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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Abstract

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

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

Dates
First available in Project Euclid: 20 February 2009

Permanent link to this document
http://projecteuclid.org/euclid.aoap/1235140346

Digital Object Identifier
doi:10.1214/08-AAP536

Mathematical Reviews number (MathSciNet)
MR2498685

Zentralblatt MATH identifier
1161.62027

Subjects
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]

Keywords
ROC tail probabilities extreme points symmetric rearrangements

Citation

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. http://projecteuclid.org/euclid.aoap/1235140346.


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