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(1)

BY HENGJIAN CUI

Write (copy) the required excises, then give solutions by yourself.

Copy excises each other is NOT permitted!

EX13-1.

1. Let X

1

, X

2

, · · · , X

n

(n > 1) iid. ∼ N(µ

0

, σ

2

) and θ = 1/σ

2

, where µ

0

is known.

(1). Find the MLE ˆ θ

M

of θ.

(2). Find the Fisher information I(θ).

(3). Suppose θ ∼ Γ(α, β), find out the Bayesian estimation ˆθ

B

of θ.

(4). Is ˆ θ

B

consistent as n → ∞? Why?

(5). Are ˆ θ

M

and ˆ θ

B

asymptotic efficiency as n → ∞? Why?

1

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