Databases Reference
In-Depth Information
A widely linear estimator is efficient if equality holds in (
6.38
). This is the case
if and
only if
the estimator of the centered error score equals the actual error, meaning that the
estimator has the representation
T
H
(
)
J
−
1
(
ˆ
T
T
(
(
y
)
− =
)
s
(
y
)
.
(6.40)
)
6.3
Fisher score and the Cramer-Rao bound
Our results for quadratic frequentist bounds so far are general. To make them applicable
we need to consider a concrete score for which
T
(
) can be computed. For this
we choose the Fisher score and compute its associated Cramer-Rao bound.
) and
J
(
Definition 6.3.
The
Fisher score
is defined as
log
p
(
y
)
T
∂
∂
log
p
(
y
)
H
∂
∂θ
1
,...,
∂
∂θ
p
(
y
)
=
=
log
p
(
y
)
,
(6.41)
where the partial derivatives are
Wirtinger derivatives
as discussed in Appendix
2
.
The
i
th component of the Fisher score is
∂
j
∂
∂
σ
i
(
y
)
=
+
.
θ
i
)
log
p
(
y
)
θ
i
)
log
p
(
y
)
(6.42)
∂
(Re
(Im
Thus, the Fisher score is a
p
-dimensional complex-valued column vector. The notation
(
∂/∂
)
p
(
y
) means (
∂/∂
)
p
(
y
), evaluated at
=
. We shall demonstrate shortly
that the expected value of the Fisher score is
E
[
(
y
)]
=
0
, so that Definition
6.3
also
defines the centered measurement score
s
(
y
).
The Fisher score has a number of properties that make it a compelling statistic for
inferring the value of a parameter and for bounding the error covariance of any estimator
for that parameter. We list these properties and annotate them here.
1. We may write the partial derivative as
∂
∂θ
i
1
p
(
y
)
∂
∂θ
i
log
p
(
y
)
=
p
(
y
)
,
(6.43)
which is a normalized measure of the sensitivity of the pdf
p
(
y
) to variations in the
parameter
θ
i
. Large sensitivity is valued, and this will be measured by the variance
of the score.
2. The Fisher score is a zero-mean random variable. This statistical property is consistent
with maximum-likelihood procedures that search for maxima of the log-likelihood
function log
p
(
y
) by searching for the zeros of (
)log
p
(
y
). Functions with steep
slopes are valued, so we value a zero-mean score with large variance.
3. The cross-correlation between the centered Fisher score and the centered error score
is the expansion-coefficient matrix
∂/∂
∂
∂
)
H
T
(
)
=
I
+
b
(
,
(6.44)
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