Biomedical Engineering Reference
In-Depth Information
where E is a symmetric positive definite (s.p.d.) matrix. We assume J to be the
measure of the estimate error or “risk.” Here,
w
depends on
z
and
w
, and it is
regarded as a stochastic process. With our definition of risk we may introduce the
Z
functional
J
.
w
/
E
.J.
e
// D
J.
e
/p
W
d
w
and in order to minimize the risk we
n
R
refer to the estimator
Z
Z
w
D arg min
J
.
w
/
J.
e
/p
W;Z
.
w
;
z
/d
z
d
w
:
n
n
R
R
By exploiting the properties of p.d.f. recalled above, we rewrite the risk to
minimize as
0
1
Z
Z
Z
@
A
p
Z
d
z
D
e
T
E
e
p
W
j
Z
d
w
J
.
w
/ D
n
J
.
w
j
z
/p
Z
d
z
;
n
n
R
R
R
for
Z
.
w
w
/
T
E.
w
w
/p
W
j
Z
d
w
:
J
.
w
j
z
/
n
R
J
.
w
/ does not involve
w
and
Since the outer integral in the definition of the cost
.
w
j
z
/.
Recall that for a generic vector
x
and a symmetric matrix A of proper size [
60
],
we have
@
x
T
A
x
@
x
J
p
Z
0, we minimize the risk by minimizing
D 2A
x
. Then the minimization of
J
.
w
j
z
/ leads to
D2E
Z
@
J
.
w
j
z
/
@
w
0 D
.
w
w
/p
W
j
Z
d
w
:
n
R
Independently of E, we have the equation
w
Z
R
Z
p
W
j
Z
d
w
D
w
p
W
j
Z
d
w
D
E
.
w
j
z
/:
n
R
n
Since
Z
R
p
W
j
Z
.
w
j
z
/d
w
D 1,wehave
n
w
D
E
.
w
j
z
/:
(6.7)
This is called
minimum variance estimator
, hereafter denoted by
w
MV
. An impor-
tant property of this estimator is that it is
unbiased
,i.e.,
E
.
e
/ D
E
.
w
MV
w
/ D 0.
Search WWH ::
Custom Search