Question

4. Use the binomial model for X with n=20; 0 ≤ θ ≤ 1, A={ a1 , a2 } where a1 means: “say θ ≤ 0.2” and a2 means: “say θ > 0.2”.
a) Let the loss be
0 , if θ ≤ 0.2 5 , if θ ≤ 0.2
L( a1 , θ) = L( a2 , θ) =
10 , if θ > 0.2 0 , if θ > 0.2
Find the Bayes procedure for a Beta prior with α=4 and β=16
b) Consider the following class of decision procedures:
t( x ) = a2 if and only if x > C where C can be any value in {0,1,2,…,n-1}
Find C such that the defined procedure is a frequentist test of size α = 0.05
c) Show that there is a Beta prior such that the corresponding Bayes procedure is the frequentist test in (b).
d) Assume the prior is Uniform on the interval 0 to 1. Show that there is a very drastic modification of the loss function above which makes the frequentist test in (b) a Bayes procedure wrt this prior.
e) Find the general form of Bayes procedure wrt “linear loss”