Let X1,...,X and Y1,...,Yn be independent samples from N(, 1) and N(, 1), and consider the hypothesis

Let X1,...,X and Y1,...,Yn be independent samples from N(ξ, 1) and N(η, 1), and consider the hypothesis H : η ≤ ξ against K : η>ξ.

There exists a UMP test, and it rejects the hypothesis when Y¯ − X¯ is too large.

[If ξ1 < η1, is a particular alternative, the distribution assigning probability 1 to the point η = ξ = (mξ1 + nη1)/(m + n) is least favorable.]