# PREVIOUS EXAM I – STAT 543 X α, θ regularity conditions are satisfied. ```PREVIOUS EXAM I – STAT 543
1. Let X1 , . . . , Xn be a random sample from a GAMMA(α, θ) distribution,
θ &gt; 0, where α is a known constant. Assume that the Cramér-Rao
regularity conditions are satisfied.
(a) Find the Cramér-Rao lower bound for estimating θ.
2. Let X1 , X2 , . . . , be a sequence of iid random variables with common pdf
&frac12; 2 −3
3x θ
if 0 &lt; x ≤ θ
f (x|θ) =
0
otherwise,
θ &gt; 0. Also let γ(θ) ≡ log(1 + θ2 ).
(a) Find the Method of Moments estimator Tn of γ(θ) based on X1 , . . . , Xn .
(b) Show that {Tn } is consistent for γ(θ). Clearly state any standard
results that you are using.
(c) Show that the maximum likelihood estimator θ̂n of θ based on X1 , . . . , Xn
is given by θ̂n = max{X1 , . . . , Xn }.
2

if y ≤ 0
0
3n
(d) Show that Pθ (θ̂n ≤ y) =
(y/θ) if 0 &lt; y ≤ θ

1
if y ≥ θ
(e) Using part(d) or otherwise, show that {θ̂n } is consistent for θ.
3
3. Let X1 , . . . , Xn be a random sample from N (2, 1/θ), θ &gt; 0. We want to
find the Bayes estimator of θ w.r.t. the prior GAMMA(α + 32 , 1), where
α &gt; 0 is a known constant.
(a) Find the posterior distribution of θ.
(b) Find the Bayes estimator under the loss function L(t, θ) = (t−θ)2 /θ2 .