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STAT 409
Homework #2
Fall 2009
(due Friday, September 11, by 4:00 p.m.)
1.
Let
X
1
, X
2
, … , X
n
be a random sample of size
n
from the distribution with
probability density function
( 29 ( 29 ( 29
θ
2
X
X
ln
1
θ
θ
;
x
x
x
f
x
f
⋅

=
=
,
x
> 1,
θ
> 1.
a)
We already know
(
Homework 1
) that the maximum likelihood estimator of
θ
is
∑
=
+
=
n
i
i
x
n
1
ln
2
1
θ
ˆ
.
Is
θ
ˆ
a consistent estimator for
θ
?
Justify your answer
.
b)
We already know
(
Homework 1
) that if
θ
> 2
then the method of moments
estimator of
θ
is
1
1
2
θ
~


=
x
x
.
Is
θ
~
a consistent estimator for
θ
?
Justify your answer
.
(
Assume
θ
> 3
).
2.
Let
X
1
, X
2
, … , X
n
be a random sample from the distribution with probability
density function
( 29
θ
2
θ
x
e
x
x
f

=
x
> 0
θ
> 0.
a)
We already know
(
Homework 1
) that the maximum likelihood estimator of
θ
is
θ
ˆ
=
∑
=
n
i
i
n
1
X
.
Is
θ
ˆ
a consistent estimator for
θ
?
Justify your answer
.
b)
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This note was uploaded on 10/12/2011 for the course STATISTICS stat 410 taught by Professor Stepanov during the Spring '11 term at University of Illinois, Urbana Champaign.
 Spring '11
 Stepanov
 Probability

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