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

TL;DR

This paper derives sharp bounds on tail probabilities for unimodal and symmetric distributions, motivated by ROC curve analysis in medical imaging, using theorems of Dubins and Riesz.

Contribution

It introduces new bounds on tail probabilities based on distribution shape assumptions, connecting probability inequalities with classical theorems.

Findings

Sharp lower bounds for unimodal distributions with finite variance.

Sharp upper bounds for symmetric densities bounded by a finite constant.

Application to bounds on the ROC curve area in medical imaging.

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).