of Quantitative FinanceNational Research Univ. Higher School of EconomicsMoscowRussia

Personalised recommendations