Homework 4 – STAT 543
On campus: Due Friday, February 9 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
Distance students: Due Friday, February 15 by 5:00 pm (TA’s email)
1. Problem 7.23, Casella and Berger (2nd Edition). Note assume sample size n ≥ 3.
2. Problem 7.24, Casella and Berger (2nd Edition)
3. Read Problem 7.62, Casella and Berger (2nd Edition). Then show the following:
(a) the posterior pdf of θ is normal with mean (1 − η)x̄n + ηµ and variance ητ 2 , for η = σ 2 /(nτ 2 +
σ2)
(b) find the Bayes estimator of θ under squared error loss
(c) Then complete parts (a),(b),(c) of problem 7.62 in the text.
Note: part (b) from problem 7.62 follows from part(a) and the form of the Bayes estimator
4. Problem 10.1, Casella and Berger (2nd Edition)
Some extra problems: for practice, not to turn it
5. Read problem 7.63, Casella and Berger (2nd Edition). You should be able to solve this using the
risk result from problem 7.62.
6. It turns out that, for the problem described in problem 7.62, (i.e., X1 , . . . , Xn iid N (θ, σ 2 ), σ 2
known), you can show that the sample mean X̄n is the minimax estimator of θ under the squared
2
error loss. To do so, we’ll use a result based on Problem 7.62: if T0µ,τ denotes the Bayes estimator
of θ based a N (µ, τ 2 ) prior for θ, then it holds that
2
“Bayes risk of T0µ,τ under N (µ, τ 2 ) prior” =
τ 2σ2
≤ min max RT (θ)
T
θ
nτ 2 + σ 2
(1)
where RT (θ) = Eθ (T − θ)2 is the risk of an estimator T = h(X1 , X2 , . . . , Xn ).1 To show that X̄ is
minimax, using the result in (1), try the following steps:
(a) Since (1) holds for any τ 2 > 0, show that
σ2
≤ min max RT (θ)
T
θ
n
must hold (ie, try taking limits).
(b) Show that the risk of the sample mean is constant: RX̄n (θ) =
σ2
n
for any θ.
(c) By (a)-(b), argue that X̄n must be the minimax estimator.
1
In defining minimax risk minT maxθ RT (θ), it is sometimes necessary to replace “min” and “max” with “inf” and “sup”
for technical reasons; this doesn’t change the nature of the problem though. If you are interested seeing how to show (1), please
talk with me; however, this is technical and not something you would be responsible for.
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