STAT 410
Fall 2011
Homework #6
(due Friday, October 14, by 3:00 p.m.)
1  4.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
(
)
(
) (
)
θ
1
1
θ
θ
;
X
x
x
f

+
=
⋅
,
0 <
x
< 1,
θ
> –
1.
1.
a)
Obtain the method of moments estimator of
θ
,
θ
~
.
b)
Obtain the maximum likelihood estimator of
θ
,
θ
ˆ
.
2.
a)
What is the probability distribution of
W = –
ln
(
1 – X
)
?
b)
What is the probability distribution of
Y =
(
)
∑
=


n
i
i
1
X
1
ln
=
∑
=
n
i
i
1
W
?
c)
Is the maximum likelihood estimator of
θ
,
θ
ˆ
,
an unbiased estimator of
θ
?
If
θ
ˆ
is not an unbiased estimator of
θ
,
construct an unbiased estimator
θ
ˆ
ˆ
of
θ
based
on
θ
ˆ
.
3.
For an estimator
θ
ˆ
of
θ
, define the
Mean Squared Error
of
θ
ˆ
by
MSE
(
θ
ˆ
)
=
E
[
(
θ
ˆ
–
θ
)
2
]
.
We showed in class that
MSE
(
θ
ˆ
)
=
E
[
(
θ
ˆ
–
θ
)
2
]
=
(
E
(
θ
ˆ
)
–
θ
)
2
+ Var
(
θ
ˆ
)
=
(
bias
(
θ
ˆ
)
)
2
+ Var
(
θ
ˆ
)
.
Find
MSE
(
θ
ˆ
)
for the maximum likelihood estimator of
θ
.
4.
Consider
θ
ˆ
=
(
)
(
)
∑
∑
=
=




n
n
i
i
i
i
n
n
1
2
1
2
X
1
X
1
3
.
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 Fall '08
 Monrad
 Statistics, Normal Distribution, Probability, probability density function, maximum likelihood estimator, 2 K

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