Comparing two poisson parameters: what to do when the optimal isn't done

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Abstract
Suppose two Poisson processes with rates γ1 and γ2 are observed for fixed times tl and t2. This paper considers hypothesis tests and confidence intervals for the parameter ρ = γ2/γ1. Uniformly most powerful unbiased tests and uniformly most accurate unbiased confidence intervals exist for ρ, but they require randomization and so are not used in practice. Several alternative procedures have been proposed. In the context of one-sided hypothesis tests these procedures are reviewed and compared on numerical grounds and by use of the conditionality and repeated sampling principles. It is argued that a conditional binomial test which rejects with conditional level closest to but not necessarily less than, the nominal a is the most reasonable. This test is different from the usual conditional binomial test which rejects with conditional level closeset to but less than or equal to the nominal α Numerical results indicate that an approximate procedure based on the Poisson variance stabilizing transformation has properties similar to the preferred conditional binomial test. Values for λ1 = t1λ1 required to achieve a specified power are considered. These results are also discussed in terms of test inversion to obtain confidence intervals.

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