This preview shows page 1. Sign up to view the full content.
HW 4 Solutions
8.8
a.
Note that
1
ˆ
θ
,
2
ˆ
θ
,
3
ˆ
θ
and
5
ˆ
θ
are simple linear combinations of
Y
1
,
Y
2
, and
Y
3
.
So,
it is easily shown that all four of these estimators are
unbiased
.
More explicitly, consider
.
Then
From Ex. 6.81 it was shown that
4
ˆ
θ
has an exponential distribution with mean θ/3, so this
estimator is biased.
b.
It is easily shown that
V
(
1
ˆ
θ
) = θ
2
,
V
(
2
ˆ
θ
) = θ
2
/2,
V
(
3
ˆ
θ
) = 5θ
2
/9, and
V
(
5
ˆ
θ
) = θ
2
/9, so the
estimator
5
ˆ
θ
is unbiased and has the smallest variance.
8.15
Using standard techniques, it can be shown that
E
(
Y
) =(3/2)β,
E
(
Y
2
) = 3β
2
.
Also
it is easiliy shown that
Y
(1)
follows the Pareto family with density function
)
1
3
(
3
)
1
(
3
)
(
+

β
=
n
n
y
n
y
g
,
y
≥ β.
Thus,
E
(
Y
(1)
) =
(
29
β

1
3
3
n
n
and
2
2
3
3
2
)
1
(
)
(
β
=

n
n
Y
E
.
a.
With
)
1
(
ˆ
Y
=
β
,
(
29
(
29
β
=
β

β
=
This is the end of the preview. Sign up
to
access the rest of the document.
This note was uploaded on 10/02/2009 for the course PSTAT 5E taught by Professor Eduardomontoya during the Spring '08 term at UCSB.
 Spring '08
 EduardoMontoya

Click to edit the document details