Databases Reference
In-Depth Information
6.2.3
Extension of the quadratic frequentist bound to improper errors and scores
The essential step in the extension of quadratic frequentist bounds to improper error and
measurement scores is to define the composite vector of
augmented
errors and scores,
[
e
T
s
T
]
T
. The covariance matrix for this composite vector is
,
e
e
H
(
y
)
Q
(
)
T
H
(
(
y
)
)
E
s
H
=
.
(6.36)
(
y
)
s
(
y
)
T
(
)
J
(
)
In this equation the constituent matrices are defined as
Q
(
Q
(
)
)
E
[
e
(
y
)
e
H
=
=
,
Q
(
)
(
y
)]
Q
∗
(
)
Q
∗
(
)
T
(
T
(
)
)
(
y
)
e
H
T
(
)
=
E
[
s
(
y
)]
=
,
(6.37)
T
∗
(
)
T
∗
(
)
J
(
J
(
)
)
(
y
)
s
H
J
(
)
=
E
[
s
(
y
)]
=
.
J
∗
(
J
∗
(
)
)
The covariance matrix
T
(
E
[
s
(
y
)
e
T
)
=
(
y
)] is the complementary sensitivity or
expansion-coefficient matrix, and
J
(
E
[
s
(
y
)
s
T
(
y
)] is the complementary covari-
ance matrix for the centered measurement score, which is called the complementary
information matrix. Furthermore,
Q
(
)
=
) is the augmented error covariance matrix,
T
(
)
is the augmented sensitivity or expansion-coefficient matrix, and
J
(
) is the augmented
information matrix. Obviously, we are interested in bounding
Q
(
) rather than
Q
(
),
but it is convenient to derive the bounds on
Q
(
) first, from which one can immediately
obtain the bound on
Q
(
).
The covariance matrix for the composite vector of augmented error score and aug-
mented measurement score is positive semidefinite, so, assuming the information matrix
to be nonsingular, we obtain the bound
Q
(
T
H
(
)
J
−
1
(
)
≥
)
T
(
). From this, we can
read out the northwest block of
Q
(
) to obtain the following result.
Result 6.3.
The frequentist error covariance is bounded as
)
T
(
T
H
(
)
J
−
1
(
)
T
T
(
Q
(
)
≥
.
(6.38)
)
T
∗
(
)
The quadratic form on the right-hand side is the general widely linear-quadratic fre-
quentist bound on the error covariance matrix for improper error and measurement
scores. Of course the underlying idea is that
Q
(
T
H
(
)
J
−
1
(
) is the augmented
error covariance matrix of the
widely linear
estimator of the error score from the mea-
surement score:
)
−
)
T
(
T
H
(
)
J
−
1
(
T
T
(
e
(
y
)
=
)
)
s
(
y
)
.
(6.39)
The bound in Result
6.3
is tighter than the bound in Result
6.2
, but for proper mea-
surement scores (i.e.,
J
(
)
=
0
) and cross-proper measurement and error scores (i.e.,
T
(
)
=
0
) the bounds are identical.
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