The Annals of Applied Probability

ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz

Eric Clarkson, J. L. Denny, and Larry Shepp

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For independent X and Y in the inequality P(XY+μ), we give sharp lower bounds for unimodal distributions having finite variance, and sharp upper bounds assuming symmetric densities bounded by a finite constant. The lower bounds depend on a result of Dubins about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz. The inequality was motivated by medical imaging: find bounds on the area under the Receiver Operating Characteristic curve (ROC).

Article information

Ann. Appl. Probab., Volume 19, Number 1 (2009), 467-476.

First available in Project Euclid: 20 February 2009

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62G32: Statistics of extreme values; tail inference 60E15: Inequalities; stochastic orderings
Secondary: 92C55: Biomedical imaging and signal processing [See also 44A12, 65R10, 94A08, 94A12]

ROC tail probabilities extreme points symmetric rearrangements


Clarkson, Eric; Denny, J. L.; Shepp, Larry. ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz. Ann. Appl. Probab. 19 (2009), no. 1, 467--476. doi:10.1214/08-AAP536.

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  • [1] Bamber, D. C. (1975). The area above the ordinal dominance graph and the area below the receiver operating characteristic graph. J. Math. Psych. 12 387–415.
  • [2] Barrett, H. H., Abbey, C. K. and Clarkson, E. (1998). Objective assessment of image quality III: ROC metrics, ideal observers and likelihood-generating functions. J. Opt. Soc. Amer. A 15 1520–1535.
  • [3] Clarkson, E. (2002). Bounds on the area under the receiver operating characteristic curve. J. Opt. Soc. Amer. A 19 1963–1968.
  • [4] Dharmadhikari, S. W. and Joag-Dev, K. (1985). The Gauss–Tchebyshev inequality for unimodal distributions. Theory Probab. Appl. 30 817–820.
  • [5] Dubins, L. E. (1962). On extreme points of convex sets. J. Math. Anal. Appl. 5 237–244.
  • [6] Feller, W. (1966). An Introduction to Probability Theory and Its Applications. Vol. II, 2nd ed. Wiley, New York.
  • [7] Hardy, G. H., Littlewood, J. E. and Pólya, G. (1952). Inequalities. Cambridge Univ. Press, Cambridge.
  • [8] Ibragimov, I. A. (1956). On the composition of unimodal distributions. Theory Probab. Appl. 1 283–288.
  • [9] Johnson, N. L. and Rogers, C. A. (1951). The moment problem for unimodal distributions. Ann. Math. Statist. 22 433–439.
  • [10] Lieb, E. H. and Loss, M. (2000). Analysis, 2nd ed. Graduate Studies in Mathematics 14. Amer. Math. Soc., Providence, RI.
  • [11] Metz, C. E. (1978). ROC methodology in radiologic imaging. Invest. Radiology 21 720–733.
  • [12] Pukelsheim, F. (1994). The three sigma rule. Amer. Statist. 48 88–91.
  • [13] Riesz, F. (1930). Sur une inégalité intégrale. J. Lond. Math. Soc. 5 162–168.
  • [14] Swets, J. A. and Pickett, R. M. (1982). Evaluation of Diagnostic Systems: Methods from Signal Detection Theory. Academic Press, New York.