181
Chapter 9: Properties of Point Estimators and Methods of Estimation
9.1
Refer to Ex. 8.8 where the variances of the four estimators were calculated.
Thus,
eff(
1
ˆ
θ
,
5
ˆ
θ
) = 1/3
eff(
2
ˆ
θ
,
5
ˆ
θ
) = 2/3
eff(
3
ˆ
θ
,
5
ˆ
θ
) = 3/5.
9.2
a.
The three estimators a unbiased since:
E
(
1
ˆ
μ
) =
()
μ
=
μ
+
μ
=
+
)
(
)
(
)
(
2
1
2
1
2
1
Y
E
Y
E
μ
=
μ
+
−
μ
−
+
μ
=
μ
4
/
)
2
(
2
)
2
(
4
/
)
ˆ
(
2
n
n
E
μ
=
=
μ
)
(
)
ˆ
(
3
Y
E
E
.
b.
The variances of the three estimators are
2
2
1
2
2
4
1
1
)
(
)
ˆ
(
σ
=
σ
+
σ
=
μ
V
)
2
(
4
8
/
16
/
)
2
(
4
)
2
(
16
/
)
ˆ
(
2
2
2
2
2
2
2
−
σ
+
σ
=
σ
+
−
σ
−
+
σ
=
μ
n
n
n
V
n
V
/
)
ˆ
(
2
3
σ
=
μ
.
Thus,
eff(
3
ˆ
μ
,
2
ˆ
μ
) =
)
2
(
8
2
−
n
n
, eff(
3
ˆ
μ
1
ˆ
μ
) =
n
/2.
9.3
a.
E
(
1
ˆ
θ
) =
E
(
Y
) – 1/2 =
θ
+ 1/2 – 1/2 =
θ
.
From Section 6.7, we can find the density
function of
2
ˆ
θ
=
Y
(
n
)
:
1
)
(
)
(
−
θ
−
=
n
n
y
n
y
g
,
θ
≤
y
≤
θ
+ 1.
From this, it is easily shown
that
E
(
2
ˆ
θ
) =
E
(
Y
(
n
)
) –
n
/(
n
+ 1) =
θ
.
b.
V
(
1
ˆ
θ
) =
V
(
Y
) =
σ
2
/
n
= 1/(12
n
).
With the density in part
a
,
V
(
2
ˆ
θ
) =
V
(
Y
(
n
)
) =
2
)
1
)(
2
(
+
+
n
n
n
.
Thus, eff(
1
ˆ
θ
,
2
ˆ
θ
) =
2
2
)
1
)(
2
(
12
+
+
n
n
n
.
9.4
See Exercises 8.18 and 6.74.
Following those, we have that
V
(
1
ˆ
θ
) = (
n
+ 1)
2
V
(
Y
(
n
)
) =
2
2
θ
+
n
n
.
Similarly,
V
(
2
ˆ
θ
) =
2
1
n
n
+
V
(
Y
(
n
)
) =
2
)
2
(
1
θ
+
n
n
.
Thus, the ratio of these variances is
as given.
9.5
From Ex. 7.20, we know
S
2
is unbiased and
V
(
S
2
) =
V
(
2
1
ˆ
σ
) =
1
2
4
−
σ
n
.
For
2
2
ˆ
σ
, note that
Y
1
–
Y
2
is normal with mean 0 and variance
σ
2
.
So,
2
2
2
1
2
)
(
σ
−
Y
Y
is chi–square with one degree of
freedom and
E
(
2
2
ˆ
σ
) =
σ
2
,
V
(
2
2
ˆ
σ
) = 2
σ
4
.
Thus, we have that eff(
2
1
ˆ
σ
,
2
2
ˆ
σ
) =
n
– 1.
9.6
Both estimators are unbiased and
V
(
1
ˆ
λ
) =
λ
/2 and
V
(
2
ˆ
λ
) =
λ
/
n
.
The efficiency is 2/
n
.
9.7
The estimator
1
ˆ
θ
is unbiased so MSE(
1
ˆ
θ
) =
V
(
1
ˆ
θ
) =
θ
2
.
Also,
2
ˆ
θ
=
Y
is unbiased for
θ
(
θ
is the mean) and
V
(
2
ˆ
θ
) =
σ
2
/
n
=
θ
2
/
n
.
Thus, we have that eff(
1
ˆ
θ
,
2
ˆ
θ
) = 1/
n
.