Digital Signal Processing Reference
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of invertible linear transformations. In [40], based on general asymptotic theory of
GLR-tests, the following result was shown:
Theorem 1 Under H
0
, n
ln
l
n
! x
p
in distribution, where p
W
k
(
k þ
1)
.
The test that rejects
H
0
whenever
n
ln
l
n
exceeds the corresponding chi-square
(1
a
)th quantile is thus GLRTwith asymptotic level
a
. This test statistic is, however,
highly sensitive to violations of the assumption of complex normality. Therefore, in
[40], a more general hypothesis was considered also
H
0
0
:
P¼
0 when
z
CE
k
(
S
,
V
,
g
) with
g
[
G
2
Hence the purpose is to test the validity of the circularity assumption when sampling
from unspecified (not necessarily normal) CES distributions with finite fourth-order
moments. Denotes by
k
i
¼ k
(
z
i
) the marginal kurtosis of the
i
th variable
z
i
. Under
H
0
0
, the marginal kurtosis coincide, so
k¼ k
1
¼¼k
k
. In addition, under
H
0
0
,
the circularity coefficient of
the marginals vanishes,
that
is,
l
(
z
i
)
¼
0 for
4
]
=s
i
2, where
s
i
¼ s
2
(
z
i
). Let
^k
be
any
consistent esti-
mate of
k
. Clearly, a natural estimate of the marginal kurtosis is the average of the
sample marginal kurtosis
i ¼
1,
...
,
k
,so
k¼ E
[
jz
i
j
k
P
i¼
1
^k
i
. Then,
in [40], an adjusted GLRT-test statistic was shown to be asymptotically robust over
the class of CES distributions with finite fourth-order moments.
^k
i
¼
(
n
P
j¼
1
jz
ij
j
4
)
= ^s
i
2, that is,
^k¼
1
Theorem 2 Under H
0
0
, '
n
W
n
ln
l
n
=
(1
þ ^k=
2)
! x
p
in distribution.
This means that by a slight adjustment, that is, by dividing the GLRT statistic
n
log
l
n
by (1
þ ^k=
2), we obtain an adjusted test statistic
'
n
of circularity that is valid—not just
at the CN distribution, but—over the whole class of CES distributions with finite
fourth-order moments. Based on the asymptotic distribution, we reject the null
hypothesis at (asymptotic)
a
-level if
P ¼
1
F
x
p
(
'
n
)
,
a
.
We now investigate the validity of the
x
p
approximation to the finite sample distri-
bution of the adjusted GLRT-test statistic
'
n
at small sample lengths graphically via
“chi-square plots”. For this purpose, let
'
n
,1
,
...
,
'
n
,
N
denote the computed values
of the adjusted GLRT-test statistic from
N
simulated samples of length
n
and let
'
n
,[1]
'
n
,[
N
]
denote the ordered sample, that is, the sample quantiles. Then
q
[
j
]
¼ F
1
((
j
0
:
5)
=N
),
j ¼
1,
...
,
N
x
p
are the corresponding theoretical quantiles (where 0
:
5in(
j
0
:
5)
=N
) is a commonly
used continuity correction). Then a plot of the points (
q
[
j
]
,
'
n
,[
j
]
) should resemble a
straight line through the origin having slope 1. Particularly, the theoretical (1
a
)th
quantile should be close to the corresponding sample quantile (e.g.
a¼
0
:
05).
Figure 2.2 depicts such chi-square plots when sampling from circular
T
k
,
n
distribution
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