An alternative proof of the optimum property of the Wilcoxon test for detecting a shift in the logistic distribution is obtained from the preceding problem by equating F(x − θ) with (1 − θ)F(x) + θF2(x), neglecting powers of θ higher than the first. This leads to the differential equation F − θF = (1 − θ)F + θF2, the solution of which is the logistic distribution.