We examine a generalization of the hierarchical (Bayes) model considered in Example 7 2 16 and Exercise 7 22 Suppose that we observe X1, , Xn, where Xi i n (i 2), i l, ,n, independent, i n(, 2), i l, ,n, independent a Show that the marginal distribution of X, is n(, 2 2) and that, marginally, X1, ,...

a We will use the results and notation from part b to do this special ...

We examine a generalization of the hierarchical (Bayes) model considered in Example 7.2.16 and Exercise 7.22. Suppose

Question:

We examine a generalization of the hierarchical (Bayes) model considered in Example 7.2.16 and Exercise 7.22. Suppose that we observe X1,... , Xn, where
Xi|θi ~ n (θi|σ2), i = l,...,n, independent,
θi ~ n(μ, τ2), i = l,...,n, independent.
a. Show that the marginal of X, is n(μ, σ2 + τ2) and that, marginally, X1,..., Xn are iid. (Empirical Bayes analysis would use the marginal of the Xis to estimate the prior parameters μ and σ2. See Miscellanea 7.5.6.)
b. Show, in general, that if
Xi|θi ~ f (x|θi),i = l,...,n, independent,
θi ~ π(θ, τ), i = l,...,n, independent.
then marginally, X1,..., Xn are iid. Distribution
The word "distribution" has several meanings in the financial world, most of them pertaining to the payment of assets from a fund, account, or individual security to an investor or beneficiary. Retirement account distributions are among the most...