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Journal of the Optical Society of America A

Journal of the Optical Society of America A


  • Vol. 16, Iss. 1 — Jan. 1, 1999
  • pp: 53–57

Bounds on the area under the ROC curve

Jeffrey H. Shapiro  »View Author Affiliations

JOSA A, Vol. 16, Issue 1, pp. 53-57 (1999)

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Upper and lower bounds are derived for the area under the receiver-operating-characteristic (ROC) curve of binary hypothesis testing. These results are compared with the area-under-the-curve (AUC) approximation and the AUC lower bound recently reported by Barrett et al. [J. Opt. Soc. Am A 15, 1520 (1998)].

© 1999 Optical Society of America

OCIS Codes
(110.3000) Imaging systems : Image quality assessment
(110.4280) Imaging systems : Noise in imaging systems

Original Manuscript: June 26, 1998
Revised Manuscript: October 5, 1998
Manuscript Accepted: October 5, 1998
Published: January 1, 1999

Jeffrey H. Shapiro, "Bounds on the area under the ROC curve," J. Opt. Soc. Am. A 16, 53-57 (1999)

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  1. H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality. III. ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998). [CrossRef]
  2. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Sec. 2.6.
  3. S. Kotz, N. L. Johnson, eds. in chief, Encyclopedia of Statistical Sciences, Vol. 5 (Wiley, New York, 1985), pp. 176–181.
  4. The upper bound in Eq. (8) is the familiar Bhattacharyya bound; see, e.g., H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Sec. 2.7.; the lower bound, which is less well known, is derived in Appendix B.
  5. Our results will still be valid if Λ is a discrete or mixed random variable, but the derivation will require inclusion of randomized tests.
  6. J. H. Shapiro, B. A. Capron, R. C. Harney, “Imaging and target detection with a heterodyne-reception optical radar,” Appl. Opt. 20, 3292–3313 (1981). [CrossRef] [PubMed]
  7. The task of evaluating Pe can be avoided—at the expense of some weakening of our AUC bounds—by using upper and lower bounds on the error probability found from the conditional semi-invariant moment-generating function of the likelihood ratio under hypothesis H0. For a recent discussion of such Pe bounds see M. V. Burnashev, “On one useful inequality in testing of hypotheses,” IEEE Trans. Inf. Theory 44, 1668–1670 (1998). In most cases, these bounds will be appreciably tighter than the Bhattarcharyya-distance results given in Eq. (8). [CrossRef]
  8. H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), p. 137.

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