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ROC and the bounds on tail probabilities via theorems of Dubins and F. Riesz. (English) Zbl 1161.62027

Summary: 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 L. E. Dubins [J. Math. Anal. Appl. 5, 237–244 (1962; Zbl 0124.37704)] about extreme points and the upper bounds depend on a symmetric rearrangement theorem of F. Riesz [Journal L.M.S. 5, 162–168 (1930; JFM 56.0232.02)]. The inequality was motivated by medical imaging: to find bounds on the area under the Receiver Operating Characteristic curve (ROC).

MSC:

62G32 Statistics of extreme values; tail inference
60E15 Inequalities; stochastic orderings
92C55 Biomedical imaging and signal processing
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