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family (of the first kind), defined for
x
∈
[0
,
1] and
α, β >
0 as
f
(
x
)=
x
α−
1
(1
x
)
β−
1
/B
(
α, β
). Variance and entropy for this subtype are given in
Sect. D.2.2. Beta densities can be symmetric, asymmetric, convex or concave.
Under certain conditions on (
α, β
) the entropy is
not
an up-saturating func-
tion of the variance, or even not a saturating function at all. For instance, for
α
=
β
and
α<
1 (U-shaped densities) the variance increases with decreasing
entropy! For
α
−
β
one can write
V
=
β/α
2
and
H
(
α, β
) = ln
Γ
(
β
)+
ψ
(
β
),
which is then an up-saturating function of
V
. The same holds for
β
α
(both
V
and
H
are symmetric in
α, β
). Numerical computation shows that
it is enough that one of the parameters is less than one-half of the other for
the result to hold.
Type II:
Δ>
0
,b
=0.
A version of type I corresponding to the density
f
(
x
)=
K
(1
x
2
/a
2
)
m
,
−
−
1. Variance and entropy are given in Sect. D.2.2.
For fixed
m
a family of PDFs with the same basic shape is obtained and the
entropy is in this case an up-saturating function of the variance. For fixed
a
the shape of the PDFs varies a lot with
m
and the entropy is not a saturating
function of the variance. Note that the uniform density is a special case of
this type and its entropy is a USF of
V
.
a<x<a, a>
0
,m>
−
=0.
There are no known formulas for the variance and entropy of the general
solution,
f
(
x
)=
K
(1+
x/a
)
ma
exp(
Type III:
Δ>
0 with
c
=0
,a,b
,a,m>
0. A subtype
is the Gamma family,
f
(
x
)=
x
k−
1
e
−x/θ
/
(
θ
k
Γ
(
k
)) for
x>
0
,k
(shape),
θ
(scale)
>
0,with
V
(
k, θ
)=
kθ
2
and
H
(
k, θ
)=
k
+ln
θ
+ln
Γ
(
k
)+(1
−
mx
),
−
a<x<
∞
k
)
ψ
(
k
).
The gamma distribution family is often used for PDF modeling. Particular
cases of the Gamma family are the exponential, the Erlang, the chi-square
−
and the Maxwell-Boltzmann families. For fixed
k
the entropy is ln(
√
V
) plus a
constant, which is an obvious up-saturating function. For fixed
θ
the entropy
canbewrittenintermsof
v
=
v/θ
2
as
H
(
v, θ
)=
v
+ln
θ
+ln
Γ
(
v
)+(1
−
v
)
ψ
(
v
).
Since
dH/dv
=1+(1
−
v
)
ψ
1
(
ν
) is a decreasing function, the entropy is also
in this case a USF of
V
.
Type IV:
Δ<
0.
The general solution is
f
(
x
)=
K
(1+(
x/a
)
2
)
−m
exp(
k
atan(
x/a
))
,a,k>
0, which describes a family of long-tailed distributions, where
a
is a scale
parameter,
k
an asymmetry parameter and
m
determines the tails (long tails
for small
m
). The Pearson Type IV is used in several areas (namely economics
and physics) whenever one needs to model empirical distributions with long
tails.
The best way to mathematically handle the Pearson Type IV family is
by performing the variable transformation tan
θ
=
x/a
[243]. The value of
K
is defined only for
m ≥
1
/
2 and is given by
K
=1
/
(
aF
(
r, k
)) with
r
=
2
m
−
2 and
F
(
r, k
)=
π/
2
π/
2
e
−kθ
cos
r
(
θ
)
dθ
(closed form expression only for
integer
r
). The variance is given by
V
=
a
2
(
r
2
+
k
2
)
/
[
r
2
(
r
−
−
1)] for
m>
3
/
2.
