Signal Processing - November 2017 - 156

is an efficient estimator w.r.t. the CMCRB, as predicted by the
theory in the section "The Mismatched ML Estimator."

Concluding remarks
The objective of this article is to provide an accessible and, at
the same time, comprehensive treatment of the fundamental
concepts about CRBs and efficient estimators in the presence
of model misspecification. Every SP practitioner is aware of
the fact that, in almost all practical applications, a certain
amount of mismatch between the true and the assumed statistical data models is inevitable. Despite its ubiquity, the assessment of performance bounds under model misspecification
appears to have received limited attention from the SP
-community, while it has been deeply investigated by the statistical community. The first aim of this tutorial is to propose to a
wide SP audience a comprehensive review of the main contributions to the mismatched estimation theory, both for the
deterministic and Bayesian frameworks, with a particular
focus on the derivation of CRB under model mismatching.
Specifically, we have described how the classical tools of the
estimation theory can be generalized to address a mismatched
scenario. First, the MCRB has been introduced and the behavior of the MML estimator investigated. Second, results related
to the deterministic estimation framework have been extended
to the Bayesian one. The existence and the asymptotic properties of a MB estimator have been discussed. Moreover, some
general ideas about the possibility to derive MBCRBs have
been provided. In the last part of the article, we showed how
to apply the theoretical findings to two well-known relevant
problems: the DOA estimation in array processing and the
estimation of the disturbance covariance matrix for adaptive
radar detection.
Of course, much work remains to be done. A question that
naturally arises is whether it is possible to derive a more general class of misspecified bounds. The first step toward this
direction has been outlined by Richmond and Horowitz in

0.2

Frobenius Norm

0.19
0.18

CMML
CMCRB
CCRB

0.17
0.16
0.15
0.14
0.13

5
10
15
20
25
30
Shape Parameter of the t-Distribution: λ

FIGURE 2. The Frobenius norms of the MSE matrix of the CMML estimator,
the CMCRB, and the CCRB for the scatter matrix estimation problem.
Simulation parameters are set as N = 16, M = 10N, and the scale parameter of the true t -distribution is h = 1.

156

[37], where a generalization of the theory to the Bhattacharyya bound, to the BB, and to the Bobrovsky-Mayer-Wolf-
Zakaï bound has been proposed. Next, as discussed in the
"-Generalization to the Bayesian Setting" section, a future area
of research is the derivation of general Bayesian lower bounds
that could be obtained by relaxing or, hopefully, removing the
constraints given in (27). Finally, a systematic and deep investigation of a general decision theory under model misspecification is required since it could lead great advantages in a huge
number of SP applications.

Acknowledgments
The work of Stefano Fortunati has been partially supported by
the Air Force Office of Scientific Research under award number FA9550-17-1-0065.

Authors
Stefano Fortunati (stefano.fortunati@iet.unipi.it) received his
Ph.D. degree in telecommunication engineering from the
University of Pisa, Italy, in 2012. He then joined the De--
partment of Ingegneria dell'Informazione at the University
of Pisa, where he is currently working as a postdoctoral
researcher. His professional expertise encompasses different
areas of statistical signal processing: estimation and detection
theory, statistical methods for data analysis, non-Gaussian signal detection and estimation, robust signal estimation and
detection, performance bounds, and compressed sensing theory with applications in radar and sonar systems. He is a
Member of the IEEE.
Fulvio Gini (f.gini@iet.unipi.it) received his Dr.-Ing. and
Ph.D. degrees in electronic engineering from the University of
Pisa, Italy, in 1990 and 1995, respectively. Currently, he is a full
professor in the Department of Information Engineering at the
University of Pisa. He has received several awards, including the
2001 and 2012 IEEE Aerospace and Electronic Systems (AES)
Society's Barry Carlton Award for Best Paper published in IEEE
Transactions on Aerospace and Electronic Systems, the 2003
IEEE Achievement Award, and the 2003 IEEE AES Society
Nathanson Award to the Young Engineer of the Year. He has
authored or coauthored 11 book chapters, approximately 125 journal papers, and 160 conference papers. He is a Fellow of the IEEE.
Maria S. Greco (m.greco@iet.unipi.it) is a full professor
in the Department of Information Engineering at the Uni--
versity of Pisa, Italy. She is a corecipient of the 2001 and 2012
IEEE Aerospace and Electronic Systems (AES) Society's
Barry Carlton Awards for Best Paper and recipient of the
IEEE AES Society 2008 Fred Nathanson Young Engineer of
the Year Award. Her research interests include statistical signal
processing, estimation, and detection theory. She has coauthored many book chapters and more than 190 journal and
conference papers. She is a Fellow of the IEEE.
Christ D. Richmond (christ.richmond@asu.edu) received
his Ph.D. degree in electrical engineering from the Massa-
chusetts Institute of Technology (MIT). His is currently an
associate professor at Arizona State University, Tempe, and a
former senior staff member at the MIT Lincoln Laboratory. He

IEEE SIGNAL PROCESSING MAGAZINE

November 2017

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