Let X1,..., Xm; Y1,..., Yn be independently, normally distributed with means and , and variances a

Let X1,..., Xm; Y1,..., Yn be independently, normally distributed with means ξ and η, and variances a σ2 and τ 2 respectively, and consider the hypothesis H : τ ≤ σ a against K : σ < τ .

(i) If



ξ and η are known, there exists a UMP test given by the rejection region

(Yj − η)2/

(Xi − ξ)2 ≥ C.

(ii) No UMP test exists when ξ and η are unknown.