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Stat1801
Probability and Statistics: Foundations of Actuarial Science
Fall 20102011
Chapter VII
Estimation
§ 7.1
Point Estimation
θ
:
unknown parameter of the population, or the corresponding pdf
()
x
f
)
:
a
point estimator
of
. It is a statistic (function of the items of a random
sample) and is a random variable. An observed value of
)
is a
point
estimate
of
.
Example 7.1
Suppose we are interested in a normal population with unknown mean
μ
and
variance
2
σ
, i.e. we only know (or assume) that the population distribution
resembles the normal bellshaped curve, but have no idea in the location and
spread of the distribution.
Then
and
2
are the unknown parameters.
We may draw a random sample from this population and calculate the statistics
X
and
2
S
, and then use their values to estimate
and
2
. Hence we may denote
X
=
)
,
2
2
S
=
)
as the point estimators of
and
2
.
§ 7.2
Desirable Properties of Estimator
The notations
ˆ
and
refer to different quantities, with
ˆ
as the sample statistic
that can be observed from the sample while
is the population parameter that is
unobservable. Usually for any parameter, there exist many different methods to do
the estimation. For example, based on a random sample from
,
0
U
,
a
reasonable estimator of
is the maximum of the sample, i.e.
( )
n
X
X
X
,...,
,
max
ˆ
2
1
=
.
However, since the population mean is
2
=
, it is also reasonable to estimate
by using the sample mean, i.e.
X
2
ˆ
=
.
P.140
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Probability and Statistics: Foundations of Actuarial Science
Fall 20102011
In order to compare the performance of different estimators, some desirable
properties of estimation are discussed below.
§ 7.2.1
Bias of Estimator
Definition
The
bias
of
an estimator
θ
)
of the parameter
is defined as
( ) ( )
θθ
−
=
)
)
E
bias
Note that the expectation is taken with respect to the sampling distribution of
)
.
We say that
)
overestimates
if
( )
0
>
)
bias
;
underestimates
if
( )
0
<
)
bias
.
Unbiased Estimator
The estimator
)
is called an unbiased estimator of the parameter
if
( )
=
)
E
for all the values of
in the parameter space,
i.e. the estimator is unbiased if
( )
0
ˆ
=
bias
for any
.
Example 7.2
Suppose we have a random sample
{ }
n
X
X
X
,...,
,
2
1
from a distribution with mean
μ
and variance
2
σ
.
1.
Sample mean is an unbiased estimator of the population mean
as
( )
=
X
E
for all
.
2.
Sample variance
()
−
X
S
2
is an unbiased estimator of the
population variance
2
∑
=
−
=
n
i
i
X
n
1
2
1
1
as
( )
2
2
=
S
E
for all
2
.
P.141
Stat1801
Probability and Statistics: Foundations of Actuarial Science
Fall 20102011
Example 7.3
Suppose
. Consider the two estimators
(
θ
,
0
~
,...,
,
2
1
U
X
X
X
iid
n
)
X
2
ˆ
1
=
and
( )
n
X
X
X
,...,
,
max
ˆ
2
1
2
=
.
Obviously
is unbiased as
1
ˆ
( )
()
μ
=
×
=
=
=
2
2
2
2
ˆ
1
X
E
E
.
For
, consider the cdf of
. For
2
ˆ
2
ˆ
<
<
y
0,
w
e
h
a
v
e
() ( )
n
n
n
n
y
y
X
P
y
X
P
y
X
P
y
X
y
X
y
X
P
y
X
X
X
P
y
P
y
F
⎟
⎠
⎞
⎜
⎝
⎛
=
≤
≤
≤
=
≤
≤
≤
=
≤
=
≤
=
L
K
2
1
2
1
2
1
2
ˆ
,
,
,
,...,
,
max
ˆ
2
The pdf of
is therefore given by
2
ˆ
() ()
⎪
⎩
⎪
⎨
⎧
<
<
=
=
−
otherwise
0
0
for
1
'
ˆ
ˆ
2
2
y
ny
y
F
y
f
n
n
and hence
1
1
ˆ
0
1
0
2
+
=
⎥
⎦
⎤
⎢
⎣
⎡
+
=
=
+
∫
n
n
n
ny
dy
ny
E
n
n
n
n
.
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This note was uploaded on 02/01/2012 for the course STAT 1801 taught by Professor Mrchung during the Fall '10 term at HKU.
 Fall '10
 MrChung
 Statistics, Probability

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