Let the probability density p of X have monotone likelihood ratio in T (x), and consider the

Let the probability density pθ of X have monotone likelihood ratio in T (x), and consider the problem of testing H : θ ≤ θ0 against θ > θ0. If the distribution of T is continuous, the p-value pˆ of the UMP test is given by pˆ = Pθ0 {T ≥ t}, where t is the observed value of T . This holds also without the assumption of continuity if for randomized tests pˆ is defined as the smallest significance level at which the hypothesis is rejected with probability 1. Show that, for any θ ≤ θ0, Pθ{ ˆp ≤ u} ≤ u for any 0 ≤ u ≤ 1.