Signal Processing - November 2017 - 149

these applications will be discussed in the "Examples of
Applications" section, while here we focus on the theoretical
implications of the condition in (8). The first immediate consequence of (8) is the fact that if the pseudotrue parameter
vector i 0 and the true parameter subvector ir belong to
the same parameter space H, then the difference vector
r _ ir - i 0 is well defined, but, in general, it is different from
a zero-vector. As shown in [10, Sect. II.D] or in [37, eq. (70)],
using r, a bound on the MSE of the estimate of the true
parameter vector ir under model misspecification can be easily established as

	
t (x), ir h _ E p "^it (x) - ir h^it (x) - ir hT ,
MSE
p ^i
1
= C p ^it (x), i 0 h + rr T $ 1 A B A -1 + rr T

M i0 i0 i0
(9)
_ LB ^ir h.
Note that, here, the lower bound (denoted as LB)
LB ^ir h = MCRB ^i 0 h + rr T is considered as a function of the
true parameter vector ir . A simple example that clarifies the
role of the inequality (9) as lower bound on the MSE is reported in "Variance Estimation."

100
MSE and Bounds

Error Covariance and Bounds

1,000

10
MML
MCRB
MCRB_est
1
-10

-5

0
5
True Mean Value

and the estimated MCRB(i 0) as a function of nr . Simulation parameters
are set as M = 10 and vr 2 = 4.

and they are equal only in the case of perfect model specification, i.e., when the true mean is equal to the assumed
mean, i.e., nr = 0.
After having established a lower bound on the mean
square error (MSE), we now investigate the properties of
the mismatched maximum likelihood (MML) estimator for
the estimation problem at hand. In particular, we can say
that the MML estimator is not consistent since, from (11), it
converges to i 0, which is different from the true variance
vr 2. More formally, we have that



M

/

m =1

a.s.

x 2m M"
i 0 = vr 2 + nr 2 ! vr 2.
"3
(S6)

However, according to (4), the MML estimator is misspecified (MS)-unbiased, since
	 E p " it MML , = E p ) M -1

M

/

m =1

MML
LB
CRB
10

1
-10

10

FIGURE S1. The error covariance of the MML estimator, the MCRB(i 0),

it MML _ it MML ^ x h = M -1

100

x 2m 3 = vr 2 + nr 2 = i 0.(S7)

-5

0
5
True Mean Value

10

FIGURE S2. The MSE of the MML estimator, the LB (vr 2), and the CRB (vr 2) ,

as a function of nr . Simulation parameters are set as M = 10 and vr 2 = 4.

Hence, according to Theorem 1, its error covariance w.r.t.
i 0, i.e., C p ^ it MML, i 0 h, is lower bounded by the MCRB in
(S4). Figure S1 shows the error covariance of the MML
estimator, the MCRB (i 0), and the sample estimate of
MCRB (i 0) obtained according to (13)-(15). As we can
see, MCRB (i 0) is a tight bound for the error variance of the
MML estimator, and the sample MCRB (i 0) accurately predicts it. Due to the particular nested structure of the true and
assumed parameter spaces of this example, we can also
evaluate the MSE of the MML estimator w.r.t. the true variance, i.e., MSE p (it MML, vr 2), and the related LB (vr 2) obtained
as shown in (9). Note that the lower bound is denoted as LB.
In Figure S2, we report the MSE of the MML estimator, the
LB (vr 2), and the classical CRB on the estimation of the variance, CRB (vr 2) , as function of the value of the true mean
value nr . As expected from (9), LB (vr 2) is a tight bound for
the MSE of the MML estimator. Finally, it can be noted that
the LB (vr 2) is equal to the CRB (vr 2) only when nr = 0, i.e.,
when the assumed mean value is equal to the true one.

IEEE SIGNAL PROCESSING MAGAZINE

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November 2017

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149
Table of Contents for the Digital Edition of Signal Processing - November 2017
