Information Technology Reference
In-Depth Information
Appendix E
Optimal Parzen Window Estimation
E.1 Parzen Window Estimation
Let us consider the problem of estimating a univariate PDF
f
(
x
) of a contin-
uous r.v.
X
based on a i.i.d. sample
X
n
=
{x
1
, ..., x
n
}
. Although unbiased
estimators do not exist in general for
f
(
x
) (see e.g., [181]), it is possible
nonetheless to define sequences of density estimators,
f
n
(
x
)
≡
f
n
(
x
;
X
n
),
asymptotically unbiased
:
n→∞
E
X
[
f
n
(
x
)] =
f
(
x
)
.
lim
(E.1)
The Parzen window estimator [169] provides such an asymptotically unbiased
estimate. It is a generalization of the shifted-histogram estimator (see e.g.,
[224]), defined as:
h
K
x
,
n
f
n
(
x
)=
1
n
1
−
x
i
(E.2)
h
i
=1
where the positive constant
h
h
(
n
) is the so-called
bandwidth
of
K
(
x
),
the
Parzen window
or
kernel function
, which is any Lebesgue measurable
function satisfying:
≡
R
|
K
(
x
)
|
<
∞
i Boundedness: sup
;
1
space, i.e.,
|
ii
K
(
x
) belongs to the
L
K
(
x
)
|
<
∞
;
iii
K
(
x
) decreases faster than 1
/x
: lim
x→∞
|
xK
(
x
)
|
=0;
iv
K
(
x
)=1.
x
2
/
2)
For reasons to be presented shortly, the Gaussian kernel,
G
(
x
)=exp(
−
/
√
2
π
, is a popular choice for kernel function
K
(
x
).
Sometimes the Parzen window estimator is written as
h
K
x
,
n
f
n
(
x
)=
1
n
1
−
X
i
(E.3)
h
i
=1
