CHAPTER 4
Section 42
41.
E
(
29
(
29
X
E
X
n
n
E
X
n
n
i
i
n
i
i
n
1
=
=
=
=
=
=
∑
∑
1
2
1
2
2
1
2
1
2
2
μ
μ
E
(
29
(
29
X
E
X
n
n
E
X
n
n
i
i
n
i
i
n
2
=
=
=
=
=
=
∑
∑
1
1
1
1
μ
μ
,
X
1
and
X
2
are unbiased estimators of
μ
.
The variances are V
(
29
X
1
2
2n
=
σ
and V
(
29
X
2
2
n
=
σ
; compare the MSE (variance in this case),
MSE
MSE
n
n
n
n
(
)
(
)
/
/
θ
θ
σ
σ
1
2
2
2
2
2
1
2
=
=
=
Since both estimators are unbiased, examination of the variances would conclude that
X
1
is the “better”
estimator with the smaller variance.
42.
E
(
29
[
]
μ
μ
θ
=
=
=
+
+
+
=
)
9
(
9
1
))
X
(
E
9
(
9
1
)
X
(
E
)
X
(
E
)
X
(
E
9
1
ˆ
9
2
1
1
E
(
29
[
]
μ
=
μ
+
μ

μ
=
+

=
θ
]
2
3
[
2
1
)
X
2
(
E
)
X
(
E
)
X
3
(
E
2
1
ˆ
4
6
1
2
a)
Both
θ
1
and
θ
2
are unbiased estimates of
μ
since the expected values of these statistics are equivalent
to
the true mean,
μ
.
b) V
(
29
(
29
2
2
9
2
1
2
9
2
1
1
9
1
)
9
(
81
1
)
X
(
V
)
X
(
V
)
X
(
V
9
1
9
X
...
X
X
V
ˆ
σ
=
σ
=
+
+
+
=
+
+
+
=
θ
9
)
ˆ
(
V
2
1
σ
=
θ
V
(
29
(
29
))
X
(
V
4
)
X
(
V
)
X
(
V
9
(
4
1
)
X
2
(
V
)
X
(
V
)
X
3
(
V
2
1
2
X
2
X
X
3
V
ˆ
4
6
1
4
6
1
2
4
6
1
2
+
+
=
+
+
=
+

=
θ
=
(
29
2
2
2
4
9
4
1
σ
+
σ
+
σ
=
)
14
(
4
1
2
σ
2
7
)
ˆ
(
V
2
2
σ
=
θ
Since both estimators are unbiased, the variances can be compared to decide
which is the better estimator.
The variance of
θ
1
is smaller than that of
θ
2
,
θ
1
is the
better estimator.
43.
Since both
θ
1
and
θ
2
are unbiased, the variances of the estimators can be examined to determine which is
the “better” estimator.
The variance of
1
ˆ
θ
is smaller than that of
θ
2
thus
θ
1
may be the better estimator.
Relative Efficiency =
5
.
0
4
2
)
ˆ
(
V
)
ˆ
(
V
)
ˆ
(
MSE
)
ˆ
(
MSE
2
1
2
1
=
=
θ
θ
=
θ
θ
1
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Since both estimators are unbiased:
Relative Efficiency =
63
2
2
/
7
9
/
)
ˆ
(
V
)
ˆ
(
V
)
ˆ
(
MSE
)
ˆ
(
MSE
2
2
2
1
2
1
=
=
=
σ
σ
θ
θ
θ
θ
45.
5
.
0
4
2
)
ˆ
(
V
)
ˆ
(
V
)
ˆ
(
MSE
)
ˆ
(
MSE
2
1
2
1
=
=
θ
θ
=
θ
θ
46.
E(
)
θ
θ
1
=
E(
)
/
θ
θ
2
2
=
Bias
E
=

(
)
θ
θ
2
=
θ
θ
2

=

θ
2
V (
)
θ
1
= 10
V (
)
θ
2
= 4
For unbiasedness, use
θ
1
since it is the only unbiased estimator. As for minimum variance and efficiency
we
have:
Relative Efficiency =
(
(
)
)
(
(
)
)
V
Bias
V
Bias
θ
θ
1
2
1
2
2
2
+
+
where, Bias for
θ
1
is 0.
Thus,
Relative Efficiency =
(
29
(
)
10
0
4
2
40
16
2
2
+
+

=
+
θ
θ
If the relative efficiency is less than or equal to 1,
θ
1
is the better estimator.
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 Spring '09
 Kapur
 Normal Distribution, Null hypothesis, Statistical hypothesis testing, θ, χ

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