Abstract
For independent X and Y in the inequality P(X≤Y+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).
Citation
Eric Clarkson. J. L. Denny. Larry Shepp. "ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz." Ann. Appl. Probab. 19 (1) 467 - 476, February 2009. https://doi.org/10.1214/08-AAP536
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