The name is absent



Skew Generalized Secant Hyperbolic. A random variable X has a skew generalized
secant hyperbolic distribution with parameters к > 0 and λ > — π is its density is

/ ɪ ʌ ___________Cl___________

∖ 1 + ɪ J a + cosh (c2 κ~ssn(x')χ) ,

where

a = cos(A),

« = 1,

a = cosh(λ)


and

Cl

sin(λ)

= ^Γc2

if λ ∈

( —7Γ, 0),

and

Cl

= C2

if λ =

0,

and

Cl

sinh(λ)

= ^^C2

if λ >

0.


The skew logistic distribution is a special case of this distribution when к ψ 1 and
ʌ = 0.

The cdf of X is given by


-P1SGSH («) =


(τ⅛∙) g(kx)

< 1 - fττ⅛) G(-)
у 1+я 2 J V к >


for X < 0,

for X > 0,


where


ɪ + ɪ tan 1 [tan (ʌ) tanh (^-æ)]


G(x) = < 1 [1 ÷ tanh (^-æ)]


ɪ + ɪ tanh 1 [tanh (ʌ) tanh (⅛Jr)]


for λ ∈ ( —7Γ, 0),
for λ = 0,
for λ > 0,


is the cdf of the symmetric generalized secant hyperbolic distribution with zero mean
and unit variance obtained when к = 1. Moreover, the moments of order r, r = 1, 2,...,
are:


E(Xr)


c1 Γ(r + 1)
2⅛+1√α2 - 1



where Lr (∙) indicates the polylogarithmic function whose primary definition is1


Iw∣ <1, w ∈ (D.


Skew Power Exponential. A random variable X has a skew exponential power
distribution (SEP) with parameters к > 0 and β > 0 is its density is

Jsep (^) = (ɪ) r = r °2 ɑə '' eχP [~cβκ 2-sr",∙''i∣.∕∙∣2-] ,

with c = Γ J Γ 1 ɪhe skew Laplace distribution is a special case of the SEP
when к ≠ 1 and β = ɪ. Likewise, when β = 1 we obtain a normal distribution.

1See at the web page http://functions.wolfram.com/10.08.02.0001.01



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