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SOLUTIONS FOR HWK 2
Problem A
(a) and (b) follow from theorems proved in class since
μ
=
θ,σ
2
=
θ
2
.
(d)
E
(
ˆ
Θ) =
1
15
μ
+
2
15
μ
+
3
15
μ
+
1
15
μ
+
8
15
μ
=
μ
. Hence
ˆ
θ
is unbiased.
(f) ) =
±
1
15
¶
2
θ
2
+
±
2
15
¶
2
θ
2
+
±
3
15
¶
2
θ
2
+
±
1
15
¶
2
θ
2
+
±
8
15
¶
2
θ
2
= 0
.
351
θ
2
.
(g) Since
n
= 5, we have
V ar
(
¯
X
) =
θ
2
/
5 = 0
.
2
θ
2
. Therefore:
re
(
¯
X
:
ˆ
Θ) =
0
.
351
θ
2
0
.
2
θ
2
= 1
.
755
>
1.
h)
¯
X
is much more eﬃcient than
ˆ
Θ.
Problem B
Let
r >
0 and let
X
be a random variable. Assume that
E
(

X

r
)
<
∞
. The
moment of order
r
,
μ
r
def
=
E
(
X
r
) (of course, the moment of order 1
μ
1
=
μ
=
E
(
X
),
is the expectation of
X
). Consider a random sample
X
1
,...,X
n
where
X
k
’s have the
same distribution as
X
. The
empirical moment of order
r M
r
def
=
1
n
∑
n
k
=1
X
r
k
.
1)
Prove that
M
r
is an unbiased estimator of
μ
r
where
r >
0 .
E
(
M
r
) =
E
(
1
n
n
X
k
=1
X
r
k
) =
1
n
E
(
n
X
k
=1
X
r
k
) =
1
n
n
X
k
=1
E
(
X
r
k
) =
1
n
n
X
k
=1
μ
r
=
1
n
nμ
r
=
μ
r
.
2)
Prove that the sequence of the empirical moments
M
(
n
)
r
=
1
n
∑
n
k
=1
X
r
k
is a
strongly consistent sequence of estimators of the moment
μ
r
.
We apply the Strong Law of Large Numbers:
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This note was uploaded on 02/24/2009 for the course STAT 415 taught by Professor Senturk,damla during the Spring '06 term at Pennsylvania State University, University Park.
 Spring '06
 SENTURK,DAMLA

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