Correcting Over-Conservatism in Exact Two-Sided Confidence Curves for Poisson Rates and Sequential Binary Tests

Exact confidence intervals (such as Garwood or Clopper-Pearson) for discrete data problems are guaranteed to bound the true parameter at some minimum desired rate given the random distribution of the data. However, the discrete nature of the data necessarily means that there are only a finite number of confidence levels which meet the nominal coverage level exactly. An acceptable compromise for one-sided intervals perhaps, but an unfortunate source of over-conservatism when this gets compounded in the creation of two-sided intervals. Many solutions to this problem have been presented over the years, either by relaxing the coverage requirements, or producing disjoint or non-nested confidence intervals. For safety critical applications these compromises may be undesirable or impractical. This manuscript describes a method to correct the over-conservatism in confidence intervals drawn for the Poisson distribution for estimating event rates, as well as a sequential binary test for efficiently accepting or rejecting a test of competing hypotheses about a probability. In both cases, the result is an exact, two-sided confidence curve that eliminates the over-conservatism present in conventional two-sided confidence intervals for these problems. These curves also protect against the issue of false confidence that stems from the use of additive belief structures such as precise probability distributions.

Keywords: Confidence, Inference, Exact, Poisson, Sequential.