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The term generalized logistic distribution is used as the name for several different families of probability distributions . For example, Johnson et al.[ 1] skew-logistic distribution . For other families of distributions that have also been called generalized logistic distributions, see the shifted log-logistic distribution , which is a generalization of the log-logistic distribution .
Definitions
The following definitions are for standardized versions of the families, which can be expanded to the full form as a location-scale family . Each is defined using either the cumulative distribution function (F ) or the probability density function (ƒ ), and is defined on (-∞,∞).
Type I
F
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1
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≡
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−
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α
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0.
{\displaystyle F(x;\alpha )={\frac {1}{(1+e^{-x})^{\alpha }}}\equiv (1+e^{-x})^{-\alpha },\quad \alpha >0.}
The corresponding probability density function is:
f
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α
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=
α
e
−
x
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1
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e
−
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α
+
1
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0.
{\displaystyle f(x;\alpha )={\frac {\alpha e^{-x}}{\left(1+e^{-x}\right)^{\alpha +1}}},\quad \alpha >0.}
This type has also been called the "skew-logistic" distribution.
Type II
F
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α
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=
1
−
e
−
α
x
(
1
+
e
−
x
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α
,
α
>
0.
{\displaystyle F(x;\alpha )=1-{\frac {e^{-\alpha x}}{(1+e^{-x})^{\alpha }}},\quad \alpha >0.}
The corresponding probability density function is:
f
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x
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α
)
=
α
e
−
α
x
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1
+
e
−
x
)
α
+
1
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α
>
0.
{\displaystyle f(x;\alpha )={\frac {\alpha e^{-\alpha x}}{(1+e^{-x})^{\alpha +1}}},\quad \alpha >0.}
Type III
f
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x
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α
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=
1
B
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α
,
α
)
e
−
α
x
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1
+
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2
α
,
α
>
0.
{\displaystyle f(x;\alpha )={\frac {1}{B(\alpha ,\alpha )}}{\frac {e^{-\alpha x}}{(1+e^{-x})^{2\alpha }}},\quad \alpha >0.}
Here B is the beta function . The moment generating function for this type is
M
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t
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=
Γ
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α
−
t
)
Γ
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α
+
t
)
(
Γ
(
α
)
)
2
,
−
α
<
t
<
α
.
{\displaystyle M(t)={\frac {\Gamma (\alpha -t)\Gamma (\alpha +t)}{(\Gamma (\alpha ))^{2}}},\quad -\alpha <t<\alpha .}
The corresponding cumulative distribution function is:
F
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x
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α
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=
(
e
x
+
1
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Γ
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α
)
e
α
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−
x
)
(
e
−
x
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1
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−
2
α
2
F
~
1
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1
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1
−
α
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α
+
1
;
−
e
x
)
B
(
α
,
α
)
,
α
>
0.
{\displaystyle F(x;\alpha )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\alpha (-x)}\left(e^{-x}+1\right)^{-2\alpha }\,_{2}{\tilde {F}}_{1}\left(1,1-\alpha ;\alpha +1;-e^{x}\right)}{B(\alpha ,\alpha )}},\quad \alpha >0.}
Type IV
f
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x
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α
,
β
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=
1
B
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α
,
β
)
e
−
β
x
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1
+
e
−
x
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α
+
β
,
α
,
β
>
0.
{\displaystyle f(x;\alpha ,\beta )={\frac {1}{B(\alpha ,\beta )}}{\frac {e^{-\beta x}}{(1+e^{-x})^{\alpha +\beta }}},\quad \alpha ,\beta >0.}
Again, B is the beta function . The moment generating function for this type is
M
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t
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=
Γ
(
β
−
t
)
Γ
(
α
+
t
)
Γ
(
α
)
Γ
(
β
)
,
−
α
<
t
<
β
.
{\displaystyle M(t)={\frac {\Gamma (\beta -t)\Gamma (\alpha +t)}{\Gamma (\alpha )\Gamma (\beta )}},\quad -\alpha <t<\beta .}
This type is also called the "exponential generalized beta of the second type".[ 1]
The corresponding cumulative distribution function is:
F
(
x
;
α
,
β
)
=
(
e
x
+
1
)
Γ
(
α
)
e
β
(
−
x
)
(
e
−
x
+
1
)
−
α
−
β
2
F
~
1
(
1
,
1
−
β
;
α
+
1
;
−
e
x
)
B
(
α
,
β
)
,
α
,
β
>
0.
{\displaystyle F(x;\alpha ,\beta )={\frac {\left(e^{x}+1\right)\Gamma (\alpha )e^{\beta (-x)}\left(e^{-x}+1\right)^{-\alpha -\beta }\,_{2}{\tilde {F}}_{1}\left(1,1-\beta ;\alpha +1;-e^{x}\right)}{B(\alpha ,\beta )}},\quad \alpha ,\beta >0.}
See also
References
^ a b Johnson, N.L., Kotz, S., Balakrishnan, N. (1995) Continuous Univariate Distributions, Volume 2 , Wiley. ISBN 0-471-58494-0 (pages 140–142)
Discrete
with finite with infinite
Continuous
supported on a supported on a supported with support
Mixed
Multivariate Directional Degenerate singular Families
