Let V1,..., Vn be independently distributed as N(0, 1), and given V1 = v1,..., Vn = vn, let Xi (i = 1,..., n) be independently distributed as N(θvi, 1).

(i) There does not exist a UMP test of H : θ = 0 against K : θ > 0.
(ii) There does exist a UMP conditional test of H against K given the ancillary (V1,..., Vn). [Buehler (1982).]