Information Technology Reference
In-Depth Information
There is no known closed form of the entropy, which can be written as
H
=
−
I
(
r, k
)
/F
(
r, k
)+ln
a
+ln
F
(
r, k
) (
I
(
r, k
) is defined in D.2.2). Therefore, in
terms of
a
,
H
(
V
) is clearly USF. Setting
k
=1, numerical computation leads
to the conclusion that
H
(
V
) is also USF in terms of
r
(and therefore of
m
,
the parameter controlling the tails).
Student's
t
-distribution is the special symmetrical case (
k
=0)of this
family. It is easy to check that the entropy of the Student's
t
distribution is
aUSFof
V
.
Type V:
Δ
=0.
Corresponds
to
the
Inverse
Gamma
family
f
(
x
)
=
1
β
α
β/x
)
/Γ
(
α
),with
x>
0
,α
(shape),
β
(scale)
>
0.The
variance and the entropy are given in Sect. D.2.2. Note that in order for
f
(
x
) to have variance
α
must be larger than 2. For fixed
α
,theentropyis
x
α
+1
exp(
−
ln(
√
V
) plus a constant, which is an obvious USF. For fixed
β
,numerical
computation shows that the entropy is also a USF of
V
.
Type VI:
Δ>
0 and
x
a
2
, the larger root.
Corresponds to
f
(
x
)=
Kx
−q
1
(
x
≥
a
)
q
2
,with
q
1
<
1
,q
2
>
−
−
1
,q
1
>q
2
−
1,
and
x
]. There is no known entropy formula for the general solution.
A particular subtype is the Beta distribution of the second kind (also known
as Beta prime distribution), with
f
(
x
)=
x
α−
1
(1+
x
)
−α−β
/B
(
α, β
) for
x
∈
]
a,
∞
0.
The variance is given in Sect. D.2.2; there is no known formula of the entropy.
Numerical computation shows that
H
(
V
) is a USF function both in terms of
α
and
β
.
Another special subtype is the
F
distribution family, whose variance and
entropy are given in Sect. D.2.2. Numerical computation shows that the en-
tropy is a USF of
V
, in relation to either of the two parameters (degrees of
freedom) of this family.
≥
Type VII:
Δ>
0
,b
=0
,c>
0
.
Corresponds to
f
(
x
)=
K
(1+
x
2
/a
2
)
−m
,with
a
=0
,m>
1
/
2. A particular
case is the Student's
t
-density already analyzed in type IV. The entropy of
Type VII distributions is a USF of the variance for fixed
a
or
m
.
Type VIII:
Δ>
0 and
a
1
=
a
2
=
m
.
Corresponds to
f
(
x
)=
K
(1 +
x/a
)
−m
,with
m>
1 and
x ∈
]
− a,
0].
These are asymmetric distributions, with hyperbolic shape sharply peaked
for high
m
. Setting w.l.o.g.
a
=1(
a
simply controls where the peak occurs)
and
x
−
]0
,
1[ (the area is infinite for
=0), one
obtains a family of distributions with variance defined for
m>
2.Numerical
computation shows that the entropy of the family is then an up-saturating
function of the variance.
∈
[
−
1+
,
0],with
∈
Type IX:
Δ>
0 and
a
1
=
a
2
=
m
.
A version of Type VIII, with
f
(
x
)=
K
(1 +
x/a
)
m
,m>
−
1 and
x
∈
]
−
a,
0]. The area is finite for
m
≥
0, which affords simpler formulas for
