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Stat 312: Lecture 03
Minimum Variance Unbiased Estimator
Moo K. Chung
[email protected]
September 9, 2004
1.
Population
of interest is a collection of measur
able objects we are studying. Let
X
1
,
···
,X
n
be
a random sample from the population. Then sam
ple mean
¯
X
and sample variance
S
2
are unbiased
estimators of population mean
μ
and population
variance
σ
2
respectively.
Proof.
Note that
S
2
=
1
n

1
X
i
=1
(
X
i

¯
X
)
2
=
1
n

1
h
n
X
i
=1
X
2
i

n
¯
X
2
i
.
Then using the fact
E
(
¯
X
)
2
=
V
¯
X
+ (
E
¯
X
)
2
=
σ
2
/n
+
μ
2
, it can be shown that
E
S
2
=
σ
2
.
2. There may be many unbiased estimators of
θ
.
Given two unbiased estimators
ˆ
θ
1
and
ˆ
θ
2
of
θ
. We
choose one that gives less variance. If
V
(
ˆ
θ
1
)
≤
V
(
ˆ
θ
2
)
,
ˆ
θ
1
is called more
efﬁcient
than
ˆ
θ
2
. An efﬁ
cient estimator has less variability so we are more
likely to make an estimate close to the true param
eter value. The following coin ﬂipping example
clearly demonstrate this.
> a<rbinom(1000,1,0.5)
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This note was uploaded on 01/31/2008 for the course STAT 312 taught by Professor Chung during the Fall '04 term at Wisconsin.
 Fall '04
 Chung
 Statistics, Variance

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