Information Technology Reference
In-Depth Information
For the Gaussian kernel
α
(
K
)=0
.
7764.The
β
(
f
) factor is usually un-
known. One could consider iteratively improving an initial estimate of
β
(
f
) as discussed in [224].
7. The optimal IMSE in the conditions of 6, attains its smallest value for the
Epanechnikov kernel defined as:
K
e
(
t
)=
√
5
√
5
4
√
5
1
5
t
2
,
3
1
−
−
≤
t
≤
.
(E.12)
0
,
otherwise
The monograph [221] indicates the eciencies (optimal IMSE ratios) of
other kernels compared to
K
e
. The eciency of the Gaussian kernel is
≈
0
.
9512.
f
n
are asymptotically
8. The deviations (residues) of the MSE-consistent
normal:
f
n
(
x
)
[
f
n
(
x
)]
σ
[
f
n
(
x
)]
−
E
lim
n→∞
P
≤ c
=
g
(
c
;0
,
1)
.
(E.13)
9. By the bounded difference inequality the following result holds:
P
≥
E
t
2
e
−t
2
/
2
n
(
|K|
)
2
|
f
n
−
f
|−
|
f
n
−
f
|
≤
.
(E.14)
10. Parzen window estimates are stable: suppose that one of the
X
i
changes
value while the other
n
1 data instances remain fixed; let
f
n
denote the
−
new perturbed estimate; then:
|
n
|
f
n
−
f
n
|≤
2
K
|
.
The above properties are discussed in detail in the following references: prop-
erties 1 through 7, in [224]; property 8 in [235]; properties 9 and 10 in [53].
The eciency of the Gaussian kernel mentioned in property 7, together
with the ease of implementation and the ease of analysis resulting from the
remarks mentioned in properties 2, 4, and 6, justify its popularity.
E.2 Optimal Bandwidth and IMSE
The choice of a kernel bandwidth adequate to the sample size
n
is of the
utmost importance in accurate Parzen window estimation of PDFs. For this
purpose, and assuming the kernel function to satisfy the conditions mentioned
in property 6, we apply formulas (E.10) and (E.11) to the computation of
the optimal bandwidth. The corresponding IMSE formula for
r
=2, is [224]:
∞
4
h
4
∞
−∞
f
(
x
)
1
nh
K
2
(
y
)
dy
+
1
2
IMSE
=
dx .
(E.15)
−∞
