Signal Processing - November 2017 - 144
is easy to evaluate for many practical problems, but it is only
asymptotically reliable. In the nonasymptotic region, which is
characterized by a low SNR and/or by a low number of measurements, the CRB can be too optimistic with respect to (w.r.t.)
the effective error covariance achievable by an estimator [44].
The second subdivision of the performance bounds is a
direct consequence of the dichotomy between the deterministic and the Bayesian estimation frameworks. In particular,
we can identify the class of deterministic lower bounds and the
class of Bayesian lower bounds [43]. Without any claim of completeness, the class of the deterministic lower bounds includes
the (global) BB [3] and two local bounds, the Bhattacharyya
bound [5] and the CRB [8], [33]. We stress that the most common forms of these bounds, including the CRB, apply only to
unbiased estimators. Versions of these bounds exist, however,
that can be applied to biased estimators whose bias function
can be determined. Concerning the Bayesian bounds, they can
be divided into two classes [34]: the Ziv-Zakaï family and the
Weiss-Weinstein family, to which the Bayesian version of the
CRB belongs. The first family is derived by relating the mean
squared error (MSE) to the probability of error in a binary
hypothesis testing problem, while the derivation of the latter
is based on the covariance inequality. For further details on
Bayesian bounds, refer to [43].
establish performance limits on the estimation error covariance in a way that indicates how the difference between the
true and assumed models affects the estimation performance.
Having established the main motivations, we can now briefly
review the literature on the estimation framework under model
misspecification, with a focus on the two classical building
blocks, i.e., the ML estimator and the CRB.
Some historical background
The first fundamental result on the behavior of the ML estimator under misspecification appeared in the statistical literature in 1967 and was provided by Huber [20]. In that
paper, the consistency and the normality of the ML estimator
were proved under very mild regularity conditions. Five
years later, Akaike [1] highlighted the link between Huber's
findings and the Kullback-Leibler divergence (KLD) [7]. He
noted that the convergence point of the ML estimator under
model misspecification could be interpreted as the point that
minimizes the KLD between the true and the assumed models. In the early 1980s, these ideas were further developed by
White [46], where the term quasi-ML estimator was introduced. Some years later, the second fundamental building
block of an estimation theory under model misspecification
was established by Vuong [45]. Vuong
was the first to derive a generalization of
The mathematical basis
the Cramér-Rao lower bound under
An estimation theory under
for a formal theory of
misspecified models. The Bayesian mismodel misspecification: Motivations
statistical inference was
specified estimation problem has been
Regardless of the differences previously
investigated in [4] and [6].
discussed, both the classical deterministic
presented by Fisher, who
Quite surprisingly and despite the wide
estimation theory and the Bayesian frameintroduced the maximum
variety of potential applications, the SP
work are based on the implicit assumption
likelihood method along
community has remained largely unaware
that the assumed data model (the pdf) and
with its main properties.
of these fundamental results. This topic
the true data model are the same, i.e., the
has only recently been rediscovered and its
model is correctly specified. However,
applications to well-known SP problems investigated [10]-[12],
much evidence from engineering practice shows that this
[14], [18], [19], [22], [28], [32], [35]-[38], [48], [50]. Of course,
assumption is often violated; the assumed model is different
every SP practitioner was aware of the misspecification probfrom the true one. There are two main reasons for model mislem, but some approaches commonly used within the SP comspecification. The first is the imperfect knowledge of the true
munity to address it differed from some of those proposed in
data model, which leads to an incorrect specification of the
the statistical literature. The effect of the misspecification has
data pdf. However, there could be cases where perfect knowlbeen modeled by adding into the true data model some random
edge of the true data model is available, but, due to an intrinsic
quantities, also called nuisance parameters, and by transformcomputational complexity or to a costly hardware implemening the estimation problem at hand into a higher dimensional
tation, it is not possible nor convenient to pursue the optimal
hybrid estimation problem. The performance degradation due
"matched" estimator. In these cases, one may prefer to derive
to the augmented level of uncertainty generated by the nuian estimator by assuming a simpler but misspecified data
sance parameters could be assessed by evaluating the true
model, e.g., the Gaussian model. Of course, this suboptimal
CRB when possible, the hybrid CRB (see, e.g., [16], [29], [31],
procedure may lead to some degradation in the overall system
and [39]), or the modified CRB (see, e.g., [2], [17], and [24]).
performance, but it ensures a simple analytical derivation and
This approach, although reasonable, is application dependent
real-time hardware implementation of the inference algorithm.
and not general at all. Other approaches include sensitivity
In such a misspecified estimation framework, the possibilanalyses [15], [44].
ity to assess the impact of the model misspecification on the
Finally, the relationship between misspecified estimation
estimation performance is of fundamental importance to
theory and robust statistics should also be noted (see [49] for
-g uarantee the reliability of the (mismatched) estimator.
a tutorial on robust statistics). As one would expect, these two
Misspecified bounds are then the perfect candidates to fulfill
frameworks share the same motivations, i.e., an imperthis task: they generalize the classical framework by
fect knowledge of the true data model. The aim of robust
allowing the assumed and true models to differ, yet they
144
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
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
Signal Processing - November 2017 - 12
Signal Processing - November 2017 - 13
Signal Processing - November 2017 - 14
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
Signal Processing - November 2017 - 27
Signal Processing - November 2017 - 28
Signal Processing - November 2017 - 29
Signal Processing - November 2017 - 30
Signal Processing - November 2017 - 31
Signal Processing - November 2017 - 32
Signal Processing - November 2017 - 33
Signal Processing - November 2017 - 34
Signal Processing - November 2017 - 35
Signal Processing - November 2017 - 36
Signal Processing - November 2017 - 37
Signal Processing - November 2017 - 38
Signal Processing - November 2017 - 39
Signal Processing - November 2017 - 40
Signal Processing - November 2017 - 41
Signal Processing - November 2017 - 42
Signal Processing - November 2017 - 43
Signal Processing - November 2017 - 44
Signal Processing - November 2017 - 45
Signal Processing - November 2017 - 46
Signal Processing - November 2017 - 47
Signal Processing - November 2017 - 48
Signal Processing - November 2017 - 49
Signal Processing - November 2017 - 50
Signal Processing - November 2017 - 51
Signal Processing - November 2017 - 52
Signal Processing - November 2017 - 53
Signal Processing - November 2017 - 54
Signal Processing - November 2017 - 55
Signal Processing - November 2017 - 56
Signal Processing - November 2017 - 57
Signal Processing - November 2017 - 58
Signal Processing - November 2017 - 59
Signal Processing - November 2017 - 60
Signal Processing - November 2017 - 61
Signal Processing - November 2017 - 62
Signal Processing - November 2017 - 63
Signal Processing - November 2017 - 64
Signal Processing - November 2017 - 65
Signal Processing - November 2017 - 66
Signal Processing - November 2017 - 67
Signal Processing - November 2017 - 68
Signal Processing - November 2017 - 69
Signal Processing - November 2017 - 70
Signal Processing - November 2017 - 71
Signal Processing - November 2017 - 72
Signal Processing - November 2017 - 73
Signal Processing - November 2017 - 74
Signal Processing - November 2017 - 75
Signal Processing - November 2017 - 76
Signal Processing - November 2017 - 77
Signal Processing - November 2017 - 78
Signal Processing - November 2017 - 79
Signal Processing - November 2017 - 80
Signal Processing - November 2017 - 81
Signal Processing - November 2017 - 82
Signal Processing - November 2017 - 83
Signal Processing - November 2017 - 84
Signal Processing - November 2017 - 85
Signal Processing - November 2017 - 86
Signal Processing - November 2017 - 87
Signal Processing - November 2017 - 88
Signal Processing - November 2017 - 89
Signal Processing - November 2017 - 90
Signal Processing - November 2017 - 91
Signal Processing - November 2017 - 92
Signal Processing - November 2017 - 93
Signal Processing - November 2017 - 94
Signal Processing - November 2017 - 95
Signal Processing - November 2017 - 96
Signal Processing - November 2017 - 97
Signal Processing - November 2017 - 98
Signal Processing - November 2017 - 99
Signal Processing - November 2017 - 100
Signal Processing - November 2017 - 101
Signal Processing - November 2017 - 102
Signal Processing - November 2017 - 103
Signal Processing - November 2017 - 104
Signal Processing - November 2017 - 105
Signal Processing - November 2017 - 106
Signal Processing - November 2017 - 107
Signal Processing - November 2017 - 108
Signal Processing - November 2017 - 109
Signal Processing - November 2017 - 110
Signal Processing - November 2017 - 111
Signal Processing - November 2017 - 112
Signal Processing - November 2017 - 113
Signal Processing - November 2017 - 114
Signal Processing - November 2017 - 115
Signal Processing - November 2017 - 116
Signal Processing - November 2017 - 117
Signal Processing - November 2017 - 118
Signal Processing - November 2017 - 119
Signal Processing - November 2017 - 120
Signal Processing - November 2017 - 121
Signal Processing - November 2017 - 122
Signal Processing - November 2017 - 123
Signal Processing - November 2017 - 124
Signal Processing - November 2017 - 125
Signal Processing - November 2017 - 126
Signal Processing - November 2017 - 127
Signal Processing - November 2017 - 128
Signal Processing - November 2017 - 129
Signal Processing - November 2017 - 130
Signal Processing - November 2017 - 131
Signal Processing - November 2017 - 132
Signal Processing - November 2017 - 133
Signal Processing - November 2017 - 134
Signal Processing - November 2017 - 135
Signal Processing - November 2017 - 136
Signal Processing - November 2017 - 137
Signal Processing - November 2017 - 138
Signal Processing - November 2017 - 139
Signal Processing - November 2017 - 140
Signal Processing - November 2017 - 141
Signal Processing - November 2017 - 142
Signal Processing - November 2017 - 143
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
Signal Processing - November 2017 - 161
Signal Processing - November 2017 - 162
Signal Processing - November 2017 - 163
Signal Processing - November 2017 - 164
Signal Processing - November 2017 - 165
Signal Processing - November 2017 - 166
Signal Processing - November 2017 - 167
Signal Processing - November 2017 - 168
Signal Processing - November 2017 - 169
Signal Processing - November 2017 - 170
Signal Processing - November 2017 - 171
Signal Processing - November 2017 - 172
Signal Processing - November 2017 - 173
Signal Processing - November 2017 - 174
Signal Processing - November 2017 - 175
Signal Processing - November 2017 - 176
Signal Processing - November 2017 - Cover3
Signal Processing - November 2017 - Cover4
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