In Example 6.8-2, take n = 30, α = 15, and β = 5.

(a) Using the squared error loss, compute the expected loss (risk function) associated with the Bayes estimator w(Y).

(b) The risk function associated with the usual estimator Y/n is, of course, θ(1 − θ)/30. Find those values of θ for which the risk function in part (a) is less than θ(1−θ)/30. In particular, if the prior mean α/(α+β) = 3/4 is a reasonable guess, then the risk function in part (a) is the better of the two (i.e., is smaller in a neighborhood of θ = 3/4) for what values of θ?

Reference Example 6.8-2

Let X1,X2, . . . ,Xn be a random sample from a gamma distribution with known α and with θ = 1/τ. Say τ has a prior pdf that is gamma with parameters α0 and θ0, so that the prior mean is α0θ0.

(a) Find the posterior pdf of τ , given that X1 = x1,X2 =x2, . . . ,Xn = xn.

(b) Find the mean of the posterior distribution found inpart (a), and write it as a function of the sample mean X and α0θ0.

(c) Explain how you would find a 95% interval estimate of τ if n = 10, α = 3, α0 = 10, and θ0 = 2.

Business Risk Business risk is a (subset of non-financial risk) and refers to strategic risks (related to a bank’s decision to enter new markets and develop new products) and reputation risk, legal risk, regulatory risk, etc. The focus here is on the potential of losses from things like: o a decrease in the competitive position of the bank o...