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Confidence intervals for the Poisson distribution

Frank C. Porter

Abstract

The Poisson probability distribution is frequently encountered in physical science measurements. In spite of the simplicity and familiarity of this distribution, there is considerable confusion among physicists concerning the description of results obtained via Poisson sampling. The goal of this paper is to mitigate this confusion by examining and comparing the properties of both conventional and popular alternative techniques. We concern ourselves in particular with the description of results, as opposed to interpretation. After considering performance with respect to several desirable properties we recommend summarizing the results of Poisson sampling with confidence intervals proposed by Garwood. We note that the p-values obtained from these intervals are well-behaved and intuitive, providing for consistent treatment. We also find that averaging intervals can be problematic if the underlying Poisson distributions are not used.

Confidence intervals for the Poisson distribution

Abstract

The Poisson probability distribution is frequently encountered in physical science measurements. In spite of the simplicity and familiarity of this distribution, there is considerable confusion among physicists concerning the description of results obtained via Poisson sampling. The goal of this paper is to mitigate this confusion by examining and comparing the properties of both conventional and popular alternative techniques. We concern ourselves in particular with the description of results, as opposed to interpretation. After considering performance with respect to several desirable properties we recommend summarizing the results of Poisson sampling with confidence intervals proposed by Garwood. We note that the p-values obtained from these intervals are well-behaved and intuitive, providing for consistent treatment. We also find that averaging intervals can be problematic if the underlying Poisson distributions are not used.

Paper Structure

This paper contains 53 sections, 1 theorem, 82 equations, 24 figures.

Table of Contents

  1. Introduction
  2. Point estimates
  3. Interval estimates
  4. Signal significance
  5. Following sections
  6. Construction of frequency statistics
  7. Confidence sets
  8. $p$-values
  9. Descriptive statistics
  10. Desirable properties of confidence intervals
  11. Exactness
  12. Connectedness
  13. Contains maximum likelihood estimator
  14. Optimal coverage
  15. Length
  16. ...and 38 more sections

Key Result

Theorem 1

(Blaker) Define function $\gamma(\mu,n)\equiv \min\left[\hbox{Prob}(N\ge n|\mu), \hbox{Prob}(N\le n|\mu)\right]$ and "acceptability function" $\alpha(\mu; n)\equiv \hbox{Prob}\left[\gamma(\mu,N)\le \gamma(\mu,n)|\mu\right]$. Then

Figures (24)

  • Figure 1: Performance of 68% confidence standard upper limits. Left: coverage probability as a function of $\mu$. The light horizontal line indicates $1-\alpha$. Right: upper limit as a function of observed counts.
  • Figure 2: Performance of Garwood intervals. Left: coverage probability as a function of $\mu$; Middle: 68% confidence interval as a function of observed counts. The plus symbols show the maximum likelihood estimator for the mean; Right: size of the 68% confidence interval as a function of observed counts. The plus symbols show the values of $2\sqrt{n}$.
  • Figure 3: Performance of Sterne and Crow&Gardner 68% confidence intervals. Left: coverage probability as a function of $\mu$; Middle: confidence interval boundaries as a function of observed counts. Circles are for Crow&Gardner, squares for Sterne. The plus symbols show the maximum likelihood estimator for the mean; Right: size of the confidence interval as a function of observed counts. Circles are for Crow&Gardner, squares for Sterne. The plus symbols show the values of $2\sqrt{n}$.
  • Figure 4: Performance of Garwood and Crow&Gardner 68% confidence intervals. Left: coverage probability as a function of $\mu$; Middle: confidence interval boundaries as a function of observed counts. Circles are for Crow&Gardner, filled squares for Garwood. The plus symbols show the maximum likelihood estimator for the mean; Right: size of the confidence interval as a function of observed counts. Circles are for Crow&Gardner, filled squares for Garwood. The plus symbols show the values of $2\sqrt{n}$.
  • Figure 5: Comparison of the symmetry of the Garwood and Crow&Gardner 68% confidence intervals. The difference between the probability of an observation in the high side tail and the probability of an observation in the low side tail is plotted as a function of $\mu$.
  • ...and 19 more figures

Theorems & Definitions (1)

  • Theorem 1