# Confidence Intervals

## Abstract

Section 5.11 presented two approximate methods to determine uncertainties of maximum likelihood estimates. Basically, either the negative log likelihood function is approximated to a parabola at the minimum, corresponding to a Gaussian PDF approximation, or the excursion of the negative log likelihood around the minimum is considered in order to obtain the possibly asymmetric uncertainty. None of those methods guarantees an exact coverage of the uncertainty interval. In many cases, the provided level of approximation is sufficient, but for measurements with a small number of events and PDF models that exhibit large deviation from the Gaussian approximation, the uncertainty determined with those approximate methods may not be sufficiently accurate.

## References

- 1.Clopper, C.J., Pearson, E.: The use of confidence or fiducial limits illustrated in the case of the binomial. Biometrika 26, 404–413 (1934)CrossRefMATHGoogle Scholar
- 2.Cousins, R., Hymes, K.E., Tucker, J.: Frequentist evaluation of intervals estimated for a binomial parameter and for the ratio of Poisson means. Nucl. Instrum. Meth. A612 388–398 (2010)ADSCrossRefGoogle Scholar
- 3.Feldman, G., Cousins, R.: Unified approach to the classical statistical analysis of small signals. Phys. Rev. D57, 3873–3889 (1998)ADSGoogle Scholar
- 4.Neyman, J.: Outline of a theory of statistical estimation based on the classical theory of probability. Philos. Trans. R. Soc. Lond. A Math. Phys. Sci. 236, 333–380 (1937)ADSCrossRefMATHGoogle Scholar