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Further Details on Exercise 7.9
ˆ
θ
=
X
(
n
)
so that
E
ˆ
θ
=
Z
xf
X
(
n
)
(
x
)
dx
=
Z
θ
0
x
nx
n

1
θ
n
dx
=
n
θ
n
Z
θ
0
x
n
dx
=
n
θ
n
θ
n
+1
n
+ 1
=
n
n
+ 1
θ
E
ˆ
θ
2
=
Z
x
2
f
X
(
n
)
(
x
)
dx
=
Z
θ
0
x
2
nx
n

1
θ
n
dx
=
n
θ
n
Z
θ
0
x
n
+1
dx
=
n
θ
n
θ
n
+2
n
+ 2
=
n
n
+ 2
θ
2
Var(
ˆ
θ
) =
n
n
+ 2
θ
2

±
n
n
+ 1
θ
²
2
=
nθ
2
(
n
+ 2) (
n
+ 1)
2
.
MSE = Bias
2
+ Var =
±

θ
n
+ 1
²
2
+
nθ
2
(
n
+ 2) (
n
+ 1)
2
=
2
θ
2
(
n
+ 1) (
n
+ 2)
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This note was uploaded on 12/15/2011 for the course STAT 5326 taught by Professor Frade during the Fall '10 term at FSU.
 Fall '10
 Frade

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