Digital Signal Processing Reference
In-Depth Information
REMARK 3.1. We also often consider the root-mean-square error on the estimation
of a scalar parameter θ
whose definition is given below:
() ( )
() ()
⎧
2
⎫
ˆ
ˆ
θ
=
θ θ
−
eqm
E
⎨
⎬
⎩
⎭
2
ˆ
ˆ
=
θ
+
θ
b
var
3.1.2.
Cramér-Rao bounds
Ideally, we look for unbiased estimators with minimum variance. This naturally
leads to the following question: is there an estimator whose variance is uniformly
(i.e. whatever be the value of
θ
)
less than the variance of all other estimators? Before
answering this question, we can ask ourselves if a lower bound of the covariance
matrix exists; we are thus led to the concept of Cramér-Rao bounds whose definition
is given below.
THEOREM 3.1 (CRAMÉR-RAO BOUNDS).
If the PDF p
(
x
N
; θ)
verifies the
regularity condition:
(
)
⎧
∂
ln
p
x
N
;
θ
⎫
⎪
⎪
=∀
0
θ
E
⎨
⎬
∂
θ
⎪
⎪
⎩
⎭
ˆ
θ
may be, its covariance
then, whatever the value of the unbiased estimator
matrix verifies
1
:
( )
ˆ
−
1
()
C
θ
−
F
θ
≥
0
N
N
where
()
F
N
θ
is the
Fisher information matrix
given by
:
2
ln
⎧
(
)
⎫
∂
px
;
θ
⎪
⎪
N
()
F
θ
=−
⎨
E
⎬
N
T
∂∂
θθ
⎪
⎪
⎩
⎭
[3.4]
(
)
(
)
⎧
∂
ln
p
x
;
θ
∂
ln
p
x
;
θ
⎫
⎪
⎪
N
N
=
⎨
E
⎬
T
∂
θ
⎪
∂
θ
⎪
⎩
⎭
1
In what follows, the notation
A
≥
B
for two hermitian matrices
A
and
Β
signifies that ∀
z
the
quadratic form
z
H
(
A - B) z
≥ 0.
Search WWH ::
Custom Search