Digital Signal Processing Reference
In-Depth Information
ˆ
θ
is said to be asymptotically unbiased if:
DEFINITION 3.1.
The estimator
{
}
ˆ
lim
θθ
−=
0
E
N
N
→∞
If we can tolerate a bias for a finite number of samples, it is understood that the
asymptotic bias must be zero, failing which, even with an infinite number of points
of a signal, we cannot find the exact value of the parameter vector. Another
important point is the
consistency.
ˆ
θ
is said to be weakly consistent if:
DEFINITION 3.2.
The estimator
{
}
ˆ
lim
P
θθ
−< = ∀
δ
1
δ
N
N
→∞
ˆ
θ
is said to be consistent in mean square if:
DEFINITION 3.3.
The estimator
⎧
(
)(
)
T
⎫
ˆ
ˆ
li
→∞
θθθθ
−
−
=
0
E
⎨
⎬
N
N
⎩
⎭
N
The second definition is generally stronger. We refer the reader to [BRO 91,
Chapter 6], [POR 94, Chapter 3] for further details on the convergences of estimator
sequences and associated properties. One of the key points of a sequence of
estimators is the speed with which the estimation errors reduce. These going towards
zero, it is natural to standardize
ˆ
N
θθ by an
N
function such that its order of
magnitude is practically independent of
N
. If there exists an increasing monotonic
sequence
d
(
N
)
such that
(
(
)
ˆ
N
dN
θθ converges in law towards a random vector
ζ, then the distribution of
ζ
measures the asymptotic behavior of
ˆ
N
θ
4
. If, for
example, ζ is a random Gaussian vector of zero mean and of covariance matrix
Γ
,
we can say that
ˆ
θ is consistent, asymptotically Gaussian and Γ is known as an
asymptotically normalized covariance matrix of
ˆ
θ . However, we must establish a
fine distinction
with:
( )
(
⎧
)(
)
T
⎫
()
2
ˆ
ˆ
Σ θ
li
→∞
dN
θ θ θ θ
−
−
E
⎨
⎬
N
N
⎩
⎭
N
which is not necessarily equal to Γ but which in practice is often the simplest
quantity to determine.
4
We implicitly consider that all components of
ˆ
θ converge at the same speed. If this is not
so,
()
(
)
ˆ
−
we
will
consider
the
asymptotic
distribution
N
θθ
where
D
N
(
)
()
() ()
()
.
D
N
=
diag
d
N
,
d
N
,
…
,
d
N
1
2
p
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