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).
@article{1235140346,
author = {Clarkson, Eric and Denny, J. L. and Shepp, Larry},
title = {ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz},
journal = {Ann. Appl. Probab.},
volume = {19},
number = {1},
year = {2009},
pages = { 467-476},
language = {en},
url = {http://dml.mathdoc.fr/item/1235140346}
}
Clarkson, Eric; Denny, J. L.; Shepp, Larry. ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz. Ann. Appl. Probab., Tome 19 (2009) no. 1, pp. 467-476. http://gdmltest.u-ga.fr/item/1235140346/