Let X = (X1,..., Xn) be a sample from the uniform distribution U(, + 1). (i)

Let X = (X1,..., Xn) be a sample from the uniform distribution U(θ, θ + 1).

(i) For testing H : θ ≤ θ0 against K : θ > θ0 at level α, there exists a UMP test which rejects when min(X1,..., Xn) > θ0 + C(α) or max(X1,..., Xn) >

θ0 + 1 for suitable C(α).

(ii) The family U(θ, θ + 1) does not have monotone likelihood ratio. [Additional results for this family are given in Birnbaum (1954b) and Pratt (1958).]

[(ii) By Theorem 3.4.1, monotone likelihood ratio implies that the family of UMP test of H : θ ≤ θ0 against K : θ > θ0 generated as α varies from 0 to 1 is independent of θ0.]