On the symmetric and skew-symmetric K-distributions

Abstract

This paper introduces the four-parameter symmetric K-distribution (SKD) and the skew-SKD as models for describing the complex dynamics of machine learning, Bayesian analysis and other fields through simplified expressions with high accuracy. Formulas for the probability density function, cumulative distribution function, moments, and cumulants are given. Finally, an order statistics analysis is provided as well as the distributions of the product and ratio of two SKD random variables are derived.

Full text

Submitted to the Annals of Statistics
ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS
BYSTYLIANOS E. TREVLAKIS1,*, GEORGE K. KARAGIANNIDIS1,†AND NE STOR
CHATZIDIAMANTIS1,‡
1Aristotle University of Thessaloniki, *trevlakis@ece.auth.gr;†geokarag@auth.gr;‡nestoras@auth.gr
This paper introduces the four-parameter symmetric K-distribution
(SKD) and the skew-SKD as models for describing the complex dynamics
of machine learning, Bayesian analysis and other fields through simplified
expressions with high accuracy. Formulas for the probability density func-
tion, cumulative distribution function, moments, and cumulants are given.
Finally, an order statistics analysis is provided as well as the distributions of
the product and ratio of two SKD random variables are derived.
1. Introduction. Mixture distributions enable the modeling of complex dynamics
through more simplified mathematical expressions. This way they offer tractable solutions
in applications where the individual weighted components of the problem exhibit different
characteristics. Mixture distributions play an important role in several scientific fields, such
as machine learning [5], Bayesian analysis [14,11], wireless communications [19,17], and
econometric models [6]. A discrete or finite mixture distribution is a linear combination of
two or more distributions, i.e.
(1) f(x) =
n
X
i=1
ωifi(x); 0 < ωi<1;
n
X
i=1
ωi= 1
where the weights ωican be considered as the probabilities of nsets of the distribution
parameters, scale, shape and location. Gaussian discrete mixtures are the most well-known
and can be applied in several fields as for example in signal processing.
If the probabilities ω1are continuous random variables (RVs) and belong to another distri-
bution, then the resulting mixture is termed continuous mixture distribution. Next, we focus
on the case where one of the parameters, denoted by Θ, is random. Then, the conditional
probability density function (PDF) fX(x|θ)is termed parental distribution and the PDF of
Θ, denoted by fΘ(θ), is termed prior distribution. In this case the joint PDF of Xand Θis
(2) f(x, θ) = fX(x|θ)fΘ(θ)
and the marginal PDF of Xis
(3) f(x) = Zf(x, θ)dθ=ZfX(x|θ)fΘ(θ)dθ.
A comprehensive summary of discrete and continuous mixture distributions can be found
in [10].
Since its introduction [9], the K distribution has proved to be remarkably useful for mod-
eling the complex dynamics of various systems, such as wireless communications channel
modeling [4,18] and radar applications [20]. Over the last couple of years, various deriva-
tive distributions of the K distribution have attracted attention due to their successful use in
machine learning-based ultrasound image reconstruction [21], as well as heat transfer [12].
MSC2020 subject classifications:Primary 60E05, 62E10; Secondary 62F15.
Keywords and phrases: Symmetric K Distribution, Skew Symmetric K Distribution, Order statistics, Product
distribution, Ratio distribution, Bayesian analysis.
1
arXiv:2107.02092v1 [math.ST] 5 Jul 2021

2
- 1 0 - 5 0 5 1 0
0 , 1
0 , 2
0 , 3
0 , 4
0 , 5
fx( x )
x
γ = 0 , α = 1 , ζ = 1 , δ = 1
γ = 0 , α = 0 .5 , ζ = 1 , δ = 1
γ = 0 , α = 3 , ζ = 1 , δ = 1
γ = - 5 , α = 1 , ζ = 1 , δ = 1
γ = - 5 , α = 1 , ζ = 1 , δ = 0 . 5
γ = - 5 , α = 1 , ζ = 1 , δ = 3
- 1 0 - 5 0 5 1 0
0 , 1
0 , 2
0 , 3
0 , 4
0 , 5
0 , 6
0 , 7
0 , 8
0 , 9
1 , 0
Fx( x )
x
γ = 0 , α = 1 , ζ = 1 , δ = 1
γ = 0 , α = 0 .5 , ζ = 1 , δ = 1
γ = 0 , α = 3 , ζ = 1 , δ = 1
γ = - 5 , α = 1 , ζ = 1 , δ = 1
γ = - 5 , α = 1 , ζ = 1 , δ = 0 . 5
γ = - 5 , α = 1 , ζ = 1 , δ = 3
Fig 1: The (a) PDF and (b) CDF of the SKD for different parameter values.
Both the symmetric counterpart of the K-distribution, namely SKD, and the skew-SKD
that are introduced in this work exhibit flexibility in a wide variety of applications, including
data fitting [2], strength-stress modeling [8], Bayesian learning [13], and more. In this paper,
we introduce the 4-parameter symmetric K-distribution (SKD) as the mixture of the parental
3-parameter reflected Gamma and the prior 2-parameter Gamma distributions. We derive the
PDF, cumulative distribution function (CDF), moments, cumulants, order statistics, as well
as the product and ratio distributions. Moreover, we present and study the skew-SKD.
Notations: Throughout, let f(·)and F(·)stand for the PDF and CDF of the given RV.
Write (a)nfor the Pochhammer Symbol. Denote by exp (·)the exponential function, Γ (·)
the Gamma function, G(·)the Meijer-G function, p
Fq(·)for the generalised hypergeometric
function, Kv(·)for the K-function, and sgn (·)the sign function.
The rest of this paper is organised as follows. Section 2introduces the four-parameter SKD
alongside the formulas for its PDF, CDF, moments, cumulants, order statistics, product and
ratio distributions. The skew-SKD is defined in Section 3. Finally, concluding remarks are
provided in 4.
2. The symmetric K-distribution. We introduce the SKD as the mixture of the parental
three-parameter reflected Gamma (RG) and the prior two-parameter Gamma distributions.
The two-parameters Gamma PDF is given by [3, eq. 17.15]
(4) f(x, δ, ζ) = δζxζ−1
Γ(ζ)exp(−δx), δ, ζ > 0,
where δand ζare the scale and shape parameters, correspondingly.Also, the three-parameters
RG PDF is given as [10]
(5) f(x, α, β, γ ) = |x−γ|α−1
2βαΓ(α)exp −|x−γ|
β, α, β > 0,
where α, β and γare the shape, scale and location parameters, correspondingly, while, for
γ= 0 and β= 1 the standard form of the RG has the PDF
f(x;α) = 1
2Γ(α)|x|α−1e−|x|
(6)
and CDF
(7) F(x, α) = 1
2−sgn(x)Γ(α, |x|)
2Γ(α)

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 3
where sgn(x)denotes the sign function and is given by
sgn(x) = 1, x ≥0
−1, x < 0.(8)
2.1. PDF. According to (3), the PDF of the SKD can be evaluated as follows:
f(x) = Z∞
0
1
2|x−γ|α−1
βαΓ(α)exp −|x−γ|
βδζβζ−1
Γ(ζ)exp(−δβ)dβ
=δζ
2Γ (α)Γ(ζ)|x−γ|α−1Z∞
0
βζ−α−1exp −|x−γ|
β−δβdβ.
(9)
Given the fact that |x−γ|, δ ∈ < and |x−γ|>0, we can express the integrands according
to [15, Eq. 2.3.16.1]. Thus, the PDF of the four-parameter SKD can be written as
(10) f(x) = δα+ζ
2
Γ (α)Γ(ζ)|x−γ|α+ζ
2−1Kα−ζ2pδ|x−γ|,
where δand γare the scale and location parameters, and α,ζare shape parameters.
For γ= 0 and δ= 1, the PDF of the standard form of the SKD is given by
(11) fs(x) = 1
Γ (α)Γ(ζ)|x|α+ζ
2−1Kα−ζ2p|x|,
2.2. CDF. The CDF of the SKD can be derived directly from the PDF through
F(x) = Zx
−∞
f(x)dx.(12)
By using (9) it can be rewritten as
F(x) =

















Zx
γ
δα+ζ
2
Γ(α)Γ(ζ)(x−γ)α+ζ
2−1Kα−ζ(2pδ(x−γ))dx
+Zγ
−∞
δα+ζ
2
Γ(α)Γ(ζ)(γ−x)α+ζ
2−1Kα−ζ(2pδ(γ−x))dx
, x ≥γ
Zx
−∞
δα+ζ
2
Γ(α)Γ(ζ)(γ−x)α+ζ
2−1Kα−ζ(2pδ(γ−x))dx , x ≤γ
.(13)
From (13), we need to calculate three integrals, namely I4,I5, and I6.I4can be transformed
based on y=x−γ > 0into
I4=δα+ζ
2
Γ(α)Γ(ζ)Zx−γ
0
yα+ζ
2−1Kα−ζ(2pδy)dy.(14)
Next, the Kvfunction can be written in terms of the Meijer-G function as in [16, eqs. 8.4.23.1]
and, thus, (14) can be equivalently written as
I4=δα+ζ
2
2Γ(α)Γ(ζ)Zx−γ
0
yα+ζ
2−1G2,0
0,2δy −
α−ζ
2,−α−ζ
2dy.(15)
After performing the integration in (15) based on [1, eqs. 26], it can be rewritten as
I4=δα+ζ
2
Γ(α)Γ(ζ)(x−γ)α+ζ
2G2,1
1,3δ(x−γ)1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2.(16)

4
Furthermore, after transforming the integral I5based on z=γ−x < 0can be written as
I5=δα+ζ
2
Γ(α)Γ(ζ)Z∞
0
zα+ζ
2−1Kα−ζ(2√δz)dz,(17)
which, by expressing the Kvfunction in terms of Meijer-G as in [16, eqs. 8.4.23.1], can be
rewritten as
I5=δα+ζ
2
2Γ(α)Γ(ζ)Z∞
0
zα+ζ
2−1G2,0
0,2δz −
α−ζ
2,−α−ζ
2dz,(18)
and, since arg δ < π and δ6= 0, we can perform the integration based on [16, eqs. 2.24.2.1].
Thus, (18) can be equivalently written as
I5=δα+ζ
2
2Γ(α)Γ(ζ)δ−α+ζ
2Γα−ζ
2+α+ζ
2,−α−ζ
2+α+ζ
2
−
=1
2Γ(α)Γ(ζ)Γα−ζ
2+α+ζ
2Γ−α−ζ
2+α+ζ
2=1
2
(19)
For the third integral, I6, after applying the transformation k=γ−x > 0and writing the
Kvfunction in terms of Meijer-G as in [16, eqs. 8.4.23.1], it can be expressed as
I6=δα+ζ
2
2Γ(α)Γ(ζ)Z∞
γ−x
kα+ζ
2−1G2,0
0,2δk −
α−ζ
2,−α−ζ
2dk,(20)
which can be rewritten as
I6=I5−δα+ζ
2
2Γ(α)Γ(ζ)Zγ−x
0
kα+ζ
2−1G2,0
0,2δk −
α−ζ
2,−α−ζ
2dk,(21)
and by performing the integration based on [1, eqs. 26], (21) can be equivalently expressed
as
1
2−δα+ζ
2
2Γ(α)Γ(ζ)(γ−x)α+ζ
2G2,1
1,3δ(γ−x)1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2.(22)
Thus, the CDF of the SKD can be written as
F(x) = 








1
2+δα+ζ
2
Γ(α)Γ(ζ)(x−γ)α+ζ
2G2,1
1,3δ(x−γ)1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2, x ≥γ
1
2−δα+ζ
2
2Γ(α)Γ(ζ)(γ−x)α+ζ
2G2,1
1,3δ(γ−x)1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2, x ≤γ
.
(23)
and after some simplifications the CDF of the four-parameter SKD can be rewritten as
F(x) = 1
2+sgn (x−γ)δα+ζ
2
2Γ(α)Γ(ζ)|x−γ|α+ζ
2G2,1
1,3δ|x−γ|1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2.(24)
Furthermore, when α6=ζ, the Meijer-G function in (24) can be expressed in terms of the
more familiar 1
F2hypergeometric function. Also, since arg δ|x−γ|< π, the CDF can be

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 5
written as [16, eq. 8.2.2.3]
F(x) =1
2+sgn (x−γ)δα+ζ
2
2Γ(α)Γ(ζ)|x−γ|α+ζ
2πcsc(π(ζ−α))
(δ|x−γ|)α−ζ
2
α(α−ζ) Γ(α−ζ)1
F2(α;α−ζ+ 1, α + 1; δ|x−γ|)
−(δ|x−γ|)ζ−α
2
ζ(ζ−α) Γ(ζ−α)1
F2(ζ;−α+ζ+ 1, ζ + 1; δ|x−γ|)!
(25)
For γ= 0 and δ= 1 the CDF of the standard form of the SKD is given by
F(x) = 1
2+sgn (x)δα+ζ
2
2Γ(α)Γ(ζ)|x|α+ζ
2G2,1
1,3|x|1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2.(26)
2.3. N-th moment. The n-th Moment of the SKD can be derived as
µn=Z∞
−∞
xnf(x)dx.(27)
By substituting (10) into (27), it can be rewritten as
µn=δα+ζ
2
Γ (α)Γ(ζ)Zγ
−∞
xn(γ−x)α+ζ
2−1Kα−ζ2pδ(γ−x)dx
+δα+ζ
2
Γ (α)Γ(ζ)Z∞
γ
xn(x−γ)α+ζ
2−1Kα−ζ2pδ(x−γ)dx,
(28)
which, by using the transformations y=x−γ > 0and k=γ−x > 0, it can be equivalently
written as
µn=δα+ζ
2
Γ (α)Γ(ζ)Z∞
0
(γ−k)nkα+ζ
2−1Kα−ζ2√δkdk
+δα+ζ
2
Γ (α)Γ(ζ)Z∞
0
(γ+y)nyα+ζ
2−1Kα−ζ2pδydy.
(29)
Next, by using [7, eqs. 1.111], (29) can be rewritten as
µn=δα+ζ
2
Γ (α)Γ(ζ)
n
X
i=0 n
iγi(−1)n−iZ∞
0
kα+ζ
2−1+n−iKα−ζ2√δkdk
+δα+ζ
2
Γ (α)Γ(ζ)
n
X
i=0 n
iγiZ∞
0
yα+ζ
2−1+n−iKα−ζ2pδydy.
(30)
Furthermore, after using the substitution η=√y=√k, (30) can be expressed as
µn=2δα+ζ
2
Γ (α)Γ(ζ)
n
X
i=0 n
iγi(−1)n−i+ 1Z∞
0
η(α+ζ+2n−2i)−1Kα−ζ2√δηdη,
(31)
and, since (α+ζ+ 2n−2i),1
2√δ>0, by employing [16, eqs. 2.16.2.2], (31) can be rewrit-
ten as
µn=δi−n
2Γ (α)Γ(ζ)
n
X
i=0 n
iγi(−1)n−i+ 1Γ (α+n−i)Γ(ζ+n−i).(32)

6
Next, the previous equation can be simplified as
µn=
n
X
k=0
Γ(n+ 1)Γ(α+n−k)Γ(ζ+n−k)
Γ(k+ 1)Γ(n−k+ 1)Γ(α)Γ(ζ)
γk(−1)n−k+ 1
2δn−k.(33)
Finally, by using basic transformation of the Γfunction, the n-th moment of the four-
parameter SKD can be expressed as
µn=
n
X
k=0
n−k∈even
nγk(α)n−k(ζ)n−k
δn−kΓ(k+ 1) (n)1−k
,(34)
with (x)ndenoting the Pochhammer symbol.
Furthermore, the first four moments and the first four central moments are presented in
Table 1.
TABL E 1
Moments and central moments of the SKD.
Moment Value
µ1γ
µ2α(α+1)ζ(ζ+1)
δ2+γ2
µ33α(α+1)γζ (ζ+1)
δ2+γ3
µ46α(α+1)γ2ζ(ζ+1)
δ2+α(α+1)(α+2)(α+3)ζ(ζ+1)(ζ+2)(ζ+3)
δ4+γ4
Central moment Value
˜µ1=µ1γ
˜µ2=µ2−µ2
1α(α+1)ζ(ζ+1)
δ2
˜µ3=µ3−3µ1µ2+ 2µ3
10
˜µ4=µ4−4µ1µ3+ 6µ2
1µ2−3µ4
1α(α+1)(α+2)(α+3)ζ(ζ+1)(ζ+2)(ζ+3)
δ4
2.4. Cumulants. In addition, the cumulants of the SKD are given by
K(t) = log Eetx ,(35)
or, in terms of the N-th moments
κn=µn−
n−1
X
m=1 n−1
mκmµn−m,(36)
which, after substituting (34), can be written as
κn=
n
X
k=0
n−k∈even
n(α)n−k(ζ)n−k(1 −n)k
k(n−k) (−1)kΓ (k)−
n−1
X
m=1
n−m
X
k=0
n−m−k∈even
n−1
mκm
×(n−m) (α)n−m−k(ζ)n−m−k(1 −n−m)k
k(n−m−k) (−1)kΓ (k).
(37)

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 7
TABL E 2
Cumulants of the SKD.
Cumulant Value
κ1= ˜µ1γ
κ2= ˜µ2α(α+1)ζ(ζ+1)
δ2
κ3= ˜µ30
κ4= ˜µ4−3˜µ2
2(α+2)(α+3)(ζ+2)(ζ+3)
α(α+1)ζ(ζ+1) α2((ζ−1)ζ−3) −α(ζ(ζ+ 11) + 15) −3(ζ+ 2)(ζ+ 3)
Thus, the first four cumulants are presented in Table 2.
Finally, based on the aforementioned analysis, the mean, variance, skewness, and kurtosis
of the SKD are given in
µ=µ1=γ,(38)
σ2=˜µ2=α(α+ 1)ζ(ζ+ 1)
δ2,(39)
γ1=˜µ3
˜µ3/2
2
= 0,and(40)
β2=˜µ4
˜µ2
2
=(α+ 2)(α+ 3)(ζ+ 2)(ζ+ 3)
α(α+ 1)ζ(ζ+ 1) .(41)
2.5. Order Statistics. Order statistics are among the most fundamental tools in non-
parametric statistics and inference. The order statistics X(1), X(2) ,···, X(n), of any random
sample of random variables, X1, X2,...,Xn, are the same random variables sorted in in-
creasing order. Assuming that any such random sample follows the SKD, the CDF of that
random sample is given by
FX(r)(x) =
n
X
j−rn
j[F(x)]j[1 −F(x)]n−j,(42)
and, after substituting (24), it can be equivalently written as
FX(r)(x) =
n
X
j−rn
j1
2+ Λ (x)j1
2−Λ (x)n−j
,(43)
with
Λ (x) = sgn (x−γ)δα+ζ
2
Γ(α)Γ(ζ)|x−γ|α+ζ
2G2,1
1,3δ|x−γ|1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2.(44)
Furthermore, the PDF of a random sample can be expressed as
fX(r)(x) = n!
(r−1)! (n−r)!f(x) [F(x)]r−1[1 −F(x)]n−r,(45)
and, after substituting (10) and (24), it can be rewritten as
fX(r)(x) = n!
(r−1)! (n−r)!
sgn (x−γ)δα+ζ
2
Γ (α)Γ(ζ)|x−γ|α+ζ
2−1
×Kα−ζ2pδ|x−γ|1
2+ Λ (x)r−11
2−Λ (x)n−r.(46)

8
2.6. Distribution of the product and ratio of two SKD RVs. In this section we derive
the PDF and CDF of the product and ratio distributions of the SKD. These two types of
distributions are extensively applied in machine learning and Bayesian analysis problems,
such as posterior distribution and density estimation.
THEOREM 2.1. The PDF of the product of two zero mean iid variables that follow the
SKD is given by
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2G4,0
0,4δ|z|−
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2.(47)
PROO F. The pdf of the product of two iid variables that follow the symmetric can be
expressed as
fZ(z) = Z∞
−∞
fX(x)fYz
x1
|x|dx,(48)
where z=xy. Next, after substituting (10) into (48), the latter can be rewritten as
fZ(z) = δα+ζ
[Γ (α)Γ(ζ)]2Z∞
−∞ |x−γ|z
x−γα+ζ
2−1
|x|
×Kα−ζ2pδ|x−γ|Kα−ζ2rδ
z
x−γdx,
(49)
Furthermore, by expressing the Kvfunction in terms of the Meijer-G function as in [16, eqs.
8.4.23.1], the previous equation can be equivalently written as
fZ(z) = δα+ζ
[2Γ (α)Γ(ζ)]2Z∞
−∞ |x−γ|z
x−γα+ζ
2−1
|x|
×G2,0
0,2δ|x−γ|−
α−ζ
2,−α−ζ
2G2,0
0,2δ
z
x−γ−
α−ζ
2,−α−ζ
2dx.
(50)
It becomes evident that the evaluation of a closed form expression for the integral pre-
sented in (50) is very challenging. However, if we assume zero mean symmetric K iid vari-
ables, (50) can be expressed as
fZ(z) = δα+ζ|z|α+ζ
2−1
[2Γ (α)Γ(ζ)]2Z∞
−∞
1
|x|G2,0
0,2δ|x|−
α−ζ
2,−α−ζ
2G2,0
0,2δ|z|
|x|−
α−ζ
2,−α−ζ
2dx,
(51)
or, equivalently
fZ(z) = δα+ζ|z|α+ζ
2−1
[2Γ (α)Γ(ζ)]2Z∞
0
1
xG2,0
0,2δx −
α−ζ
2,−α−ζ
2G2,0
0,2δ|z|
x−
α−ζ
2,−α−ζ
2dx
−Z0
−∞
1
xG2,0
0,2−δx −
α−ζ
2,−α−ζ
2G2,0
0,2−δ|z|
x−
α−ζ
2,−α−ζ
2dx,
(52)
and, after using the substitution x=−xon the second integral, (52) can be rewritten as
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2Z∞
0
1
xG2,0
0,2δx −
α−ζ
2,−α−ζ
2G2,0
0,2δ|z|
x−
α−ζ
2,−α−ζ
2dx.
(53)

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 9
Next, by using [16, eqs. 8.2.2.14], the previous equation can be expressed as
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2Z∞
0
1
xG2,0
0,2δx −
α−ζ
2,−α−ζ
2G0,2
2,0x
δ|z|
1−α−ζ
2,1 + α−ζ
2
−dx.
(54)
Moreover, by performing the integration based on [16, eqs. 2.24.1.3], the previous equation
can be rewritten as
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2G0,4
4,01
δ|z|
1−α−ζ
2,1 + α−ζ
2,1−α−ζ
2,1 + α−ζ
2
−.(55)
Finally, after inverting the argument in (55), the PDF of the product distribution is presented
in (47). This concludes the proof.
THEOREM 2.2. The CDF of the product of two zero mean iid variables that follow the
SKD is given by
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2[Γ (α)Γ(ζ)]2
+sgn (z)|z|α+ζ
2G4,1
1,5δ|z|1−α+ζ
2
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2,−α+ζ
2.
(56)
PROO F. The CDF of the product of two RVs that follow the standard SKD with identical
parameters is given by
FZ(z) = Zz
−∞
fZ(x)dx,(57)
After substituting (47) into (57), the later can be rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Zz
−∞ |x|α+ζ
2−1G4,0
0,4δ|x|−
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx.(58)
On the first hand, if z > 0the previous equation can transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z0
−∞
(−x)α+ζ
2−1G4,0
0,4δ(−x)−
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx
+Zz
0
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx,
(59)
or equivalently
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
0
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx
+Zz
0
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx.
(60)
Furthermore, we evaluate the integrals in (60) by using [1, eqs. 24, 26] and, thus, it can be
rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2[Γ (α)Γ(ζ)]2
+zα+ζ
2G4,1
1,5δz 1−α+ζ
2
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2,−α+ζ
2.
(61)

10
On the other hand, if z < 0the previous equation can transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Zz
−∞
(−x)α+ζ
2−1G4,0
0,4δ(−x)−
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx,
(62)
which can be transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
−z
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx,(63)
or equivalently
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
0
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx
−Z−z
0
xα+ζ
2−1G4,0
0,4δx −
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2dx.
(64)
Furthermore, we evaluate the integrals in (64) by using [1, eqs. 24, 26] and, thus, it can be
rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2[Γ (α)Γ(ζ)]2
−(−z)α+ζ
2G4,1
1,5δ(−z)1−α+ζ
2
α−ζ
2,−α−ζ
2,α−ζ
2,−α−ζ
2,−α+ζ
2.
(65)
Finally, by combining the two branches presented in (60) and (64), the CDF of the product
distribution is presented in (56). This concludes the proof.
THEOREM 2.3. The PDF of the ratio of two zero mean iid variables that follow the SKD
is given by
fZ(z) = δα+ζ
|z|1+α
2αζ (α+ζ)2B(2α, 2ζ)
(2α+ 2ζ)2[B(α, ζ )]22
F1
1 + α+ζ , 1 + 2α; 2 + 2α+ 2ζ; 1 −1
|z|.
(66)
PROO F. The pdf of the ratio of two iid variables that follow the SKD can be expressed as
fZ(z) = Z∞
−∞
fX(zy)fY(y)|y|dy,(67)
where z=x
y. Next, after substituting (10) into (67), the latter can be rewritten as
fZ(z) = δα+ζ
[Γ (α)Γ(ζ)]2Z∞
−∞ |y|(|y−γ||zy −γ|)α+ζ
2−1
×Kα−ζ2pδ|y−γ|Kα−ζ2pδ|zy −γ|dy,
(68)
Furthermore, by expressing the Kvfunction in terms of the Meijer-G function as in [16, eqs.
8.4.23.1], the previous equation can be equivalently written as
fZ(z) = δα+ζ
[2Γ (α)Γ(ζ)]2Z∞
−∞ |y|(|y−γ||zy −γ|)α+ζ
2−1
×G2,0
0,2δ|y−γ|−
α−ζ
2,−α−ζ
2G2,0
0,2δ|zy −γ|−
α−ζ
2,−α−ζ
2dy.
(69)

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 11
Next, we assume zero mean symmetric K iid variables and (69) can be expressed as
fZ(z) = δα+ζ|z|α+ζ
2−1
[2Γ (α)Γ(ζ)]2Z∞
0
yα+ζ−1G2,0
0,2δy −
α−ζ
2,−α−ζ
2G2,0
0,2δ|z|y−
α−ζ
2,−α−ζ
2dy
+Z0
−∞
(−y)α+ζ−1G2,0
0,2−δy −
α−ζ
2,−α−ζ
2G2,0
0,2−δ|z|y−
α−ζ
2,−α−ζ
2dy,
(70)
which, after using the substitution y=−yon the second integral can be rewritten as
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2Z∞
0
yα+ζ−1G2,0
0,2δy −
α−ζ
2,−α−ζ
2G2,0
0,2δ|z|y−
α−ζ
2,−α−ζ
2dy,
(71)
Finally, since arg δ < π and arg δ|z|< π, we can perform the integration based on [16, eqs.
2.24.1.3]. Thus, the PDF of the ratio distribution can be expressed as
fZ(z) = δα+ζ|z|α+ζ
2−1
2 [Γ (α)Γ(ζ)]2G2,2
2,2δ|z|1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2.(72)
Furthermore, after expressing Meijer-G in terms of the 2
F1hypergeometric function based
on [7, eq. 9.34.7], can be written as
fZ(z) = δα+ζ
|z|1+α
[Γ (1 + α+ζ)]2Γ (1 + 2α) Γ (1 + 2ζ)
2 [Γ (α)Γ(ζ)]2Γ (2 + 2α+ 2ζ)
×2
F11 + α+ζ , 1 + 2α; 2 + 2α+ 2ζ; 1 −1
|z|.
(73)
and, after performing basic transformation of the Gamma function, it can be transformed as
shown in (66). This concludes the proof.
THEOREM 2.4. The CDF of the ratio of two zero mean iid variables that follow the SKD
is given by
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2Γ (1 + α) Γ (1 + ζ)Γ(α)Γ(ζ)
+sgn (z)|z|α+ζ
2G2,3
3,3δ|z|1−3α+ζ
2,1−α+3ζ
2,1−α+ζ
2
1−α−ζ
2,1 + α−ζ
2,−α+ζ
2.
(74)
PROO F. The cdf of the ratio of two RVs that follow the standard SKD with identical
parameters is given by
FZ(z) = Zz
−∞
fZ(x)dx,(75)
After substituting (72) into (75), the later can be rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Zz
−∞ |x|α+ζ
2−1G2,2
2,2δ|x|1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx.(76)

12
On the first hand, if z > 0the previous equation can transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z0
−∞
(−x)α+ζ
2−1G2,2
2,2δ(−x)1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx
+Zz
0
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx,
(77)
or equivalently
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
0
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx
+Zz
0
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx.
(78)
Furthermore, we evaluate the integrals in (78) by using [1, eqs. 24, 26] and, thus, it can be
rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2Γ (1 + α) Γ (1 + ζ)Γ(α)Γ(ζ)
+zα+ζ
2G2,3
3,3δz 1−3α+ζ
2,1−α+3ζ
2,1−α+ζ
2
1−α−ζ
2,1 + α−ζ
2,−α+ζ
2.
(79)
On the other hand, if z < 0the previous equation can transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Zz
−∞
(−x)α+ζ
2−1G2,2
2,2δ(−x)1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx,(80)
which can be transformed into
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
−z
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx,(81)
or equivalently
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2Z∞
0
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx
−Z−z
0
xα+ζ
2−1G2,2
2,2δx 1−3α+ζ
2,1−α+3ζ
2
1−α−ζ
2,1 + α−ζ
2dx.
(82)
Furthermore, we evaluate the integrals in (82) by using [1, eqs. 24, 26] and, thus, it can be
rewritten as
FZ(z) = δα+ζ
2 [Γ (α)Γ(ζ)]2hδα+ζ
2Γ (1 + α) Γ (1 + ζ)Γ(α)Γ(ζ)
−(−z)α+ζ
2G2,3
3,3δ(−z)1−3α+ζ
2,1−α+3ζ
2,1−α+ζ
2
1−α−ζ
2,1 + α−ζ
2,−α+ζ
2.
(83)
Finally, by combining the two branches presented in (79) and (83), the CDF of the product
distribution is presented in (74). This concludes the proof.

ON THE SYMMETRIC AND SKEW-SYMMETRIC K-DISTRIBUTIONS 13
3. The skew-symmetric K-distribution. The skew-SKD is a continuous probability
distribution that generalizes the SKD to enable non-zero skewness values. To achieve this, an
extra parameter, λ, that regulated the skewness of the distribution is introduced. The PDF of
the skew-SKD can be obtained by
φ(x) = 2f(x)F(λx),(84)
which, after using (10) and (24), can be rewritten as
φ(x) = δα+ζ
2
Γ (α)Γ(ζ)|x−γ|α+ζ
2−1Kα−ζ2pδ|x−γ|"1
+sgn (λx −γ)δα+ζ
2
Γ(α)Γ(ζ)|λx −γ|α+ζ
2G2,1
1,3δ|λx −γ|1−α+ζ
2
α−ζ
2,−α−ζ
2,−α+ζ
2#.
(85)
4. Conclusions. In this work we have introduced a four-parameter distribution that is
derived as a mixture of the parental three-parameter reflected Gamma distribution by using
the two-parameter Gamma as prior. The proposed distribution is termed Symmetric K distri-
bution due to its dependence on the K function. Furthermore, closed-form expressions of the
basic metrics of the SKD are derived, namely PDF, CDF, moments, cumulants, order statis-
tics, as well as product and ratio distributions. Finally, the skew-SKD is calculated in order
to enable non-zero skewness values. The derived distributions exhibit great promise for ap-
plications in various fields, including machine learning, Bayesian learning, communications
and econometric models.
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