Signal Processing - November 2017 - 145
a matched case, since there exists ir = (nr , vr 2) ! R # R + such
that p X (x m) = fX (x m ir ) = N (nr , vr 2). Suppose now that the
collected data are distributed according to a univariate Laplace
distribution with a location parameter cr and a scale parameter
br , i.e., x m + p X (x m) = L (cr, br ). Due to, perhaps, misleading
and incomplete information on the experiment at hand or due
to the need to derive a simple algorithm, we decide to adopt a
parametric Gaussian model F to characterize the collected
data. Unlike the previous example, this is obviously a mismatched case, since there does not exist any i = (i 1, i 2) for
which the assumed Gaussian model is equal to the true
Laplace model.
Many practical examples of model misspecification
can be found in everyday engineering practices. Just to list
a few, recent papers have investigated the application of this
misspecified model framework to
■■ the direction-of-arrival (DOA) estimation problem in sensor
arrays [22], [36], [37] and multiple-input, multiple-output
(MIMO) radars [35]
Description of a misspecified model problem
Let x 1, f, x M be a set of N-dimensional (generally complex)
■■ the covariance matrix estimation problem in non-Gaussian disrandom vectors representing the outcome of a measurement
turbance [10], [12], [18]
process. Let x m ! C N be a single observation vector with pdf
■■ radar-communication systems coexistence [38]
p X (x m) belonging to a possibly parametric
■■
waveform parameter estimation in the
family, or model, P that characterizes the
presence of uncertainty in the propagaThe concept of efficiency
tion model [32]
observed random experiment. As d- iscussed
is strictly related to
in the section "A Formal Theory of
■■
the time-of-arrival estimation problem
the existence of some
Statistical Inference Under Misspecified
for ultra-wideband signals in the presModels," in almost all practical applicaence of interference [19].
lower bounds on the
tions, the true pdf p X (x m) is either not perIn
"The Misspecified CRB" and "The
performance of any
Mismatched ML Estimator" sections, the
fectly known, or it does not admit a simple
estimator designed for a
parameter vector i is assumed to be an
derivation or easy im--plementation of the
specific inference task.
estimation algorithm. Thus, instead of
unknown and deterministic real vector. The
extension to the Bayesian case is discussed
p X (x m), in the mismatched estimation
in the "Generalization to the Bayesian Setting" section. Supframework, we adopt a different parametric pdf, say,
pose, for inference purposes, we collect M independent, idenfX (x m i), with i ! H 1 R d, to characterize the statistical
behavior of the data vector x m . Potential estimation algorithms
tically distributed (i.i.d.) measurement vectors x = {x m} mM= 1,
where x m + p X (x m). Due to the independence, the true joint
may be derived from the misspecified parametric pdf
pdf of the data set x can be expressed as the product of the
fX (x m i), belonging to a parametric model F , and not from
M
the true pdf p X (x m). Moreover, we assume that fX (x m i)
marginal pdf as p X (x) = % m = 1 p X (x m). The assumed joint
M
could differ from p X (x m) for every i ! H. Since this assumppdf of the data set is instead fX ^x ih = % m = 1 fX ^x m ih.
tion represents the division between the classical matched and
This misspecified model framework raises three importhe misspecified parametric estimation theories, some additiontant questions:
al comments are warranted. The matched estimation theory
■■ Is it still possible to derive lower bounds on the error covarirequires the existence of at least a parameter vector ir ! H for
ance of any mismatched estimator of the parameter vector i?
which the pdf assumed by the SP practitioner is equal to
■■ How will the classical statistical properties of an estimator,
the true one. Mathematically, we can say that the classie.g., unbiasedness, consistency, and efficiency, change in
cal matched theory holds true if, for some ir ! H,
this misspecified model framework?
p X (x m) = fX (x m ir ) or, equivalently, if p X (x m) ! F . For
■■ How meaningful are the parameter estimates under extreme
example, suppose the collected data, i.e., the outcomes of a
cases of mismatch?
random experiment, are distributed according to a univariate Gaussian distribution with the mean value nr and variance
The misspecified CRB
2
2
vr , i.e., x m + p X (x m) = N (nr , vr ), m = 1, f, M. Moreover,
This section introduces a version of the CRB accounting for
possible model misspecification, i.e., the misspecified CRB
-suppose that the assumed paramet-ric m o d e l for data
(MCRB), which can be considered a generalization of the usual
inference is the Gaussian parametric model, i.e.,
CRB. In particular, as we will show later, the CRB is obtained
F = $ fX fX (x m i) = N (i 1, i 2) 6i ! R # R +., where R + is
when the model is correctly specified. We start by providing the
the set of positive real numbers. This -situation clearly represents
e- stimation theory is to develop estimation algorithms that are
capable of achieving good performance over a large set of
allowable input data models, even if they are suboptimal
under any nominal (or true) model. Even though the development of robust estimators is certainly vital in many SP
applications, for some of these, the mathematical derivation and consequent implementation may be too involved or
too time and hardware intensive. In these cases, as discussed
before, one may prefer to apply the classical, nonrobust estimation theory by assuming a simplified, hence, misspecified,
statistical model for the data.
The first aim of this article is to summarize the most relevant
existing works in the statistical literature using a formalism that
is more familiar to the SP community. The second is to show
the potential application of misspecified estimation theory, in
both the deterministic and Bayesian contexts, for various classical SP problems.
IEEE SIGNAL PROCESSING MAGAZINE
|
November 2017
|
145
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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