Homework 3 – STAT 543
On campus: Due Friday, February 2 by 5:00 pm (TA’s office);
you also may turn in the assignment in class on the same Friday
Distance students: Due Wednesday, February 7 by 12:00 pm (TA’s email)
1. Suppose that the random variables Y1 , . . . , Yn satisfy
Yi = βxi + εi ,
i = 1, . . . , n
where x1 , . . . , xn are fixed constants and ε1 , . . . , εn are iid N (0, σ 2 ); here we assume σ 2 > 0 is known.
(a) Find the MLE of β.
(b) Find the distribution of the MLE.
(c) Find the CRLB for estimating β. (Hint: you’ll have to work with the joint distribution f (y1 , . . . , yn |β)
directly, since Y1 , . . . , Yn are not iid.)
(d) Show the MLE is the UMVUE of β.
2. Problem 7.40, Casella & Berger (You may assume 0 < p < 1 is the parameter space).
Pn
3. Problem 7.41, Casella & Berger Hint: By Jensen’s inequality, it holds that (
2
i=1 ai /n)
≤
Pn
2
i=1 ai /n.
4. Prove that: for an estimator T of a parameter function it holds that
M SEθ (T ) = Varθ (T ) + [bθ (T )]2 ,
where M SEθ (T ) = E{[T − γ(θ)]2 } is the mean-squared error of T and bθ (T ) = T − γ(θ) is the bias of T .
5. Problem 7.49, Casella & Berger
6. Let X1 , . . . , Xn be iid Bernoulli(θ), θ ∈ (0, 1). Find the Bayes estimator of θ with respect to the uniform(0,1)
prior under the loss function
(t − θ)2
L(t, θ) =
θ(1 − θ)
1