It is desired to test the null hypothesis that a sequential sample has a U(0, 2)

common density versus the alternative hypothesis that the common density is U(1, 3). The decision loss is 0 for a correct decision and 20 for an incorrect decision. It is possible, at any stage, to take observations in batches. A batch of m observations costs m1/2. The prior probability of each hypothesis is . Let do denote the procedure which makes an immediate Bayes decision. Let d denote the procedure which takes successive samples (batches) of size m, stopping and deciding Ho (deciding H₁) if any observation in the batch is less than one (greater than two), and taking another batch otherwise.

(a) Why can attention be restricted to consideration of do and the d?

(b) Show that the Bayes risk of dm is m/2 r(7,

d) =1-2-m

(c) Show that d₂ is the optimal procedure.