Section 7.6
We had a brief introduction/review to random sample (iid
rvs), estimator, unbiased estimator, and consistent
estimator.
We now discuss
a method of finding estimators which
possess good properties.
1.
Maximum Likelihood Estimation
This method yields estimators that have many desirable properties;
both finite as well as large sample properties.
The basic idea to find an estimator
)
(
ˆ
x
θ
which is the most likely
given the data
)
,...,
(
1
n
X
X
X
=
.
Example 1.
Let
)
1
(
π
,
X ~ B
, Bernoulli population with density
1
,
0
,
)
1
(
)

(
)

(
1
x
=

=
=
=

x
x
p
x
X
P
x
π
π
π
π
where
)
1
(
=
=
X
P
π
.
We want to estimate population proportion
π
based on a random
sample of size
n
.
That is,
n
X
X
,...,
1
are independent and
identically distributed
random variables.
1
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For
}
1
,
0
{
∈
i
x
, we have
]
,...,
[
1
1
n
n
x
X
x
X
P
=
=
]
[
.
.
.
]
[
1
1
n
n
x
X
P
x
X
P
=
=
=
n
n
x
x
x
x




=
1
1
)
1
(
...
)
1
(
1
1
π
π
π
π
=
,
)
1
(
1
1
∑

∑

n
i
n
i
x
n
x
π
π
is called the joint density of
n
X
X
,...,
1
.
Write the above density as
,
)
1
(
)
1
(
)

(
1
1
s
n
s
x
n
x
n
i
n
i
x
L



=
∑

∑
=
π
π
π
π
π
where
∑
=
n
i
x
s
1
.
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 Summer '08
 Palaniappan
 Normal Distribution, Maximum likelihood, Estimation theory, µ, mle, 2 2 L

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