5.9 In Theorem 5.7, verify E c(|X| 2) |X| 2 X ( X) = E c(|X|...

5.9 In Theorem 5.7, verify Eθ

c(|X|

2)

|X|

2 X

(θ − X) = Eθ

c(|X|

2)

|X|

2 tr() − 2 c(|X|

2)

|X|

4 X

X + 2 c

(|X|

2)

|X|

2 X

X

"

.

[Hint: There are several ways to do this:

(a) Write Eθ

c(|X|

2)

|X|

2 X

(θ − X) = Eθ

c(Y

Y Y

Y Y

(η − Y)

=

i Eθ



c(Y

Y Y

Y j

Yjσj i(ηi − Yi)



where  = {σij } and Y = −1/2X ∼ N(−1/2θ , I ) = N(η, I ). Now apply Stein’s lemma.

(b) Write  = PDP

, where P is an orthogonal matrix (P

P = I ) and D=diagonal matrix of eigenvalues of , D = diagonal{di}. Then, establish that Eθ

c(|X|

2)

|X|

2 X

(θ − X) =

j Eθ

c(

i diZ2 i )

i diZ2 i

djZj (η∗

j − Zj )

where Z = P −1/2X and η ∗ = P −1/2θ . Now apply Stein’s lemma.