Signal Processing - November 2017 - 148

parameter space X = H # C, where # indicates the Cartesian
product. Under this restriction, the true parametric model can
be expressed as
P = " p X p X ^x m i, c h is a pdf 6 ^i, c h ! H # C ,, (8)

	

while the assumed model is F = " fX fX ^x m ih is a pdf
6i ! H} as before. Note that fX ^x m ih could differ from
the true p X (x m i, c) 6i ! H and 6c ! C. Moreover, the
nested parameter vector assumption includes, as a special

case, the scenario in which the true parameter space and the
assumed one are equal, i.e., X / H. This case arises, for
example, in array processing applications in which both the
true and the assumed pdfs of the acquired data vectors can be
parameterized by the angles of arrival of a certain number of
sources [37]. A practical example of the more general nested
model assumption is the estimation of the disturbance covariance matrix in adaptive radar detection [10]. In this misspecified estimation problem, both the unknown true data pdf and
the assumed one can be parameterized by a scaled version of
the covariance matrix and by the disturbance power. Both of

Variance Estimation
Here is an illustrative example to clarify the use and
derivation of the misspecified Cramér- Rao bound
(MCRB). Building upon the examples discussed in
[10], we investigate the problem of estimating the
variance of a Gaussian-distributed data set under the
misspecification of the mean value. Let x = {x m} M
m =1
be a set of M independent, identically distributed
(i.i.d.) univariate data sampled from a Gaussian
probabilit y densit y function (pdf) with the mean
value nr and variance vr 2, i.e., p X (x m) / N (nr , vr 2) ! P
with nr ! 0. Due to perhaps an imperfect knowl edge about the data generation process, the user
assumes a zero -mean parametric Gaussian model
F = {fX | fX (x m i) = N (0, i) 6i ! H 3 R +}, i.e., the user
misspecifies the mean value. Note that, as long as
nr ! 0, the true but unknown pdf p X (x m) does not belong
to the assumed model F . Moreover, the reader can easily recognize this mismatched scenario as a simple
instance of the particular case discussed in the section
"An Interesting Case: A Lower Bound on the MSE via the
MCRB." In fact, it is immediately verified that the para--
meter space H 3 R + that characterizes the assumed
model is a subset of the true parameter space, i.e.,
[nr , vr 2] ! X = R 0 # H, where R 0 indicates the set of all
the real numbers excluding 0.
According to the theory presented in the section "The
Misspecified CRB," we first must check whether the
assumed model F is regular with respect to (w.r.t.) P; in
other words, we have to prove the existence of the pseudotrue parameter i 0 (Assumption 1) and the nonsingularity of the matrix A i 0 defined in (2) (Assumption 2). Note
that, for the problem at hand, A i 0 is a scalar quantity, so
we have to prove that A i 0 ! 0. The pseudotrue parameter
i 0 is defined in (1). Following [7], the Kullback-Leibler
divergence (KLD) can be expressed as
	

148

D (p X fX | i) =

nr 2
r2
r2
+ 1 c v - 1 - ln v m .(S1)
i
2i 2 i

The minimum is obtained for i 0 = vr 2 + nr 2, which, according to (1), represents the pseudotrue parameter. Since the
pseudotrue parameter exists and is unique, Assumption 1
is satisfied. We can now check Assumption 2. To this end,
from (2), A i0 can be evaluated as
A i0 _ E p )


2 2 ln fX (x m i)
2i 2

i = i0

3 = 1 2 - 13 E p " x 2m , =- 1 2 ,
2i 0 i 0
2i 0
(S2)

yielding a denominator different from zero since vr 2 ! R +;
consequently, Assumption 2 is verified as well. Now we
can evaluate the MCRB in (5) for the estimation problem
at hand. First, the scalar B i0 can be easily evaluated from
(3) as
2 ln fX (x m i) 2
m i = i0 1
B i0 _ E p 'c
2i
	

2
4
, - 2i 0 E p " x m2 ,
i0 + Ep "xm
2vr 4 + 4vr 2 nr 2 (S3)
=
=
.
4i 40
4i 40
Finally, from (5), we get
2

2

r 4 4vr nr
.(S4)
MCRB (i 0) = 2v +
M
M

	

Since this misspecified scenario belongs to the particular
class of nested parametric models, as discussed in the section "An Interesting Case: A Lower Bound on the MSE via
the MCRB," we can also rewrite the MCRB in (S4) as a
function of the true variance vr 2. This can be easily done
by introducing the (scalar) r _ vr 2 - i 0 =-nr 2 and, consequently, according to (9), by evaluating the LB (vr 2) as
2

2

r 4 4vr nr
+ nr 4 .(S5)
LB (vr 2) = 2v +
M
M

	

It can be noted that the lower bound in (S5) is always
greater than the classical CRB given by CRB (vr 2) = 2vr 4 M

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November 2017

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