CHAPTER 7
Section 72
71.
E
()
µ
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
=
∑
∑
=
=
n
n
X
E
n
n
X
E
X
n
i
i
n
i
i
2
2
1
2
1
2
2
1
2
1
1
E
=
=
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
=
⎟
⎟
⎟
⎟
⎠
⎞
⎜
⎜
⎜
⎜
⎝
⎛
=
∑
∑
=
=
n
n
X
E
n
n
X
E
X
n
i
i
n
i
i
1
1
1
1
2
,
1
X
and
2
X
are unbiased estimators of
µ
.
The variances are V
2n
2
1
σ
=
X
and V
n
2
2
=
X
; compare the MSE (variance in this case),
2
1
2
/
2
/
)
ˆ
(
)
ˆ
(
2
2
2
1
=
=
=
n
n
n
n
MSE
MSE
Θ
Since both estimators are unbiased, examination of the variances would conclude that
X
1
is the
“better” estimator with the smaller variance.
72.
E
[]
=
=
=
+
+
+
=
)
7
(
7
1
))
(
7
(
7
1
)
(
)
(
)
(
7
1
ˆ
7
2
1
1
X
E
X
E
X
E
X
E
"
E
=
+
−
=
+
+
=
]
2
[
2
1
)
(
)
(
)
2
(
2
1
ˆ
7
6
1
2
X
E
X
E
X
E
a) Both
and
are unbiased estimates of
µ
since the expected values of these statistics are
1
ˆ
2
ˆ
equivalent to the true mean,
µ
.
b) V
2
2
7
2
1
2
7
2
1
1
7
1
)
7
(
49
1
)
(
)
(
)
(
7
1
7
...
ˆ
=
=
+
+
+
=
⎥
⎦
⎤
⎢
⎣
⎡
+
+
+
=
X
V
X
V
X
V
X
X
X
V
"
7
)
ˆ
(
2
1
=
V
V
))
(
)
(
)
(
4
(
4
1
)
(
)
(
)
2
(
2
1
2
2
ˆ
4
6
1
4
6
1
2
4
6
1
2
X
V
X
V
X
V
X
V
X
V
X
V
X
X
X
V
+
+
=
+
+
=
⎥
⎦
⎤
⎢
⎣
⎡
+
−
=
=
(
)
1
4
4
222
σσσ
++
=
1
4
6
2
σ
2
3
)
ˆ
(
2
2
=
V
Since both estimators are unbiased, the variances can be compared to decide to select the better
estimator. The variance of
is smaller than that of
,
is the better estimator.
1
ˆ
2
ˆ
1
ˆ
73.
Since both
and
are unbiased, the variances of the estimators can be examined to determine
the “better” estimator.
The variance of
is smaller than that of
thus
may be the better
estimator.
1
ˆ
2
ˆ
±
θ
2
1
ˆ
θ
±
θ
2
Relative Efficiency =
5
.
2
4
10
)
ˆ
(
)
(
)
ˆ
(
)
ˆ
(
2
1
2
1
=
=
=
V
V
MSE
MSE
L
71
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Since both estimators are unbiased:
Relative Efficiency =
21
2
2
/
3
7
/
)
ˆ
(
)
ˆ
(
)
ˆ
(
)
ˆ
(
2
2
2
1
2
1
=
=
Θ
Θ
=
Θ
Θ
σ
V
V
MSE
MSE
75.
5
.
2
4
10
)
ˆ
(
)
ˆ
(
)
ˆ
(
)
ˆ
(
2
1
2
1
=
=
=
Θ
V
V
MSE
MSE
76.
θ
=
)
ˆ
(
1
E
2
/
)
ˆ
(
2
=
E
−
=
)
ˆ
(
2
E
Bias
=
θ
θ
2
−
=
−
θ
2
V
= 10
V
= 4
)
ˆ
(
1
)
ˆ
(
2
For unbiasedness, use
since it is the only unbiased estimator.
1
ˆ
As for minimum variance and efficiency we have:
Relative Efficiency =
2
2
2
1
2
1
)
)
ˆ
(
(
)
)
ˆ
(
(
Bias
V
Bias
V
+
+
where bias for
θ
1
is 0.
Thus,
Relative Efficiency =
(
)
()
10
0
4
2
40
16
2
2
+
+
−
⎛
⎝
⎜
⎞
⎠
⎟
⎛
⎝
⎜
⎜
⎞
⎠
⎟
⎟
=
+
θ
θ
If the relative efficiency is less than or equal to 1,
is the better estimator.
1
ˆ
Use
, when
1
ˆ
40
16
1
2
+
≤
θ
40
16
2
≤+
θ
24
2
≤θ
θ ≤ −
4 899
.
or
θ ≥
4899
.
If
−<
then use
.
<
.
θ
.
2
ˆ
For unbiasedness, use
.
For efficiency, use
when
1
ˆ
1
ˆ
θ ≤ −
4 899
.
or
θ ≥
.
and use
when
.
2
ˆ
<
..
θ
77.
No bias
=
)
ˆ
(
1
E
)
ˆ
(
12
)
ˆ
(
1
1
MSE
V
=
=
No bias
=
)
ˆ
(
2
E
)
ˆ
(
10
)
ˆ
(
2
2
MSE
V
=
=
Bias
[note that this includes (bias
≠
)
ˆ
(
3
E
6
)
ˆ
(
3
=
MSE
2
)]
To compare the three estimators, calculate the relative efficiencies:
2
.
1
10
12
)
ˆ
(
)
ˆ
(
2
1
=
=
MSE
MSE
,
since rel. eff. > 1 use
as the estimator for
θ
2
ˆ
2
6
12
)
ˆ
(
)
ˆ
(
3
1
=
=
MSE
MSE
,
since rel. eff. > 1 use
as the estimator for
θ
3
ˆ
8
.
1
6
10
)
ˆ
(
)
ˆ
(
3
2
=
=
MSE
MSE
,
since rel. eff. > 1 use
as the estimator for
θ
3
ˆ
Conclusion:
is the most efficient estimator with bias, but it is biased.
is the best “unbiased” estimator.
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 Spring '08
 Harris
 Normal Distribution, Standard Deviation, Variance, θ θ

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