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the variance and entropy (given in Sect. D.2.2). The entropy is a USF of the
variance.
Type X:
Δ
=0and
b
=
c
=0.
Corresponds to the generalized exponential family,
f
(
x
)=
σe
−
(
x−m
)
/σ
,
with
x
[ and
σ>
0. As already noted for type III, the entropy is an
up-saturating function of the respective variance (
H
is insensitive to
m
;see
also remark on property (ii) of Sect. D.1).
Type XI:
Δ
=0and
b
=
c
=
m
=0.
Corresponds to
f
(
x
)=
Kx
−α
,with
α>
1
,x
∈
[
m,
∞
[
, >
0.Variance
(defined for
α>
3) and entropy formulas are easily obtained (see Sect. D.2.2).
The entropy is a USF of the variance. A special case of this type is the Pareto
family of distributions.
∈
[
,
∞
<
1,
and contains as special case the Beta distributions with
α
=
β
.Whatwesaid
for type I also apply here.
Type XII: is essentially a version of type I with
m
1
=
m
2
=
m,
|
m
|
D.2.2 List of PDFs with USF
H
(
V
)
The following list covers by alphabetical order most of the PDF families
described in the literature, namely those having closed form integrals for the
variance and the entropy. In a few cases where no formulas were available we
had to resort to numerical computation. The list is by no means intended to
be an exhaustive list of PDFs with up-saturating
H
(
V
) (where up-saturating
is understood in relation to all parameters of interest except when stated
otherwise).
1. Beta distribution (also known as Beta distribution of the first kind).
PDF:
f
(
x
;
α, β
)=
x
α
−
1
(1
−x
)
β
−
1
B
(
α,β
)
;
x
∈
[0
,
1];
α, β >
0.
V
:
αβ
(
α
+
β
)
2
(
α
+
β
+1)
.
H
:ln(
B
(
α, β
))
2)
ψ
(
α
+
β
).
Comment: A special case of Pearson Type I, corresponding to symmetric
distributions in
α, β
.
H
(
V
) is USF only for
α
−
(
α
−
1)
ψ
(
α
)
−
(
β
−
1)
ψ
(
β
)+(
α
+
β
−
2
α
.
2. Beta Prime distribution (also known as Beta distribution of the second
kind).
PDF:
f
(
x
;
α, β
)=
x
α
−
1
(1+
x
)
−
α
−
β
B
(
α,β
)
2
β
or
β
;
x>
0;
α, β >
0.
1)
(
β−
2)(
β−
1)
2
;
β>
2.
H
:(noknownformula).
Comment: a special case of Pearson Type VI.
α
(
α
+
β
−
V
:
3. Chi distribution.
PDF:
f
(
x
;
k
)=
2
1
−
k/
2
x
k
−
1
e
−
x
2
/
2
Γ
(
k/
2)
;
x
≥
0
,k>
0.
