Economics 506
FALL 2009
FINAL EXAM
Total Points: 200 (SOLUTION)
Name: __Ali Toossi_________________________
e
mail:_______________________
1) [Total points: 80]
Consider the following density function
=


elsewhere
0
0
,
0
1
)
(
1
y
e
ry
y
f
r
y
r
θ
where
r
is a
known
positive constant.
a) [10 points]
use the method of transformation to find the density function of U = Y
r
u
r
u
r
r
r
r
Y
U
r
r
r
e
u
r
e
u
r
u
r
u
f
u
f
u
r
dr
dY
u
Y
Y
U
r
r





=
=
=
=
=
⇒
=
1
1
)
(
1
1
)
(
)
(
1
1
1
)
(
1
1
1
1
1
1
1
1
1
Thus distribution of U is exponential for U >0
b)[10 points]
Let
Y
1
, Y
2
, …, Y
n
denote a random sample from the density function given
above. Find a sufficient statistics for θ.
The likelihood function is
(
29
(
29
∑
∏
=

=

θ

θ
=
θ
n
i
r
i
r
n
i
i
n
n
y
y
r
L
1
1
1
/
exp
)
(
.
a sufficient statistic for θ is
∑
=
n
i
r
i
Y
1
.
c) [10 points]
Use method of moment generating function to find the distribution of the
sufficient statistics you found in part (b).
We know from part (a) that
r
Y
has exponential, therefore each
r
i
Y
has exponential
distribution. Now:
n
Y
Y
Y
Y
t
t
t
t
t
M
t
M
t
M
t
M
r
n
r
r
r
i





=



=
=
∑
)
1
(
)
1
(
)
1
(
)
1
(
)
(
)
(
)
(
)
(
1
1
1
2
1
.
Therefore the distribution of
∑
=
n
i
r
i
Y
1
is Gamma with parameters n &
θ.
(Use the back side if you need to)
1
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View Full DocumentEconomics 506
FALL 2009
FINAL EXAM
d) [10 points]
Let
Y
1
, Y
2
, …, Y
n
denote a random sample from the density function given
above. Find the maximum likelihood estimator (MLE) of θ
The log–likelihood is
(
29
∑
∏
=
=
θ


+
+
θ

=
θ
n
i
r
i
n
i
i
y
y
r
r
n
n
L
1
1
/
ln
)
1
(
ln
ln
)
(
ln
.
By taking a derivative w.r.t. θ and equating to 0, we find
∑
=
=
θ
n
i
r
i
n
Y
1
1
ˆ
.
e)
[5 points]
Is the MLE you found in part (d) a MVUE (minimum variance unbiased
estimator) for θ?
Note that
θ
ˆ
is a function of the sufficient statistic.
We showed that Y
r
has
exponential distribution thus
θ
=
)
(
r
Y
E
, therefore
θ
=
=
=
=
∑
∑
∑
=
=
=
n
i
n
i
r
i
n
i
r
i
n
n
Y
E
n
Y
E
E
1
1
1
1
1
)
(
1
)
(
)
ˆ
(
This means that
θ
ˆ
is also unbiased. Hence it is MVUE for θ.
f)
[10 points]
Find the minimum variance of an unbiased estimator of
,
using
Rao
Cramer
inequality? Confirm that it is the same as the variance of the MLE you found in
part (d).
n
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