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Parzen window method with the Gaussian kernel [246, 245]. The estimate
f
n
(
x
) is then written as (see Appendix E):
G
x − x
i
h
=
n
n
1
nh
1
n
f
n
(
x
)=
G
h
(
x
−
x
i
)
,
(F.2)
i
=1
i
=1
where
G
h
(
x
) is the Gaussian kernel of bandwidth
h
(same role as the standard
deviation of the Gaussian PDF)
√
2
πh
exp
2
h
2
.
x
2
1
G
h
(
x
)=
−
(F.3)
f
n
(
x
) in the formula of
H
R
2
(
x
), one obtains the
Substituting the estimate
integral estimate:
1
n
x
i
)
2
ln
+
∞
−∞
n
H
R
2
(
X
)=
−
G
h
(
x
−
dx
=
i
=1
⎡
⎤
(F.4)
n
x
i
)
2
n
2
+
∞
−∞
1
⎣
⎦
.
=
−
ln
G
h
(
x
−
dx
i
=1
Since we have a finite sum we may interchange sum and integration and write
this integral estimate as:
⎡
⎤
+
∞
n
n
1
n
2
H
R
2
(
X
)=
⎣
⎦
.
−
ln
G
h
(
x
−
x
i
)
G
h
(
x
−
x
j
)
dx
(F.5)
−∞
i
=1
j
=1
We now use the following theorem (a stronger version is proved in [245]):
Theorem F.1.
Let
g
(
x
;
a, σ
)
and
g
(
x
;
b, σ
)
be two Gaussian functions with
equal variance. The integral of their product is a Gaussian whose mean is
the difference of the means,
a
−
b
, and whose variance is the double of the
original variance,
2
σ
2
.
Proof.
We have
2
πσ
2
exp
.
a
)
2
+(
x
b
)
2
1
1
2
(
x
−
−
g
(
x
;
a, σ
)
g
(
x
;
b, σ
)=
−
(F.6)
σ
2
By adding and subtracting (
a
+
b
)
2
/
2 to the numerator of the exponent, we
express it as 2(
x
(
a
+
b
)
/
2)
2
+(
a
b
)
2
/
2. Therefore:
−
−
