Information Technology Reference
In-Depth Information
Tabl e E. 1
Optimal
h
and
IMSE
(per
n
) for the Gamma distribution with
b
=1
hIMSEσ
=
b
√
a
1 0.5000 0.8918 0.7908
a
2
1.0000
2 1.2500 0.7425 0.9498
1.4142
3 0.1875 1.0851 0.6499
1.7321
4 0.0313 1.5528 0.4542
2.0000
We now proceed to simplify these formulas, using the following simplified
notation:
α, β
instead of
α
(
K
),
β
(
f
) ;
I
K
=
K
2
;
I
2
=
f
2
.Theoptimal
h
and IMSE can then be written as:
h
=
I
K
I
2
0
.
2
nh
+
h
4
I
2
5
h
4
4
IMSE
=
I
K
n
−
0
.
2
;
=
I
2
.
(E.16)
4
For the Gaussian kernel we have:
h
=
0
.
282095
I
2
0
.
2
n
−
0
.
2
;
IMSE
=0
.
454178
I
0
.
2
n
−
0
.
8
.
(E.17)
Thus, in order to compute the optimal bandwidth and IMSE when using a
Gaussian kernel, the only thing that remains to be done is to compute
I
2
.For
instance, for the Gaussian density with standard deviation
σ
, we compute:
f
(
y
)
2
dy
0
.
212
σ
−
5
|
|
≈
⇒
β
(
f
)
≈
1
.
3637
σ.
(E.18)
Hence:
IMSE
=
0
.
332911
σ
h
=1
.
0592
σn
−
0
.
2
;
n
−
0
.
8
.
(E.19)
The diculty with formulas (E.17) is the need to compute
I
2
dependent
on the second derivative of
f
(
x
) which may be unknown. For symmetrical
PDFs, reasonably close to the Gaussian PDF, formulas (E.19) are satisfactory.
For other distributions one has to compute the integral
I
2
.Asanexample,
Table E.1 shows the optimal
h
and
IMSE
,per
n
, for a few cases of the
gamma distribution:
1
b
a
Γ
(
a
)
x
a−
1
e
−x/b
γ
(
x
;
a, b
)=
for
x>
0
,
with
a>
0(shape)
,b>
0(scale)
.
(E.20)
Note that
a
=1corresponds to the special exponential case of the gamma
distribution.
Let us consider the
a
=2case. For equal
σ
one obtains for the normal
distribution, per
n
,
h
=1
.
4979 (with
IMSE
=0
.
2354). The lower
h
for the
