Exercise 12.9.8. Consider a two group allocation problem in which the prior probabilities are (1) =(2) = 0.5 and the sampling distributions are exponential, namely

Exercise 12.9.8. Consider a two group allocation problem in which the prior probabilities are π(1) =π(2) = 0.5 and the sampling distributions are exponential, namely f (y|i) =θie−θiy, y ≥ 0.

Find the optimal allocation rule. Assume a cost structure where c(i| j) is zero for i = j and one otherwise. The total probability of misclassification for an allocation rule is precisely the Bayes risk of the allocation rule under this cost structure.

Let δ (y) be an allocation rule. The frequentist risk for the true population j is R( j,δ)=

c(δ (y)| j) f (y| j)dy and the Bayes risk is r(p,δ)=Σt j=1 R( j,δ )π( j). See Berger (1985, Section 1.3) for more on risk functions. Find the total probability of misclassification for the optimal rule.