On comparing two poisson intensity functions

Given realizations of two possion processes with unknown intensities A(·) and F(·) observed over the interval (t₁,t₂), we suppose that it is desired to distinution between H₀ Ξ(·)/λ(·) is constant on (t₁,t₂) versus H₊:Ξ(·)/λ(·) increases on (t₁,t₂). We propose a decision rule which uses the percentage points of the Mann-Whitney U-distribution. We show that the decision rule is unbiased and that the set of alternatives in H₊ can be weakly ordered, specifically: if Ξ(·)/λ(·), β(·)/λ(·) and Ξ(·)/β(·) are increasing on (t₁, t₂) then P{H₀ is rejected |Ξ(·)}≧P{H₀ is rejected|B(·)}≧P{H₀ is rejected|H₀}.