Signal Processing - November 2017 - 146

required regularity conditions and the notion of unbiasedness
for mismatched estimators.

Regular models
As with the classical CRB, to guarantee the existence of the
MCRB, some regularity conditions on the assumed pdf need
to be imposed. Specifically, the assumed parametric model F
must be regular w.r.t. P, i.e., the family to which the true pdf
belongs. The complete list of assumptions that F must satisfy
to be regular w.r.t. P are given in [45] and briefly recalled in
[10]. Most of them are rather technical and facilitate an order
reversal of the integral and derivative operators. Nevertheless,
there are two assumptions that need to be discussed here due
to their importance in the development of the theory. The first
condition that must be satisfied is Assumption 1.

Assumption 1
There exists a unique interior point i 0 of H such that
	 i 0 = argmin " -E p " ln fX ^x m ih,, = argmin " D ^ p X fX ih,,
i!H
i!H

(1)
where E p {$} indicates the expectation operator of a vec--toror scalar-valued function w.r.t. the pdf p X (x m) and
D ^ p X fX ih _ # ln ^ p X (x m) fX ^x m ihh p X (x m) dx m i s t h e
KLD [7] between the true and the assumed pdfs. As indicated
by (1), i 0 can be interpreted as the point that minimizes the
KLD between p X (x m) and fX (x m i), and it is called the
pseudotrue parameter vector [45], [46].
After having defined the pseudotrue parameter vector i 0
in this assumption, let A i 0 be the matrix whose entries are
defined as

	

6A i 0@ij _ 6E p " d i d Ti ln fX ^x m i 0 h,@ij

= Ep '


2 2 ln f ^x ih
1,
X
m
2i i 2i j
i = i0

(2)

where d i u (i 0) and d i d iT u (i 0) indicate, respectively, the
gradient (column) vector and the symmetric Hessian matrix of
the scalar function u evaluated at i 0 . The second fundamental
condition that must be satisfied by the assumed model F to
be regular w.r.t. P is Assumption 2.

Assumption 2

The matrix A i 0 is nonsingular.
The pseudotrue parameter vector i 0 plays a fundamental
role in estimation theory for misspecified models. Roughly
speaking, it identifies the pdf fX (x m i 0) in the assumed parametric model F that is closest, in the KLD sense, to the true
pdf. As the next sections will clarify, it can be interpreted as the
counterpart of the true parameter vector of the classical
matched theory. Regarding the matrix A i 0 , its negative represents a generalization of the classical Fisher information matrix
(FIM) to the misspecified model framework. To clarify this,
we first define the matrix B i 0 as
146

	

6B i 0@ij _ 6E p " d i ln fX ^x m i 0 h d iT ln fX ^x m i 0 h,@ij

= Ep )

2 ln fX ^x m ih
2i i

i = i0

$

2 ln fX ^x m ih
2i j

i = i0


3 . (3)

As with matrix A i 0, we recognize in B i 0 the second po--s-
si--ble generalization of the FIM. Vuong [45] showed that
if  p X (x m) = fX (x m ir ) for some ir ! H, then i 0 = ir and
B ir =-A ir , where ir is the true parameter vector of the
-classical matched theory. The last equation shows that, under
the correct model specification, the two expressions of the FIM
are equal, as expected [44]. This provides evidence of the fact
that the misspecified estimation theory is consistent with the
classical one. The reader, however, should note that the equality
between the pseudotrue parameter vector and the true one does
not imply in any way the equality between the true and the
assumed pdfs and, consequently, between the matrices B i 0 and
-A i 0 . After having established the necessary regularity conditions, we can introduce the class of misspecified-unbiased
(MS-unbiased) estimators.

The MS-unbiasedness property
The first generalization of the classical unbiasedness property
to mismatched estimators was proposed by Vuong [45].
Specifically, let it (x) be an estimator of the pseudotrue
parameter vector i 0, i.e., a function of the M available i.i.d.
observation vectors x = {x m} mM= 1, derived under the misspecified parametric model F . Then, it (x) is said to be an
MS-unbiased estimator if and only if
E p " it (x) , =

	

# it (x) p X (x) dx = i 0, (4)

where i 0 is the pseudotrue parameter vector defined in (1).
The link with the classical matched unbiasedness property is
obvious: if the parametric model F is correctly specified, i 0
is equal to the vector ir ! H such that p X (x m) = fX (x m ir ).
C o n s e q u e n t l y, (4) can be r ew r i t t e n as E p {it (x)} =
# it (x) fX (x ir ) dx = ir , which is the classical definition of the
unbiasedness property. At this point, we are ready to introduce
the explicit expression for the MCRB.

A covariance inequality in the presence
of misspecified models
In this section, we present the MCRB as introduced by Vuong
in his seminal paper [45]. An alternative derivation was proposed by Richmond and Horowitz in [36] and [37]. A comparison between the derivation given in [45] and the one proposed
in [36] and [37] has been provided in [10].

Theorem 1
In Theorem 1 [45], let F be a misspecified parametric model
that is regular w.r.t. P. Let it (x) be an MS-unbiased estimator
derived under the misspecified model F from a set of M i.i.d.
observation vectors x = {x m} mM= 1 . Then, for every possible
p X (x m) ! P,
1
C p ^it (x), i 0 h $ 1 A B A -1 _ MCRB ^i 0 h, (5)
M i0 i0 i0

	

IEEE SIGNAL PROCESSING MAGAZINE

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November 2017

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Table of Contents for the Digital Edition of Signal Processing - November 2017

Signal Processing - November 2017 - Cover1
Signal Processing - November 2017 - Cover2
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