The Annals of Statistics

The problem of low counts in a signal plus noise model

Hsiuying Wang and Michael Woodroofe

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Consider the model $X = B + S$, where $B$and $S$ are independent Poisson random variables with means $\mu$ and $\nu$, $\nu$ is unknown, but $\mu$ is known. The model arises in particle physics and some recent articles have suggested conditioning on the observed bound on $B$; that is, if $X = n$ is observed, then the suggestion is to base inference on the conditional distribution of $X$ given $B \leq n$. This conditioning is non-standard in that it does not correspond to a partition of the sample space. It is examined here from the view point of decision theory and shown to lead to admissible formal Bayes procedures.

Article information

Ann. Statist., Volume 28, Number 6 (2000), 1561-1569.

First available in Project Euclid: 12 March 2002

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 62C15: Admissibility
Secondary: 62F03: Hypothesis testing 62P35: Applications to physics

Admissibility ancillary statistic Bayesian solutions confidence intervals neutrino oscillations $p$-values risk


Woodroofe, Michael; Wang, Hsiuying. The problem of low counts in a signal plus noise model. Ann. Statist. 28 (2000), no. 6, 1561--1569. doi:10.1214/aos/1015957470.

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  • Berger, J. (1985). Statistical Decision Theory and Bayesian Analysis, 2nd ed. Springer, New York.
  • Billingsley, P. (1995). Probability and Measure, 3rd ed. Wiley, New York.
  • Cousins, R. (2000). Application of conditioning to the Gaussian with boundary problem in the unified approach to confidence intervals. Phys. Rev. D. To appear.
  • Dawid, A. P. and Stone, M. (1982). The functional model basis of fiducial inference (with discussion). Ann. Statist. 10 1054-1067.
  • Feldman, G. and Cousins, R. (1998). Unified approach to the classical statistical analysis of small signals. Phys. Rev. D 57 3873-3889.
  • Hwang, J. T., Casella, G., Robert, C. Wells, M. and Farrell, R. (1992). Estimation of accuracy in testing. Ann. Statist. 20 490-509.
  • Lehmann, E. (1986). Testing Statistical Hypotheses, 2nd ed. Wadsworth, Belmont, CA.
  • Read, A. L. (2000). Modified frequentist analysis of search methods (the CLs method). Unpublished manuscript.
  • The Particle Data Group (1992). Probability, statistics, and Monte Carlo. Phys. Rev. D 45 III.32III.42.
  • Roe, B. P. and Woodroofe, M. (1999). Improved probability method for estimating signal in the presence of background. Phys. Rev. D 60 3009-3015.
  • Zeitnitz, B. et al. (1998). Neutrino oscillation results from KARMEN. Progress in Particle and Nuclear Physics 40 169-181.