Asymptotic Probability of Error
Under certain regularity conditions, the M-estimators defined by (4.14) or (4.15) are consistent and
asymptotically Gaussian [170]; that is (here we denote 
as the estimate of q based on N chip
samples),
Equation 4.41
where the asymptotic covariance matrix is given by
Equation 4.42
and where (4.42) follows
from (4.32) and (4.40).
We can also compute the Fisher information matrix for the
parameters q at
the underlying noise distribution. Define the likelihood score vector as
Equation 4.43
The Fisher information matrix is then given by
Equation 4.44
It is known that the maximum likelihood estimate based on
i.i.d. samples is asymptotically unbiased and the asymptotic covariance matrix
is J(q)–1 [377]. As discussed earlier, the
maximum likelihood estimate of q corresponds to having y(x) = –f '(x)/f(x). Hence we can
deduce that the asymptotic covariance matrix 
= J(q)–1, when y(x) = –f'(x)/ f(x). To verify this,
substitute y(x) = –f'(x)/f(x) into (4.42);
we obtain
Equation 4.45
where we have assumed that f'(–
) = f'() = 0.
Next we consider the asymptotic probability of error for the
class of decorrelating detectors defined by (4.15) as the processing gain
N 
. Using the asymptotic normality condition (4.41), 
. The
asymptotic probability of error for the kth user
is then given by
Equation 4.46
where u is the asymptotic variance
given by
Equation 4.47
Hence for the class of M-decorrelators defined by (4.15), their asymptotic
probabilities of detection error are determined by the parameter u. We next compute u for the three
decorrelating detectors discussed in Section 4.2.3, under the
Gaussian mixture noise model (4.3).
Linear Decorrelating Detector
The asymptotic variance for the linear decorrelator is given
by
Equation 4.48
That is, asymptotically, the performance of the linear
decorrelating detector is determined completely by the noise variance,
independent of the noise distribution. However, as will be seen later, the noise
distribution does affect substantially the finite sample performance of the
linear decorrelating detector.
Maximum-Likelihood Decorrelating Detector
The variance of the estimate used in the maximum-likelihood
decorrelating detector achieves the Fisher information covariance matrix, and we
have
Equation 4.49
In fact, (4.49) gives
the minimum achievable u2. To see this, we
use the Cauchy–Schwarz inequality, to yield
Equation 4.50
where the last equality follows from the fact that y(u)f(u) 0, as |u| . To see this, we use (4.3) and (4.21) to obtain
Equation 4.51
Hence it follows from (4.50) that
Equation 4.52
Minimax Decorrelating Detector
For the minimax decorrelating detector, we have
Equation 4.53
Equation 4.54
where 1W(x) denotes the indicator function of the set W, and dx denotes
the Dirac delta function at x. After some
algebra, we obtain
Equation 4.55
Equation 4.56
The asymptotic variance 
of the
minimax decorrelating detector is obtained by substituting (4.55) and (4.56)
into (4.47).
In Fig. 4.2 we plot the
asymptotic variance u2 of the
maximum-likelihood decorrelator and the minimax robust decorrelator as a
function of 
and k under the Gaussian mixture noise model (4.3). The total
noise variance is kept constant as and k vary
[i.e., 
]. From the two plots we see that the two nonlinear detectors
have very similar asymptotic performance. Moreover, in this case the asymptotic
variance u2 is a decreasing function of
either or
k when one of them is fixed. The asymptotic variances
of both nonlinear decorrelators are strictly less than that of the linear
decorrelator, which corresponds to a plane at u2 = s2 = (0.1)2. In Fig. 4.3 we plot the asymptotic variance u2 of the three
decorrelating detectors as a function of k with fixed and in Fig. 4.4 we plot the asymptotic variance u2 of the three
decorrelating detectors as a function of with fixed k. As before, the total variance of the noise for both
figures is fixed at s2 = (0.1)2. From these figures
we see that the asymptotic variance of the minimax decorrelator is very close to
that of the maximum-likelihood decorrelator for the cases of small contamination
(e.g., 
0.1), while both of the detectors can outperform
the linear detector by a substantial margin.
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