Abstract
Suppose that, given ω=(ω1,ω2)∈ℜ2, X1,X2,… and
Y1,Y2,… are independent random variables and their respective
distribution functions Gω1 and Gω2 belong to a one parameter
exponential family of distributions. We derive approximations
to the posterior probabilities of ω lying in closed convex subsets
of the parameter space under a general prior density. Using
this, we then approximate the Bayes posterior risk for testing
the hypotheses H0:ω∈Ω1 versus H1:ω∈Ω2 using a zero-one
loss function, where Ω1 and Ω2 are disjoint closed convex subsets
of the parameter space.