Digital Signal Processing Reference
In-Depth Information
since
P
F
n
(
x
)
¼
1
=n 8x ¼ x
i
,
i ¼
1,
...
,
n
. The value of
T
a
v
e
at
F
1
,
t
is
ð
xd
[(1
1
)
F þ1D
t
](
x
)
¼
(1
1
)
ð
xdF
(
x
)
þ1
ð
xdD
t
(
x
)
ð
xdF
1
,
t
(
x
)
¼
T
a
v
e
(
F
1
,
t
)
¼
¼
(1
1
)
T
(
F
)
þ1t
Hence
IF
(
t
;
T
a
v
e
,
F
)
¼
@
@1
T
a
v
e
(
F
1
,
t
)
j
1¼
0
¼ t m
since
T
a
v
e
(
F
)
¼ m
(as the expected value of the symmetric c.d.f
F
is equal to the sym-
metry center
m
of
F
). The IF for the median
T
med
(
) is well-known to be
<
1
2
f
(
m
)
,
t
,
m
IF(
t
;
T
med
,
F
)
¼
(2
:
16)
0,
t ¼ m
:
1
2
f
(
m
)
,
t
.
m
,
If the c.d.f.
F
is the c.d.f. of the standard normal distribution
F
(i.e.,
m¼
0), then the
above IF expressions can be written as
IF(
t
;
T
a
v
e
,
F
)
¼ t
,
r
p
2
IF(
t
;
T
med
,
F
)
¼
sign(
t
)
:
These are depicted in Figure 2.3. The median has bounded IF for all possible values
of the contamination
t
, where as large outlier
t
can have a large effect on the mean.
B
EXAMPLE 2.4
IF of the covariance matrix
. Let
T
(
F
)
¼C
(
F
) be our statistical functional of
interest. The value of
C
(
F
)
¼
Ð
zz
H
dF
(
z
) at the
1
-point-mass distribution is
ð
zz
H
d
[(1
1
)
F þ1D
t
](
z
)
¼
(1
1
)
ð
zz
H
dF
(
z
)
þ1
ð
zz
H
dD
t
(
z
)
C
(
F
1
,
t
)
¼
¼
(1
1
)
C
(
F
)
þ1tt
H
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