Stat1801
Probability and Statistics: Foundations of Actuarial Science
Fall 2010-2011
Chapter VII
Estimation
§ 7.1
Point Estimation
θ
:
unknown parameter of the population, or the corresponding pdf
( )
x
f
θ
)
:
a
point estimator
of
θ
. It is a statistic (function of the items of a random
sample) and is a random variable. An observed value of
θ
)
is a
point
estimate
of
θ
.
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 bell-shaped curve, but have no idea in the location and
spread of the distribution.
Then
μ
and
2
σ
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
μ
and
2
σ
. Hence we may denote
X
=
μ
)
,
2
2
S
=
σ
)
as the point estimators of
μ
and
2
σ
.
§ 7.2
Desirable Properties of Estimator
The notations
θ
ˆ
and
θ
refer to different quantities, with
θ
ˆ
as the sample statistic
that can be observed from the sample while
θ
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
(
)
θ
,
0
U
, a
reasonable estimator of
θ
is the maximum of the sample, i.e.
(
)
n
X
X
X
,...,
,
max
ˆ
2
1
=
θ
.
However, since the population mean is
2
θ
μ
=
, it is also reasonable to estimate
θ
by using the sample mean, i.e.
X
2
ˆ
=
θ
.
P.140

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