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M4056 Lecture Notes.
Monday, October 4, 2010
We now look at methods for evaluating the quality of estimators. Ultimately, we will show
that MLEs have many desirable properties (even though they may be biased).
The following refers to 7.3.1, page 330–331.
To be able to compare MLEs with other estimators, we need some general measures of
quality that we may apply to many estimators. Here is possibly the most useful one:
Defnition 7.3.1.
Suppose we have a sample
X
1
, . . ., X
n
from a
f
X
(
x

θ
) (where we
imagine
θ
to be unknown). Let
W
=
W
(
X
1
, . . ., X
n
) be a statistic (which we want to use
as an estimator for
θ
). Then, the mean squared error (MSE) of
W
is E
θ
(
W

θ
)
2
.
The subscript on the E reminds us that the expectation is dependent on the value of the
parameter. The MSE can be viewed as a (numbervalued) function of the parameter.
(When calculating the MSE, always use the same value of
θ
in both places.) Note that
E
θ
(
W

θ
)
2
= Var
θ
W
+ (E
θ
W

θ
)
2
.
(
*
)
The quantity E
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 Fall '08
 Staff
 Statistics

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