Signal Processing - November 2017 - 147

where
	

C p ^it (x), i 0h _ E p "^it (x) - i 0h^it (x) - i 0h , (6)
T

is the error covariance matrix of the mismatched estimator
it (x) where the matrices A i 0 and B i 0 have been defined in
(2) and (3), respectively.
The following comments are in order. The major implication of Theorem 1 is that it is still possible to establish a
lower bound on the error covariance matrix of an (MS-unbiased) estimator, even if it is derived under a misspecified
data model, i.e., it is derived under a pdf fX (x m i) that could
differ from the true pdf p X (x m) for every value of i in the
parameter space H. An important question that may arise
under a misspecified model framework is which vector in the
assumed parameter space H should be used to evaluate the
effectiveness of a mismatched estimator, particularly when no
true parameter vector exists, i.e., p X (x m) ! fX (x m i), for all
i ! H? It is certainly reasonable to use the parameter value
that minimizes the distance, in a given sense, between the
assumed misspecified pdf fX (x m i) and the true pdf p X (x m).
Theorem 1 shows that, if one uses the KLD as a measure of
distance and by assuming that the misspecified model F is
regular w.r.t. the true model P, this parameter vector exists,
and it is the pseudotrue parameter vector i 0 defined in (1).
Specifically, the MCRB is a lower bound on the error covariance matrix of any MS-unbiased estimator, where the error
is defined as the difference between the estimator and the
pseudotrue parameter vector. Moreover, if the model F is
correctly specified, then, as stated before, i 0 = ir , such that
p X (x m) = fX (x m ir ) and B i 0 = B ir =-A ir . Consequently,
the inequality in (5) becomes the classical (matched) CRB
inequality for u- nbiased estimators
1
1 A -r 1 _ CRB ^ir h.
	E p "^it (x) - ir h^it (x) - ir h , $ 1 B r =M i
M i
(7)
T

The second point concerns the matter of how to exploit
Theorem 1 in practice. The MCRB is a generalization of the
classical CRB to the misspecified model framework and can
play a similar role. Specifically, the MCRB can be used to
assess the performance of any mismatched estimator, and it
plays the same key role as the classical CRB in any feasibility
study, but with the added flexibility to assess performance
under modeling errors. For example, consider the recurring
scenario in which the SP practitioner is aware of the true data
pdf p X (x m), but, to fulfill some operational constraints, the
user is forced to derive the required estimator by exploiting a
simpler, but m
- isspecified, model. In this scenario, the MCRB
in (5) can be directly applied to assess the potential estimation loss due to the mismatch between the assumed and the
true models.
This scenario can be extended to the case in which the
SP practitioner is not completely aware of the functional
form of the true data pdf, but the user is still able to infer
some of its properties, e.g., from empirical data or parameter estimates based on such data. Such knowledge can be

used to motivate surrogate models for the true data pdf,
which in turn can be exploited to conduct a system analysis
and -performance assessment. To clarify this point, consider
the case in which the SP practitioner, to derive a simple
inference algorithm, decides to assume a Gaussian model to
describe the data behavior. However, thanks to a preliminary
data analysis, the user knows that the data share a heavytailed distribution, e.g., due to the presence of impulsive
non-Gaussian noise. Then, the user could choose as true
data pdf a heavy-tailed distribution, e.g., the t--d istribution,
and, consequently, exploit the MCRB to assess how ignoring the heavy-tailed and impulsive nature of the data affects
the performance of the estimation algorithm based on a
Gaussian model. This explains that, although the chosen
"true" pdf (in this example, the t-distribution) may not be
the exact true data pdf, it can still serve as a useful surrogate for the purposes of system analysis and design by
means of the MCRB.
The MCRB can also be used to predict potential weaknesses (i.e., a breakdown of the estimation performance) of
a system. Suppose one has a system/estimator derived under
a certain modeling assumption, but it is of practical interest
to predict how well this system will react in the presence of
different true input data distributions, perhaps characterizing
operational scenarios that the system can undergo. Clearly, the
MCRB is well suited to address this task.
Another important question may arise analyzing Theorem 1. To evaluate the pseudotrue parameter vector i 0 in (1)
and then the MCRB in (5), we need to know the true data pdf
p X (x m), since it is required to evaluate the expectation operators. How can we calculate the MCRB in all the practical cases
in which we haven't any a priori knowledge of the functional
form of p X (x m) ? An answer to this fundamental question is
given in the section "A Consistent Sample Estimate of the
MCRB," where we show that consistent estimators for both the
pseudotrue parameter vector i 0 and the MCRB can be derived
from the acquired data set.
The proposed MCRB can be easily extended to misspecified estimation problems that require equality constraints. We
refer the reader to [11] for a comprehensive treatise on this
problem. Additionally, with regard to the possibility of generalizing the previously discussed results to the case of complex unknown parameter vectors, the extension to the complex
fields can be achieved in two equivalent ways. We can always
map a complex parameter vector into a real one simply by
stacking its real and imaginary parts, as, e.g., in [35], or we
could exploit the so-called Wirtinger calculus, as discussed in
[13] and [37].

An interesting case: A lower bound on
the MSE via the MCRB
In this section, we focus on a mismatched case that is of great
interest in many practical applications. Specifically, we consider the case in which the parameter vector of the assumed
model F is nested in the one of the true parametric model P,
i.e., the assumed parameter space H is a subspace of the true

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
Signal Processing - November 2017 - 1
Signal Processing - November 2017 - 2
Signal Processing - November 2017 - 3
Signal Processing - November 2017 - 4
Signal Processing - November 2017 - 5
Signal Processing - November 2017 - 6
Signal Processing - November 2017 - 7
Signal Processing - November 2017 - 8
Signal Processing - November 2017 - 9
Signal Processing - November 2017 - 10
Signal Processing - November 2017 - 11
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Signal Processing - November 2017 - 15
Signal Processing - November 2017 - 16
Signal Processing - November 2017 - 17
Signal Processing - November 2017 - 18
Signal Processing - November 2017 - 19
Signal Processing - November 2017 - 20
Signal Processing - November 2017 - 21
Signal Processing - November 2017 - 22
Signal Processing - November 2017 - 23
Signal Processing - November 2017 - 24
Signal Processing - November 2017 - 25
Signal Processing - November 2017 - 26
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Signal Processing - November 2017 - 28
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Signal Processing - November 2017 - 30
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Signal Processing - November 2017 - 131
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Signal Processing - November 2017 - 140
Signal Processing - November 2017 - 141
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Signal Processing - November 2017 - 144
Signal Processing - November 2017 - 145
Signal Processing - November 2017 - 146
Signal Processing - November 2017 - 147
Signal Processing - November 2017 - 148
Signal Processing - November 2017 - 149
Signal Processing - November 2017 - 150
Signal Processing - November 2017 - 151
Signal Processing - November 2017 - 152
Signal Processing - November 2017 - 153
Signal Processing - November 2017 - 154
Signal Processing - November 2017 - 155
Signal Processing - November 2017 - 156
Signal Processing - November 2017 - 157
Signal Processing - November 2017 - 158
Signal Processing - November 2017 - 159
Signal Processing - November 2017 - 160
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Signal Processing - November 2017 - 166
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Signal Processing - November 2017 - 168
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Signal Processing - November 2017 - 170
Signal Processing - November 2017 - 171
Signal Processing - November 2017 - 172
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Signal Processing - November 2017 - 175
Signal Processing - November 2017 - 176
Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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