Let X1,...,Xm and Y1,...,Yn be positive, independent random variables distributed with densities f(x/) and g(y/ ) respectively.

Let X1,...,Xm and Y1,...,Yn be positive, independent random variables distributed with densities f(x/σ) and g(y/τ ) respectively. If f and g have monotone likelihood ratios in (x, σ) and (y, τ ) respectively, there exists a UMP conditional test of H : τ /σ ≤ ∆0 against τ /σ > ∆0 given the ancillary statistics Ui = Xi/Xm and Vj = Yj/Yn (i = 1,...,m − 1; j = 1,...,n − 1).