Bernoulli

Generalized Neyman-Pearson lemma via convex duality

Jaksa Cvitanic and Ioannis Karatzas

Full-text: Open access

PDF File (347 KB)

Abstract

We extend the classical Neyman-Pearson theory for testing composite hypotheses versus composite alternatives, using a convex duality approach, first employed by Witting. Results of Aubin and Ekeland from non-smooth convex analysis are used, along with a theorem of Komlós, in order to establish the existence of a max-min optimal test in considerable generality, and to investigate its properties. The theory is illustrated on representative examples involving Gaussian measures on Euclidean and Wiener space.

Article information

Source
Bernoulli, Volume 7, Number 1 (2001), 79-97.

Dates
First available in Project Euclid: 29 March 2004

Permanent link to this document
https://projecteuclid.org/euclid.bj/1080572340

Mathematical Reviews number (MathSciNet)
MR1811745

Zentralblatt MATH identifier
1054.62056

Keywords
hypothesis testing Komlós theorem non-smooth convex analysis normal cones optimal generalized tests saddle-points stochastic games subdifferentials

Citation

Cvitanic, Jaksa; Karatzas, Ioannis. Generalized Neyman-Pearson lemma via convex duality. Bernoulli 7 (2001), no. 1, 79--97. https://projecteuclid.org/euclid.bj/1080572340


Export citation

References

  • [1] Aubin, J. and Ekeland, I. (1984) Applied Nonlinear Analysis. New York: Wiley.
    Mathematical Reviews (MathSciNet): MR749753
  • [2] Baumann, V. (1968) Eine parameterfreie Theorie der ungünstigsten Verteilungen fü r das Testen von Hypothesen. Z. Wahrscheinlichkeitstheorie Verw. Geb., 11, 61-73.
    Digital Object Identifier: doi:10.1007/BF00538386
  • [3] Benes, V.E. (1971) Existence of optimal stochastic control laws. SIAM J. Control, 8, 446-475.
    Mathematical Reviews (MathSciNet): MR300726
    Digital Object Identifier: doi:10.1137/0309034
  • [4] Cvitanic¬, J. (2000) Minimizing expected loss of hedging in incomplete and constrained markets. SIAM J. Control Optim. To appear.
  • [5] Cvitanic¬, J. and Karatzas, I. (1999) On dynamic measures of risk. Finance Stochastics, 3, 451-482.
    Digital Object Identifier: doi:10.1007/s007800050071
  • [6] Ferguson, T.S. (1967) Mathematical Statistics: A Decision-Theoretic Approach. New York and London: Academic Press.
    Mathematical Reviews (MathSciNet): MR215390
    Zentralblatt MATH: 0153.47602
  • [7] Föllmer, H. and Leukert, P. (1999) Quantile hedging. Finance Stochastics, 3, 251-273.
    Zentralblatt MATH: 0977.91019
    Digital Object Identifier: doi:10.1007/s007800050062
  • [8] Föllmer, H. and Leukert, P. (2000) Partial hedging with minimal shortfall risk. Finance Stochastics, 4, 117-146.
    Digital Object Identifier: doi:10.1007/s007800050008
  • [9] Heath, D. (1993) A continuous-time version of Kulldorff 's result. Unpublished Manuscript, Cornell University.
  • [10] Huber, P. and Strassen, V. (1973) Minimax tests and the Neyman-Pearson lemma for capacities. Ann. Statist., 1, 251-263.
    Mathematical Reviews (MathSciNet): MR356306
    Digital Object Identifier: doi:10.1214/aos/1176342363
    Project Euclid: euclid.aos/1176342363
  • [11] Karatzas, I. (1997) Adaptive control of a diffusion to a goal, and a parabolic Monge-Ampère-type equation. Asian J. Math., 1, 295-313.
  • [12] Karatzas, I. and Shreve, S.E. (1991) Brownian Motion and Stochastic Calculus (2nd edition). New York: Springer-Verlag.
    Mathematical Reviews (MathSciNet): MR1121940
  • [13] Komlós, J. (1967) A generalization of a problem of Steinhaus. Acta Math. Acad. Sci. Hungar., 18, 217-229.
    Digital Object Identifier: doi:10.1007/BF02020976
  • [14] Krafft, O. and Witting, H. (1967) Optimale Tests und ungünstigste Verteilungen. Z. Wahrscheinlichkeitstheorie Verw. Geb., 7, 289-302.
    Digital Object Identifier: doi:10.1007/BF01844447
  • [15] Lehmann, E.L. (1952) On the existence of least-favorable distributions. Ann. Math. Statist., 23, 408- 416.
    Mathematical Reviews (MathSciNet): MR50231
    Zentralblatt MATH: 0047.38401
    Digital Object Identifier: doi:10.1214/aoms/1177729385
    Project Euclid: euclid.aoms/1177729385
  • [16] Lehmann, E.L. (1986) Testing Statistical Hypotheses (2nd edition). New York: Wiley.
    Mathematical Reviews (MathSciNet): MR852406
  • [17] Schwartz, M. (1986) New proofs of a theorem of Komlós. Acta Math. Hungar., 47, 181-185.
    Digital Object Identifier: doi:10.1007/BF01949141
  • [18] Spivak, G. (1998) Maximizing the probability of perfect hedge. Doctoral dissertation, Department of Statistics, Columbia University. Abstract can also be found in the ISI/STMA publication
    URL: Link to item
  • [19] Vajda, I. (1989) Theory of Statistical Inference and Information. Dordrecht: Kluwer.
  • [20] Witting, H. (1985) Mathematische Statistik I. Stuttgart: B.G. Teubner.
    Mathematical Reviews (MathSciNet): MR943833
    Zentralblatt MATH: 0581.62001
  • [21] Xu, G.-L. and Shreve, S.E. (1992) A duality method for optimal consumption and investment under short-selling prohibition. II: Constant market coefficients. Ann. Appl. Probab., 2, 314-328. Abstract can also be found in the ISI/STMA publication
    URL: Link to item
    Mathematical Reviews (MathSciNet): MR1161057
    Zentralblatt MATH: 0773.90017
    Digital Object Identifier: doi:10.1214/aoap/1177005706
    Project Euclid: euclid.aoap/1177005706

Bernoulli Society for Mathematical Statistics and Probability

Bernoulli

New content alerts

Email RSS ToC RSS Article
  • You have access to this content.
  • You have partial access to this content.
  • You do not have access to this content.
Turn Off MathJax

What is MathJax?

More like this

+ See more