Optimal confidence intervals for the expectation of a Poisson random variable

Confidence intervals are a very useful tool for making inferences on the expectation, λ, of a Poisson random variable. When making inferences about λ using a confidence interval, one would hope the process used to create the interval has the level of confidence it promises and also produces confidence intervals that are as short as possible so as to hone in on the true value of λ. Many confidence procedures have been developed in an attempt to achieve these two goals; that is, to be the "shortest" strict confidence procedure. I discuss several of these methods through the perspective of coverage probability functions. I also introduce three new methods which are optimal according to Kabaila and Byrne's Inability to be Shortened property. One method is derived by first creating a specialized coverage probability function through an exhaustive graphical examination of all Poisson probability functions for a set of consecutive values. Subsequently, confidence intervals for λ with any desired confidence level can then be formed for all possible values of the observed Poisson random variable. The process for creating the coverage probability function of a Poisson confidence procedure and the resulting confidence intervals will be described in detail. Then from the insight gained from deriving this method two other high-performance methods are produced. Finally, the collection of intervals derived from these three methods will be compared with those derived from existing methods with respect to coverage and several different measures of length.