ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz
Abstract
For independent $X$ and $Y$ in the inequality $P(X\leq Y+\mu)$, 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).
 Publication:

arXiv eprints
 Pub Date:
 March 2009
 arXiv:
 arXiv:0903.0518
 Bibcode:
 2009arXiv0903.0518C
 Keywords:

 Mathematics  Probability;
 62G32;
 60E15 (Primary) 92C55. (Secondary)
 EPrint:
 Published in at http://dx.doi.org/10.1214/08AAP536 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)