# On the symmetric and skew-symmetric K-distributions

Stylianos E. Trevlakis, Nestor D. Chatzidiamantis, and
George K. Karagiannidis

Department of Electrical and Computer Engineering,
Aristotle University of Thessaloniki, Thessaloniki, 54124, Greece

###### Abstract

We propose a family of four-parameter distributions that contain the
K-distribution as special case. The family is derived as a mixture
distribution that uses the three-parameter reflected Gamma distribution as
parental and the two-parameter Gamma distribution as prior. Properties of the
proposed family are investigated as well; these include probability density
function, cumulative distribution function, moments, and cumulants. The family
is termed symmetric K-distribution (SKD) based on its resemblance to the
K-distribution as well as its symmetric nature. The standard form of the SKD,
which often proves to be an adequate model, is also discussed. Moreover, an
order statistics analysis is provided as well as the distributions of the
product and ratio of two independent and identical SKD random variables are
derived. Finally, a generalisation of the proposed family, which enables non-
zero skewness values, is investigated, while both the SKD and the skew-SKD are
proven capable of describing the complex dynamics of machine learning,
Bayesian analysis and other fields through simplified expressions with high
accuracy.

Keywords: Bayesian methods, Mathematical statistics, Mixture distributions,
Order statistics.

##  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, Bayesian
analysis, wireless communications, and econometric models (e.g., Boulogeorgos
et al. (2020); Pedersen et al. (2015); Karagiannidis et al. (2019); Withers
and Nadarajah (2012); Trevlakis et al. (2020); Compiani and Kitamura (2016)).
A discrete or finite mixture distribution is a linear combination of two or
more distributions, i.e.

$f(x)=\sum_{i=1}^{n}\omega_{i}f_{i}(x);0<\omega_{i}<1;\sum_{i=1}^{n}\omega_{i}=1$
(1)

where the weights $\omega_{i}$ can be considered as the probabilities of $n$
sets 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 $\omega_{1}$ are continuous random variables (RVs) and
belong to another distribution, then the resulting mixture is termed
continuous mixture distribution. Next, we focus on the case where one of the
parameters, denoted by $\Theta$, is random. Then, the conditional probability
density function (PDF) $f_{X}(x|\theta)$ is termed parental distribution and
the PDF of $\Theta$, denoted by $f_{\Theta}(\theta)$, is termed prior
distribution. In this case the joint PDF of $X$ and $\Theta$ is

$f(x,\theta)=f_{X}(x|\theta)f_{\Theta}(\theta)$ (2)

and the marginal PDF of $X$ is

$f(x)=\int f(x,\theta)\mathrm{d}\theta=\int
f_{X}(x|\theta)f_{\Theta}(\theta)\mathrm{d}\theta.$ (3)

A comprehensive summary of discrete and continuous mixture distributions can
be found in Johnson et al. (1994).

Since its introduction by Jakeman and Pusey (1976), the K-distribution has
proved to be remarkably useful for modeling the complex dynamics of various
systems, such as wireless communications channel modeling and radar
applications (e.g., Bithas et al. (2006); Wang et al. (2019); Zhao et al.
(2016)). Over the last couple of years, various derivative distributions of
the K distribution have attracted attention due to their successful use in
machine learning-based ultrasound image reconstruction, as well as heat
transfer (e.g., Zhou et al. (2020); Liu et al. (2020)).

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, strength-stress modeling,
Bayesian learning, and more (e.g., Azzalini and Capitanio (1999); Gupta and
Brown (2001); Mudholkar and Hutson (2000)). 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 $f\left(\cdot\right)$ and $F\left(\cdot\right)$
stand for the PDF and CDF of the given RV. Write $\left(a\right)_{n}$ for the
Pochhammer Symbol. Denote by $\exp\left(\cdot\right)$ the exponential
function, $\Gamma\left(\cdot\right)$ the Gamma function, $G\left(\cdot\right)$
the Meijer-G function, ${}_{p}\\!F_{q}\left(\cdot\right)$ for the generalised
hypergeometric function, $K_{v}\left(\cdot\right)$ for the K-function, and
$\mathrm{sgn}\left(\cdot\right)$ the sign function.

The rest of this paper is organised as follows. Section 2 introduces 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.

Figure 1: The (a) PDF and (b) CDF of the SKD for different parameter values.

##  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 (Balakrishnan et al., 2016, eq. 17.15)

$f(x,\delta,\zeta)=\frac{\delta^{\zeta}x^{\zeta-1}}{\Gamma(\zeta)}\exp(-\delta
x),\quad\delta,\zeta>0,$ (4)

where $\delta$ and $\zeta$ are the scale and shape parameters,
correspondingly. Also, the three-parameters RG PDF is given as in Johnson et
al. (1994)

$f(x,\alpha,\beta,\gamma)=\frac{|x-\gamma|^{\alpha-1}}{2\beta^{\alpha}\Gamma(\alpha)}\exp\left(-\frac{|x-\gamma|}{\beta}\right),\quad\alpha,\beta>0,$
(5)

where $\alpha,\beta$ and $\gamma$ are the shape, scale and location
parameters, correspondingly, while, for $\gamma=0$ and $\beta=1$ the standard
form of the RG has the PDF

$\displaystyle f(x;\alpha)=\frac{1}{2\Gamma(\alpha)}|x|^{\alpha-1}e^{-|x|}$
(6)

and CDF

$F(x,\alpha)=\frac{1}{2}-\mathrm{sgn}(x)\frac{\Gamma(\alpha,|x|)}{2\Gamma(\alpha)}$
(7)

where $\mathrm{sgn}(x)$ denotes the sign function and is given by

$\displaystyle\mathrm{sgn}(x)=\left\\{\begin{array}[]{ll}1,&x\geq 0\\\
-1,&x<0\end{array}\right..$ (10)

###  2.1 PDF

According to (3), the PDF of the SKD can be evaluated as follows:

$\displaystyle\begin{split}f(x)&=\int_{0}^{\infty}\frac{1}{2}\frac{|x-\gamma|^{\alpha-1}}{\beta^{\alpha}\Gamma(\alpha)}\exp\left(-\frac{|x-\gamma|}{\beta}\right)\frac{\delta^{\zeta}\beta^{\zeta-1}}{\Gamma(\zeta)}\exp(-\delta\beta)d\beta\\\
&=\frac{\delta^{\zeta}}{2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\left|x-\gamma\right|^{\alpha-1}\int_{0}^{\infty}\beta^{\zeta-\alpha-1}\exp\left(-\frac{|x-\gamma|}{\beta}-\delta\beta\right)d\beta.\end{split}$
(11)

Given the fact that $|x-\gamma|,\delta\in\Re$ and $|x-\gamma|>0$, we can
express the integrands according to (Prudnikov, 1986, Eq. 2.3.16.1). Thus, the
PDF of the four-parameter SKD can be written as

$f(x)=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\left|x-\gamma\right|^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta\left|x-\gamma\right|}\right),$
(12)

where $\delta$ and $\gamma$ are the scale and location parameters, and
$\alpha$, $\zeta$ are shape parameters.

For $\gamma=0$ and $\delta=1$, the PDF of the standard form of the SKD is
given by

$f_{s}(x)=\frac{1}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\left|x\right|^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\left|x\right|}\right),$
(13)

###  2.2 CDF

The CDF of the SKD can be derived directly from the PDF through

$\displaystyle F\left(x\right)=\int_{-\infty}^{x}f\left(x\right)dx.$ (14)

By using (11) it can be rewritten as

$\displaystyle
F\left(x\right)=\left\\{\begin{array}[]{LL}\begin{array}[]{LL}\int_{\gamma}^{x}&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}\left(x-\gamma\right)^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}(2\sqrt{\delta(x-\gamma)})dx\\\
&+\int_{-\infty}^{\gamma}\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}\left(\gamma-x\right)^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}(2\sqrt{\delta(\gamma-x)})dx\end{array}&,\quad
x\geq\gamma\\\
\int_{-\infty}^{x}\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}\left(\gamma-x\right)^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}(2\sqrt{\delta(\gamma-x)})dx&,\quad
x\leq\gamma\end{array}\right..$ (19)

From (19), we need to calculate three integrals, namely $I_{4}$, $I_{5}$, and
$I_{6}$. $I_{4}$ can be transformed based on $y=x-\gamma>0$ into

$\displaystyle
I_{4}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}\int_{0}^{x-\gamma}y^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}(2\sqrt{\delta
y})dy.$ (20)

Next, the $K_{v}$ function can be written in terms of the Meijer-G function as
in (Prudnikov et al., 1986, eq. 8.4.23.1) and, thus, (20) can be equivalently
written as

$\displaystyle
I_{4}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}\int_{0}^{x-\gamma}y^{\frac{\alpha+\zeta}{2}-1}G_{0,2}^{2,0}\left[\delta
y\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dy.$ (23)

After performing the integration in (23) based on (Adamchik and Marichev,
1990, eq. 26), it can be rewritten as

$\displaystyle
I_{4}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}(x-\gamma)^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[\delta(x-\gamma)\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right].$
(26)

Furthermore, after transforming the integral $I_{5}$ based on $z=\gamma-x<0$
can be written as

$\displaystyle
I_{5}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}\int_{0}^{\infty}z^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}(2\sqrt{\delta
z})dz,$ (27)

which, by expressing the $K_{v}$ function in terms of Meijer-G as in
(Prudnikov et al., 1986, eq. 8.4.23.1), can be rewritten as

$\displaystyle
I_{5}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}\int_{0}^{\infty}z^{\frac{\alpha+\zeta}{2}-1}G_{0,2}^{2,0}\left[\delta
z\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dz,$ (30)

and, since $\arg\delta<\pi$ and $\delta\neq 0$, we can perform the integration
based on (Prudnikov et al., 1986, eq. 2.24.2.1). Thus, (30) can be
equivalently written as

$\displaystyle\begin{split}I_{5}=&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}\delta^{-\frac{\alpha+\zeta}{2}}\Gamma\left[\begin{array}[]{cc}\frac{\alpha-\zeta}{2}+\frac{\alpha+\zeta}{2},-\frac{\alpha-\zeta}{2}+\frac{\alpha+\zeta}{2}\\\
-\end{array}\right]\\\
=&\frac{1}{2\Gamma(\alpha)\Gamma(\zeta)}\Gamma\left[\frac{\alpha-\zeta}{2}+\frac{\alpha+\zeta}{2}\right]\Gamma\left[-\frac{\alpha-\zeta}{2}+\frac{\alpha+\zeta}{2}\right]=\frac{1}{2}\end{split}$
(31)

For the third integral, $I_{6}$, after applying the transformation
$k=\gamma-x>0$ and writing the $K_{v}$ function in terms of Meijer-G as in
(Prudnikov et al., 1986, eq. 8.4.23.1), it can be expressed as

$\displaystyle
I_{6}=\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}\int_{\gamma-x}^{\infty}k^{\frac{\alpha+\zeta}{2}-1}G_{0,2}^{2,0}\left[\delta
k\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dk,$ (34)

which can be rewritten as

$\displaystyle
I_{6}=I_{5}-\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}\int_{0}^{\gamma-x}k^{\frac{\alpha+\zeta}{2}-1}G_{0,2}^{2,0}\left[\delta
k\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dk,$ (37)

and by performing the integration based on (Adamchik and Marichev, 1990, eq.
26), (37) can be equivalently expressed as

$\displaystyle\frac{1}{2}-\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}(\gamma-x)^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[\delta(\gamma-x)\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right].$
(40)

Thus, the CDF of the SKD can be written as

$\displaystyle
F\left(x\right)=\left\\{\begin{array}[]{LL}\frac{1}{2}+\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}(x-\gamma)^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[\delta(x-\gamma)\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right]&,\quad
x\geq\gamma\\\
\frac{1}{2}-\frac{\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}(\gamma-x)^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[\delta(\gamma-x)\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right]&,\quad
x\leq\gamma\end{array}\right..$ (47)

and after some simplifications the CDF of the four-parameter SKD can be
rewritten as

$\displaystyle
F\left(x\right)=\frac{1}{2}+\frac{\mathrm{sgn}\left(x-\gamma\right)\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}|x-\gamma|^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left(\delta|x-\gamma|\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right).$
(50)

Furthermore, when $\alpha\neq\zeta$, the Meijer-G function in (50) can be
expressed in terms of the more familiar ${}_{1}\\!F_{2}$ hypergeometric
function. Also, since $\arg\delta\left|x-\gamma\right|<\pi$, the CDF can be
written as (Prudnikov et al., 1986, eq. 8.2.2.3)

$\displaystyle\begin{split}F\left(x\right)=&\frac{1}{2}+\frac{\mathrm{sgn}\left(x-\gamma\right)\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}|x-\gamma|^{\frac{\alpha+\zeta}{2}}\pi\csc(\pi(\zeta-\alpha))\\\
&\left(\frac{(\delta\left|x-\gamma\right|)^{\frac{\alpha-\zeta}{2}}}{\alpha\left(\alpha-\zeta\right)\Gamma(\alpha-\zeta)}\,_{1}\\!F_{2}(\alpha;\alpha-\zeta+1,\alpha+1;\delta\left|x-\gamma\right|)\right.\\\
&\left.-\frac{(\delta\left|x-\gamma\right|)^{\frac{\zeta-\alpha}{2}}}{\zeta\left(\zeta-\alpha\right)\Gamma(\zeta-\alpha)}\,_{1}\\!F_{2}(\zeta;-\alpha+\zeta+1,\zeta+1;\delta\left|x-\gamma\right|)\right)\end{split}$
(51)

For $\gamma=0$ and $\delta=1$ the CDF of the standard form of the SKD is given
by

$\displaystyle
F\left(x\right)=\frac{1}{2}+\frac{\mathrm{sgn}\left(x\right)\delta^{\frac{\alpha+\zeta}{2}}}{2\Gamma(\alpha)\Gamma(\zeta)}|x|^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[|x|\ \vline\ \begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right].$
(54)

###  2.3 N-th moment

The n-th Moment of the SKD can be derived as

$\displaystyle\mu_{n}=\int_{-\infty}^{\infty}x^{n}f\left(x\right)dx.$ (55)

By substituting (12) into (55), it can be rewritten as

$\displaystyle\begin{split}\mu_{n}=&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\int_{-\infty}^{\gamma}x^{n}\left(\gamma-x\right)^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta\left(\gamma-x\right)}\right)dx\\\
&+\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\int_{\gamma}^{\infty}x^{n}\left(x-\gamma\right)^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta\left(x-\gamma\right)}\right)dx,\end{split}$
(56)

which, by using the transformations $y=x-\gamma>0$ and $k=\gamma-x>0$, it can
be equivalently written as

$\displaystyle\begin{split}\mu_{n}=&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\int_{0}^{\infty}\left(\gamma-k\right)^{n}k^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta
k}\right)dk\\\
&+\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\int_{0}^{\infty}\left(\gamma+y\right)^{n}y^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta
y}\right)dy.\end{split}$ (57)

Next, by using (Gradshteyn and Ryzhik, 2014, eq. 1.111), (57) can be rewritten
as

$\displaystyle\begin{split}\mu_{n}=&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\sum_{i=0}^{n}\left(\begin{array}[]{c}n\\\
i\end{array}\right)\gamma^{i}\left(-1\right)^{n-i}\int_{0}^{\infty}k^{\frac{\alpha+\zeta}{2}-1+n-i}K_{\alpha-\zeta}\left(2\sqrt{\delta
k}\right)dk\\\
&+\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\sum_{i=0}^{n}\left(\begin{array}[]{c}n\\\
i\end{array}\right)\gamma^{i}\int_{0}^{\infty}y^{\frac{\alpha+\zeta}{2}-1+n-i}K_{\alpha-\zeta}\left(2\sqrt{\delta
y}\right)dy.\end{split}$ (58)

Furthermore, after using the substitution $\eta=\sqrt{y}=\sqrt{k}$, (58) can
be expressed as

$\displaystyle\mu_{n}=\frac{2\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\sum_{i=0}^{n}\left(\begin{array}[]{c}n\\\
i\end{array}\right)\gamma^{i}\left(\left(-1\right)^{n-i}+1\right)\int_{0}^{\infty}\eta^{\left(\alpha+\zeta+2n-2i\right)-1}K_{\alpha-\zeta}\left(2\sqrt{\delta}\eta\right)d\eta,$
(61)

and, since $\left(\alpha+\zeta+2n-2i\right),\frac{1}{2\sqrt{\delta}}>0$, by
employing (Prudnikov et al., 1986, eq. 2.16.2.2), (61) can be rewritten as

$\displaystyle\mu_{n}=\frac{\delta^{i-n}}{2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\sum_{i=0}^{n}\left(\begin{array}[]{c}n\\\
i\end{array}\right)\gamma^{i}\left(\left(-1\right)^{n-i}+1\right)\Gamma\left(\alpha+n-i\right)\Gamma\left(\zeta+n-i\right).$
(64)

Next, the previous equation can be simplified as

$\displaystyle\mu_{n}=\sum_{k=0}^{n}\frac{\Gamma(n+1)\Gamma(\alpha+n-k)\Gamma(\zeta+n-k)}{\Gamma(k+1)\Gamma(n-k+1)\Gamma(\alpha)\Gamma(\zeta)}\frac{\gamma^{k}\left((-1)^{n-k}+1\right)}{2\delta^{n-k}}.$
(65)

Finally, by using basic transformation of the $\Gamma$ function, the n-th
moment of the four-parameter SKD can be expressed as

$\displaystyle\mu_{n}=\sum_{\begin{subarray}{c}k=0\\\
n-k\in\text{even}\end{subarray}}^{n}\frac{n\gamma^{k}\left(\alpha\right)_{n-k}\left(\zeta\right)_{n-k}}{\delta^{n-k}\Gamma(k+1)\left(n\right)_{1-k}},$
(66)

with $\left(x\right)_{n}$ denoting the Pochhammer symbol.

Furthermore, the first four moments and the first four central moments are
presented in Table 1.

Table 1: Moments and central moments of the SKD. Moment | Value
---|---
$\mu_{1}$ | $\gamma$
$\mu_{2}$ | $\frac{\alpha(\alpha+1)\zeta(\zeta+1)}{\delta^{2}}+\gamma^{2}$
$\mu_{3}$ | $\frac{3\alpha(\alpha+1)\gamma\zeta(\zeta+1)}{\delta^{2}}+\gamma^{3}$
$\mu_{4}$ | $\frac{6\alpha(\alpha+1)\gamma^{2}\zeta(\zeta+1)}{\delta^{2}}+\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)\zeta(\zeta+1)(\zeta+2)(\zeta+3)}{\delta^{4}}+\gamma^{4}$
Central moment | Value
$\tilde{\mu}_{1}=\mu_{1}$ | $\gamma$
$\tilde{\mu}_{2}=\mu_{2}-\mu_{1}^{2}$ | $\frac{\alpha(\alpha+1)\zeta(\zeta+1)}{\delta^{2}}$
$\tilde{\mu}_{3}=\mu_{3}-3\mu_{1}\mu_{2}+2\mu_{1}^{3}$ | $0$
$\tilde{\mu}_{4}=\mu_{4}-4\mu_{1}\mu_{3}+6\mu_{1}^{2}\mu_{2}-3\mu_{1}^{4}$ | $\frac{\alpha(\alpha+1)(\alpha+2)(\alpha+3)\zeta(\zeta+1)(\zeta+2)(\zeta+3)}{\delta^{4}}$

###  2.4 Cumulants

In addition, the cumulants of the SKD are given by

$\displaystyle
K\left(t\right)=\log\left(\mathrm{E}\left[e^{tx}\right]\right),$ (67)

or, in terms of the N-th moments

$\displaystyle\kappa_{n}=\mu_{n}-\sum_{m=1}^{n-1}\left(\begin{array}[]{c}n-1\\\
m\end{array}\right)\kappa_{m}\mu_{n-m},$ (70)

which, after substituting (66), can be written as

$\displaystyle\begin{split}\kappa_{n}=\sum_{\begin{subarray}{c}k=0\\\
n-k\in\text{even}\end{subarray}}^{n}&\frac{n\left(\alpha\right)_{n-k}\left(\zeta\right)_{n-k}\left(1-n\right)_{k}}{k\left(n-k\right)\left(-1\right)^{k}\Gamma\left(k\right)}-\sum_{m=1}^{n-1}\\!\\!\\!\\!\\!\\!\sum_{\begin{subarray}{c}k=0\\\
n-m-k\in\text{even}\end{subarray}}^{n-m}\\!\\!\\!\\!\\!\\!\left(\begin{array}[]{c}n-1\\\
m\end{array}\right)\kappa_{m}\\\
&\times\frac{\left(n-m\right)\left(\alpha\right)_{n-m-k}\left(\zeta\right)_{n-m-k}\left(1-n-m\right)_{k}}{k\left(n-m-k\right)\left(-1\right)^{k}\Gamma\left(k\right)}.\end{split}$
(71)

Thus, the first four cumulants are presented in Table 2.

Table 2: Cumulants of the SKD. Cumulant | Value
---|---
$\kappa_{1}=\tilde{\mu}_{1}$ | $\gamma$
$\kappa_{2}=\tilde{\mu}_{2}$ | $\frac{\alpha(\alpha+1)\zeta(\zeta+1)}{\delta^{2}}$
$\kappa_{3}=\tilde{\mu}_{3}$ | $0$
$\kappa_{4}=\tilde{\mu}_{4}-3\tilde{\mu}_{2}^{2}$ | $\frac{(\alpha+2)(\alpha+3)(\zeta+2)(\zeta+3)}{\alpha(\alpha+1)\zeta(\zeta+1)}\left(\alpha^{2}((\zeta-1)\zeta-3)-\alpha(\zeta(\zeta+11)+15)-3(\zeta+2)(\zeta+3)\right)$

Finally, based on the aforementioned analysis, the mean, variance, skewness,
and kurtosis of the SKD are given in

$\displaystyle\mu=$ $\displaystyle\mu_{1}=\gamma,$ (72)
$\displaystyle\sigma^{2}=$
$\displaystyle\tilde{\mu}_{2}=\frac{\alpha(\alpha+1)\zeta(\zeta+1)}{\delta^{2}},$
(73) $\displaystyle\gamma_{1}=$
$\displaystyle\frac{\tilde{\mu}_{3}}{\tilde{\mu}_{2}^{3/2}}=0,\text{ and}$
(74) $\displaystyle\beta_{2}=$
$\displaystyle\frac{\tilde{\mu}_{4}}{\tilde{\mu}_{2}^{2}}=\frac{(\alpha+2)(\alpha+3)(\zeta+2)(\zeta+3)}{\alpha(\alpha+1)\zeta(\zeta+1)}.$
(75)

###  2.5 Order Statistics

Order statistics are among the most fundamental tools in non-parametric
statistics and inference. The order statistics
$X_{(1)},X_{(2)},\cdots,X_{(n)}$, of any random sample of random variables,
$X_{1},X_{2},\dots,X_{n}$, are the same random variables sorted in increasing
order. Assuming that any such random sample follows the SKD, the CDF of that
random sample is given by

$\displaystyle
F_{\mathrm{X}_{\left(r\right)}}\left(x\right)=\sum_{j-r}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\left[F\left(x\right)\right]^{j}\left[1-F\left(x\right)\right]^{n-j},$
(78)

and, after substituting (50), it can be equivalently written as

$\displaystyle
F_{\mathrm{X}_{\left(r\right)}}\left(x\right)=\sum_{j-r}^{n}\left(\begin{array}[]{c}n\\\
j\end{array}\right)\left[\frac{1}{2}+\Lambda\left(x\right)\right]^{j}\left[\frac{1}{2}-\Lambda\left(x\right)\right]^{n-j},$
(81)

with

$\displaystyle\Lambda\left(x\right)=\frac{\mathrm{sgn}\left(x-\gamma\right)\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}|x-\gamma|^{\frac{\alpha+\zeta}{2}}\quad
G_{1,3}^{2,1}\left[\delta|x-\gamma|\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right].$
(84)

Furthermore, the PDF of a random sample can be expressed as

$\displaystyle
f_{\mathrm{X}_{\left(r\right)}}\left(x\right)=\frac{n!}{\left(r-1\right)!\left(n-r\right)!}f\left(x\right)\left[F\left(x\right)\right]^{r-1}\left[1-F\left(x\right)\right]^{n-r},$
(85)

and, after substituting (12) and (50), it can be rewritten as

$\displaystyle\begin{split}f_{\mathrm{X}_{\left(r\right)}}\left(x\right)=&\frac{n!}{\left(r-1\right)!\left(n-r\right)!}\frac{\mathrm{sgn}\left(x-\gamma\right)\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}|x-\gamma|^{\frac{\alpha+\zeta}{2}-1}\\\
&\times
K_{\alpha-\zeta}\left(2\sqrt{\delta|x-\gamma|}\right)\left[\frac{1}{2}+\Lambda\left(x\right)\right]^{r-1}\left[\frac{1}{2}-\Lambda\left(x\right)\right]^{n-r}\end{split}.$
(86)

###  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.

####  2.6.1 Product PDF

######  Theorem 1.

The PDF of the product of two zero mean iid variables that follow the SKD is
given by

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}G_{0,4}^{4,0}\left(\delta\left|z\right|\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right).$
(89)

###### Proof.

The pdf of the product of two iid variables that follow the symmetric can be
expressed as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\int_{-\infty}^{\infty}f_{\mathrm{X}}\left(x\right)f_{\mathrm{Y}}\left(\frac{z}{x}\right)\frac{1}{\left|x\right|}dx,$
(90)

where $z=xy$. Next, after substituting (12) into (90), the latter can be
rewritten as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}&\int_{-\infty}^{\infty}\frac{\left(\left|x-\gamma\right|\left|\frac{z}{x}-\gamma\right|\right)^{\frac{\alpha+\zeta}{2}-1}}{\left|x\right|}\\\
&\times
K_{\alpha-\zeta}\left(2\sqrt{\delta\left|x-\gamma\right|}\right)K_{\alpha-\zeta}\left(2\sqrt{\delta\left|\frac{z}{x}-\gamma\right|}\right)dx,\end{split}$
(91)

Furthermore, by expressing the $K_{v}$ function in terms of the Meijer-G
function as in (Prudnikov et al., 1986, eq. 8.4.23.1), the previous equation
can be equivalently written as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{\left[2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{\infty}\frac{\left(\left|x-\gamma\right|\left|\frac{z}{x}-\gamma\right|\right)^{\frac{\alpha+\zeta}{2}-1}}{\left|x\right|}\\\
&\times G_{0,2}^{2,0}\left[\delta\left|x-\gamma\right|\ \vline\
\begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]G_{0,2}^{2,0}\left[\delta\left|\frac{z}{x}-\gamma\right|\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dx.\end{split}$
(92)

It becomes evident that the evaluation of a closed form expression for the
integral presented in (92) is very challenging. However, if we assume zero
mean symmetric K iid variables, (92) can be expressed as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{\left[2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{\infty}\frac{1}{\left|x\right|}G_{0,2}^{2,0}\left[\delta\left|x\right|\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]G_{0,2}^{2,0}\left[\frac{\delta\left|z\right|}{\left|x\right|}\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dx,\end{split}$
(93)

or, equivalently

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{\left[2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}&\left[\int_{0}^{\infty}\frac{1}{x}G_{0,2}^{2,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(\frac{\delta\left|z\right|}{x}\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&-\left.\int_{-\infty}^{0}\frac{1}{x}G_{0,2}^{2,0}\left(-\delta x\ \vline\
\begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(-\frac{\delta\left|z\right|}{x}\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right],\end{split}$
(94)

and, after using the substitution $x=-x$ on the second integral, (94) can be
rewritten as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{0}^{\infty}\frac{1}{x}G_{0,2}^{2,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(\frac{\delta\left|z\right|}{x}\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx.$ (99)

Next, by using (Prudnikov et al., 1986, eq. 8.2.2.14), the previous equation
can be expressed as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{0}^{\infty}\frac{1}{x}G_{0,2}^{2,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{2,0}^{0,2}\left(\frac{x}{\delta\left|z\right|}\
\vline\
\begin{array}[]{cc}1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\\\
-\end{array}\right)dx.$ (104)

Moreover, by performing the integration based on (Prudnikov et al., 1986, eq.
2.24.1.3), the previous equation can be rewritten as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}G_{4,0}^{0,4}\left(\frac{1}{\delta\left|z\right|}\
\vline\
\begin{array}[]{cc}1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2},1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\\\
-\end{array}\right).$ (107)

Finally, after inverting the argument in (107), the PDF of the product
distribution is presented in (89). This concludes the proof. ∎

####  2.6.2 Product CDF

######  Theorem 2.

The CDF of the product of two zero mean iid variables that follow the SKD is
given by

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}\right.\\\
&\left.+\mathrm{sgn}\left(z\right)\left|z\right|^{\frac{\alpha+\zeta}{2}}G_{1,5}^{4,1}\left(\delta\left|z\right|\
\vline\ \begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(108)

###### Proof.

The CDF of the product of two RVs that follow the standard SKD with identical
parameters is given by

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\int_{-\infty}^{z}f_{\mathrm{Z}}\left(x\right)dx,$
(109)

After substituting (89) into (109), the later can be rewritten as

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{z}\left|x\right|^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta\left|x\right|\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx.$
(112)

On the first hand, if $z>0$ the previous equation can transformed into

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{-\infty}^{0}\left(-x\right)^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta\left(-x\right)\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.+\int_{0}^{z}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta x\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right],\end{split}$
(113)

or equivalently

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{0}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.+\int_{0}^{z}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta x\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right].\end{split}$
(114)

Furthermore, we evaluate the integrals in (114) by using (Adamchik and
Marichev, 1990, eq. 24, 26) and, thus, it can be rewritten as

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}\right.\\\
&\left.+z^{\frac{\alpha+\zeta}{2}}G_{1,5}^{4,1}\left(\delta z\ \vline\
\begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(115)

On the other hand, if $z<0$ the previous equation can transformed into

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{z}\left(-x\right)^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta\left(-x\right)\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx,$
(118)

which can be transformed into

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-z}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx,$
(121)

or equivalently

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{0}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta
x\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.-\int_{0}^{-z}x^{\frac{\alpha+\zeta}{2}-1}G_{0,4}^{4,0}\left(\delta x\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dx\right].\end{split}$
(122)

Furthermore, we evaluate the integrals in (122) by using (Adamchik and
Marichev, 1990, eq. 24, 26) and, thus, it can be rewritten as

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}\right.\\\
&\left.-\left(-z\right)^{\frac{\alpha+\zeta}{2}}G_{1,5}^{4,1}\left(\delta\left(-z\right)\
\vline\ \begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(123)

Finally, by combining the two branches presented in (114) and (122), the CDF
of the product distribution is presented in (108). This concludes the proof. ∎

####  2.6.3 Ratio PDF

######  Theorem 3.

The PDF of the ratio of two zero mean iid variables that follow the SKD is
given by

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{\left|z\right|^{1+\alpha}}\frac{2\alpha\zeta\left(\alpha+\zeta\right)^{2}B\left(2\alpha,2\zeta\right)}{\left(2\alpha+2\zeta\right)_{2}\left[B\left(\alpha,\zeta\right)\right]^{2}}\
_{2}\\!F_{1}\\!\\!\left(\\!\\!1+\alpha+\zeta,1+2\alpha;2+2\alpha+2\zeta;1\\!-\\!\frac{1}{\left|z\right|}\right).$
(124)

###### Proof.

The pdf of the ratio of two iid variables that follow the SKD can be expressed
as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\int_{-\infty}^{\infty}f_{\mathrm{X}}\left(zy\right)f_{\mathrm{Y}}\left(y\right)\left|y\right|dy,$
(125)

where $z=\frac{x}{y}$. Next, after substituting (12) into (125), the latter
can be rewritten as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}&\int_{-\infty}^{\infty}\left|y\right|\left(\left|y-\gamma\right|\left|zy-\gamma\right|\right)^{\frac{\alpha+\zeta}{2}-1}\\\
&\times
K_{\alpha-\zeta}\left(2\sqrt{\delta\left|y-\gamma\right|}\right)K_{\alpha-\zeta}\left(2\sqrt{\delta\left|zy-\gamma\right|}\right)dy,\end{split}$
(126)

Furthermore, by expressing the $K_{v}$ function in terms of the Meijer-G
function as in (Prudnikov et al., 1986, eq. 8.4.23.1), the previous equation
can be equivalently written as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{\left[2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{\infty}\left|y\right|\left(\left|y-\gamma\right|\left|zy-\gamma\right|\right)^{\frac{\alpha+\zeta}{2}-1}\\\
&\times G_{0,2}^{2,0}\left[\delta\left|y-\gamma\right|\ \vline\
\begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]G_{0,2}^{2,0}\left[\delta\left|zy-\gamma\right|\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right]dy.\end{split}$
(127)

Next, we assume zero mean symmetric K iid variables and (127) can be expressed
as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{\left[2\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{0}^{\infty}y^{\alpha+\zeta-1}G_{0,2}^{2,0}\left(\delta
y\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(\delta\left|z\right|y\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dy\right.\\\
&+\left.\int_{-\infty}^{0}\left(-y\right)^{\alpha+\zeta-1}G_{0,2}^{2,0}\left(-\delta
y\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(-\delta\left|z\right|y\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dy\right],\end{split}$
(128)

which, after using the substitution $y=-y$ on the second integral can be
rewritten as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{0}^{\infty}y^{\alpha+\zeta-1}G_{0,2}^{2,0}\left(\delta
y\ \vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)G_{0,2}^{2,0}\left(\delta\left|z\right|y\
\vline\ \begin{array}[]{cc}-\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2}\end{array}\right)dy,$ (133)

Finally, since $\arg\delta<\pi$ and $\arg\delta\left|z\right|<\pi$, we can
perform the integration based on (Prudnikov et al., 1986, eq. 2.24.1.3). Thus,
the PDF of the ratio distribution can be expressed as

$\displaystyle
f_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}\left|z\right|^{\frac{\alpha+\zeta}{2}-1}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}G_{2,2}^{2,2}\left(\delta\left|z\right|\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right).$ (136)

Furthermore, after expressing Meijer-G in terms of the ${}_{2}\\!F_{1}$
hypergeometric function based on (Gradshteyn and Ryzhik, 2014, eq. 9.34.7),
can be written as

$\displaystyle\begin{split}f_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{\left|z\right|^{1+\alpha}}\frac{\left[\Gamma\left(1+\alpha+\zeta\right)\right]^{2}\Gamma\left(1+2\alpha\right)\Gamma\left(1+2\zeta\right)}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}\Gamma\left(2+2\alpha+2\zeta\right)}\\\
&\times\
_{2}\\!F_{1}\left(1+\alpha+\zeta,1+2\alpha;2+2\alpha+2\zeta;1-\frac{1}{\left|z\right|}\right).\end{split}$
(137)

and, after performing basic transformation of the Gamma function, it can be
transformed as shown in (124). This concludes the proof. ∎

####  2.6.4 Ratio CDF

######  Theorem 4.

The CDF of the ratio of two zero mean iid variables that follow the SKD is
given by

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\Gamma\left(1+\alpha\right)\Gamma\left(1+\zeta\right)\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right.\\\
&\left.+\mathrm{sgn}\left(z\right)\left|z\right|^{\frac{\alpha+\zeta}{2}}G_{3,3}^{2,3}\left(\delta\left|z\right|\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2},1-\frac{\alpha+\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(138)

###### Proof.

The cdf of the ratio of two RVs that follow the standard SKD with identical
parameters is given by

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\int_{-\infty}^{z}f_{\mathrm{Z}}\left(x\right)dx,$
(139)

After substituting (136) into (139), the later can be rewritten as

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{z}\left|x\right|^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta\left|x\right|\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx.$ (142)

On the first hand, if $z>0$ the previous equation can transformed into

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{-\infty}^{0}\left(-x\right)^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta\left(-x\right)\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.+\int_{0}^{z}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta x\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right],\end{split}$
(143)

or equivalently

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{0}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta
x\ \vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.+\int_{0}^{z}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta x\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right].\end{split}$
(144)

Furthermore, we evaluate the integrals in (144) by using (Adamchik and
Marichev, 1990, eq. 24, 26) and, thus, it can be rewritten as

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\Gamma\left(1+\alpha\right)\Gamma\left(1+\zeta\right)\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right.\\\
&\left.+z^{\frac{\alpha+\zeta}{2}}G_{3,3}^{2,3}\left(\delta z\ \vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2},1-\frac{\alpha+\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(145)

On the other hand, if $z<0$ the previous equation can transformed into

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-\infty}^{z}\left(-x\right)^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta\left(-x\right)\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx,$ (148)

which can be transformed into

$\displaystyle
F_{\mathrm{Z}}\left(z\right)=\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\int_{-z}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta
x\ \vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx,$ (151)

or equivalently

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\int_{0}^{\infty}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta
x\ \vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right.\\\
&\left.-\int_{0}^{-z}x^{\frac{\alpha+\zeta}{2}-1}G_{2,2}^{2,2}\left(\delta x\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2}\end{array}\right)dx\right].\end{split}$
(152)

Furthermore, we evaluate the integrals in (152) by using (Adamchik and
Marichev, 1990, eq. 24, 26) and, thus, it can be rewritten as

$\displaystyle\begin{split}F_{\mathrm{Z}}\left(z\right)=&\frac{\delta^{\alpha+\zeta}}{2\left[\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right]^{2}}\left[\delta^{\frac{\alpha+\zeta}{2}}\Gamma\left(1+\alpha\right)\Gamma\left(1+\zeta\right)\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)\right.\\\
&\left.-\left(-z\right)^{\frac{\alpha+\zeta}{2}}G_{3,3}^{2,3}\left(\delta\left(-z\right)\
\vline\
\begin{array}[]{cc}1-\frac{3\alpha+\zeta}{2},1-\frac{\alpha+3\zeta}{2},1-\frac{\alpha+\zeta}{2}\\\
1-\frac{\alpha-\zeta}{2},1+\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(153)

Finally, by combining the two branches presented in (145) and (153), the CDF
of the product distribution is presented in (138). This concludes the proof. ∎

##  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,
$\lambda$, that regulated the skewness of the distribution is introduced. The
PDF of the skew-SKD can be obtained by

$\displaystyle\phi\left(x\right)=2f\left(x\right)F\left(\lambda x\right),$
(154)

which, after using (12) and (50), can be rewritten as

$\displaystyle\begin{split}\phi\left(x\right)=&\frac{\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma\left(\alpha\right)\Gamma\left(\zeta\right)}\left|x-\gamma\right|^{\frac{\alpha+\zeta}{2}-1}K_{\alpha-\zeta}\left(2\sqrt{\delta\left|x-\gamma\right|}\right)\Bigg{[}1\Bigg{.}\\\
&\left.+\frac{\mathrm{sgn}\left(\lambda
x-\gamma\right)\delta^{\frac{\alpha+\zeta}{2}}}{\Gamma(\alpha)\Gamma(\zeta)}|\lambda
x-\gamma|^{\frac{\alpha+\zeta}{2}}G_{1,3}^{2,1}\left(\delta|\lambda x-\gamma|\
\vline\ \begin{array}[]{cc}1-\frac{\alpha+\zeta}{2}\\\
\frac{\alpha-\zeta}{2},-\frac{\alpha-\zeta}{2},-\frac{\alpha+\zeta}{2}\end{array}\right)\right].\end{split}$
(155)

##  4 Conclusions

Our work on the SKD and skew-SKD was motivated by our interest on machine
learning and, in particular, on Bayesian inference models. In more detail, the
promising potential of distributions with adaptable tail nature in Bayesian
learning applications, such as sparse signal reconstruction, incentivised our
efforts towards finding more general distribution families that can more
accurately model such complex dynamics. Specifically, we 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 distribution due to its
dependence on the K-function as well as its similarities to the
K-distribution. Closed-form expressions of the basic metrics of the SKD are
derived, namely PDF, CDF, moments, cumulants, order statistics, as well as
product and ratio distributions. In addition, the skew-SKD is calculated in
order to enable non-zero skewness values. The derived distributions exhibit
great promise for applications in various fields, including machine learning,
Bayesian learning, communications and econometric models.

## References

  * Adamchik and Marichev (1990) Adamchik, V. S. and O. I. Marichev (1990).  The algorithm for calculating integrals of hypergeometric type functions and its realization in REDUCE system.  In Proceedings of the international symposium on Symbolic and algebraic computation - ISSAC '90. ACM Press.
  * Azzalini and Capitanio (1999) Azzalini, A. and A. Capitanio (1999).  Statistical applications of the multivariate skew normal distribution.  Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61(3), 579–602.
  * Balakrishnan et al. (2016) Balakrishnan, N., N. Johnson, and S. Kotz (2016).  Continuous Univariate Distributions.  Wiley Series in Probability and Statistics. John Wiley & Sons Incorporated.
  * Bithas et al. (2006) Bithas, P. S., N. C. Sagias, P. T. Mathiopoulos, G. K. Karagiannidis, and A. A. Rontogiannis (2006).  On the performance analysis of digital communications over generalized-k fading channels.  IEEE Communications Letters 10(5), 353–355.
  * Boulogeorgos et al. (2020) Boulogeorgos, A.-A. A., S. E. Trevlakis, S. A. Tegos, V. K. Papanikolaou, and G. K. Karagiannidis (2020).  Machine learning in nano-scale biomedical engineering.  IEEE Transactions on Molecular, Biological and Multi-Scale Communications.
  * Compiani and Kitamura (2016) Compiani, G. and Y. Kitamura (2016, 10).  Using Mixtures in Econometric Models: A Brief Review and Some New Results.  The Econometrics Journal 19(3), C95–C127.
  * Gradshteyn and Ryzhik (2014) Gradshteyn, I. S. and I. M. Ryzhik (2014).  Table of Integrals, Series, and Products.  Elsevier LTD, Oxford.
  * Gupta and Brown (2001) Gupta, R. C. and N. Brown (2001).  Reliability studies of the skew-normal distribution and its application to a strength-stress model.  Communications in Statistics-Theory and Methods 30(11), 2427–2445.
  * Jakeman and Pusey (1976) Jakeman, E. and P. Pusey (1976).  A model for non-rayleigh sea echo.  IEEE Transactions on antennas and propagation 24(6), 806–814.
  * Johnson et al. (1994) Johnson, N. L., S. Kotz, and N. Balakrishnan (1994).  Continuous Univariate Distributions.  Wiley-Interscience.
  * Karagiannidis et al. (2019) Karagiannidis, G. K., A. Papathanasiou, P. D. Diamantoulakis, A. Saratzis, and N. Saratzis (2019).  A low complexity and cost method to diagnose arterial stenosis using lightwave wearables.  In 2019 IEEE 19th International Conference on Bioinformatics and Bioengineering (BIBE), pp. 675–680.
  * Liu et al. (2020) Liu, Y., G. Liu, F. Liu, and J.-l. Consalvi (2020).  Effects of the k-value solution schemes on radiation heat transfer modelling in oxy-fuel flames using the full-spectrum correlated k-distribution method.  Applied Thermal Engineering 170, 114986.
  * Mudholkar and Hutson (2000) Mudholkar, G. S. and A. D. Hutson (2000).  The epsilon–skew–normal distribution for analyzing near-normal data.  Journal of Statistical Planning and Inference 83(2), 291–309.
  * Pedersen et al. (2015) Pedersen, N. L., C. N. Manchón, M.-A. Badiu, D. Shutin, and B. H. Fleury (2015).  Sparse estimation using bayesian hierarchical prior modeling for real and complex linear models.  Signal processing 115, 94–109.
  * Prudnikov (1986) Prudnikov, A. (1986).  Integrals and Series: Volume 1: Elementary Functions; Volume 2: Special Functions.  Taylor & Francis.
  * Prudnikov et al. (1986) Prudnikov, A., Y. Brychkov, I. Brychkov, and O. Marichev (1986).  Integrals and Series: More special functions.  Integrals and Series. Gordon and Breach Science Publishers.
  * Trevlakis et al. (2020) Trevlakis, S. E., A. A. A. Boulogeorgos, N. D. Chatzidiamantis, G. K. Karagiannidis, and X. Lei (2020).  Electrical vs optical cell stimulation: A communication perspective.  IEEE Access 8, 192259–192269.
  * Wang et al. (2019) Wang, Z., H. Zhao, S. Wang, J. Zhang, and M.-S. Alouini (2019).  Secrecy analysis in swipt systems over generalized-$k$ fading channels.  IEEE Communications Letters 23(5), 834–837.
  * Withers and Nadarajah (2012) Withers, C. S. and S. Nadarajah (2012).  A generalized suzuki distribution.  Wireless Personal Communications 62(4), 807–830.
  * Zhao et al. (2016) Zhao, H., Y. Tan, G. Pan, Y. Chen, and N. Yang (2016).  Secrecy outage on transmit antenna selection/maximal ratio combining in mimo cognitive radio networks.  IEEE Transactions on Vehicular Technology 65(12), 10236–10242.
  * Zhou et al. (2020) Zhou, Z., J. Fang, A. Cristea, Y.-H. Lin, Y.-W. Tsai, Y.-L. Wan, K.-M. Yeow, M.-C. Ho, and P.-H. Tsui (2020).  Value of homodyned k distribution in ultrasound parametric imaging of hepatic steatosis: An animal study.  Ultrasonics 101, 106001.


WOODSTOCK'97 El Paso, Texas USA

A Fisher Kernel Approach to Video Genre Categorization
(Produces the permission block, and
copyright information). For use with
SIG-ALTERNATE.CLS. Supported by ACM.
[Extended Abstract]
A full version of this paper is available as
Author's Guide to Preparing ACM SIG Proceedings Using
$2_\epsilon$ and BibTeX at

Ben TrovatoDr. Trovato insisted his name be first.

Institute for Clarity in Documentation

1932 Wallamaloo Lane

Wallamaloo, New Zealand

G.K.M. TobinThe secretary disavows
any knowledge of this author's actions.

Institute for Clarity in Documentation

P.O. Box 1212

Dublin, Ohio 43017-6221

Lars ThørväldThis author is the
one who did all the really hard work.

The Thørväld Group

1 Thørväld Circle

Hekla, Iceland

Lawrence P. Leipuner

Brookhaven Laboratories

Brookhaven National Lab

P.O. Box 5000

Sean Fogarty

NASA Ames Research Center

Moffett Field

California 94035

Charles Palmer

Palmer Research Laboratories

8600 Datapoint Drive

San Antonio, Texas 78229

Additional authors: John Smith (The Thørväld Group,
email: ) and Julius P. Kumquat
(The Kumquat Consortium, email: ).

This paper provides a sample of a  document which conforms,
somewhat loosely, to the formatting guidelines for
ACM SIG Proceedings. It is an alternate style which produces
a tighter-looking paper and was designed in response to
concerns expressed, by authors, over page-budgets.
It complements the document Author's (Alternate) Guide to
Preparing ACM SIG Proceedings Using $2_\epsilon$ and Bib.
This source file has been written with the intention of being
compiled under $2_\epsilon$ and BibTeX.

The developers have tried to include every imaginable sort
of “bells and whistles", such as a subtitle, footnotes on
title, subtitle and authors, as well as in the text, and
every optional component (e.g. Acknowledgments, Additional
Authors, Appendices), not to mention examples of
equations, theorems, tables and figures.

To make best use of this sample document, run it through and BibTeX, and compare this source code with the printed
output produced by the dvi file. A compiled PDF version
is available on the web page to help you with the
`look and feel'.

H.4Information Systems ApplicationsMiscellaneous

D.2.8Software EngineeringMetrics[complexity measures, performance measures]

§ INTRODUCTION

The proceedings are the records of a conference.
ACM seeks to give these conference by-products a uniform,
high-quality appearance.  To do this, ACM has some rigid
requirements for the format of the proceedings documents: there
is a specified format (balanced  double columns), a specified
set of fonts (Arial or Helvetica and Times Roman) in
certain specified sizes (for instance, 9 point for body copy),
a specified live area (18 $\times$ 23.5 cm [7" $\times$ 9.25"]) centered on
the page, specified size of margins (1.9 cm [0.75"]) top, (2.54 cm [1"]) bottom
and (1.9 cm [.75"]) left and right; specified column width
(8.45 cm [3.33"]) and gutter size (.83 cm [.33"]).

The good news is, with only a handful of manual
settings[Two of these, the
and  commands, you have
already used; another, , will
be used in your very last run of  to ensure
balanced column heights on the last page.], the  document
class file handles all of this for you.

The remainder of this document is concerned with showing, in
the context of an “actual” document, the  commands
specifically available for denoting the structure of a
proceedings paper, rather than with giving rigorous descriptions
or explanations of such commands.

§ PROPOSED FISHER KERNEL FOR VIDEO CATEGORIZATION

The Fisher kernel was designed as a framework to combine the benefits of generative and
discriminative approaches. The general idea is to represent a signal as the gradient of the
probability density function that is a learned generative model of that signal. Intuitively, such
Fisher vector representation  measures how to modify the parameters of the probability density
function in order to best fit the signal, similar to the measurements in a gradient descent
algorithm for fitting a generative model. The Fisher vector is subsequently used in a discriminative
classifier. In this paper we follow <cit.> and use a Gaussian Mixture Model (GMM) followed by
a linear SVM.

Let $X = \{ x_1, x_2, ..., x_T \}$ be a set of T multimodal video descriptors.
Now X can be represented by its gradient vector with respect to a Gaussian Mixture Model $u_\lambda$
with parameters $\lambda$:

\begin{equation}
G(X)_\lambda =\frac{1}{T} \bigtriangledown_{\lambda} log \left ( u_\lambda (X) ) \right
\end{equation}

Intuitively, the gradient of the log-likelihood describes the direction in which the
model parameters should be modified to fit best the data. We will use this property
to learn the user's feedback using a small amount of document samples. The
dimensionality of this vector depends only on the number of parameters in $\lambda$, and
not on the number of patches T <cit.>.

A natural kernel on these gradients is <cit.>:

\begin{equation} K(X,Y) = {{G}_{\lambda}^{X'} }   {{F}_{\lambda}^{-1}}   {{G}_{\lambda}^{Y} } \end{equation}

where $F_{\lambda}$ is the Fisher information matrix of $u_{\lambda}$ <cit.>

\begin{equation} F_{\lambda} = E_{x \sim u_{\lambda}} [ \nabla _ {\lambda} log(u_{\lambda}(x)) \nabla _ {\lambda} log(u_{\lambda}(x)) ] \end{equation}

Let us turn now to the application of this general framework
to the RF problem. In our case, X is the sequence
of features vectors for a set of video documents. The
probability distribution function (pdf) p
models the feature space of videos. In order to characterize
the feature space, we use the Gaussian Mixture Model (GMM).

In this case, the probability parameters can be subdivided
into mixture weights, mean vectors and covariance matrices.
We assume that the covariance matrices are diagonal and any
distribution can be approximated with an arbitrary precision, by a weighted
sum of Gaussians with diagonal covariances <cit.>.

The gradient vector is, by definition, the concatenation of
the partial derivatives with respect to the model parameters. Let $\mu_{i}$ and
$\sigma_{i}$ be the mean and the standard deviation of $i$'s Gaussian centroid,
$\gamma(i)$ be the soft assignment of descriptor $x_t$ to
Gaussian $i$, and let D denote the dimensionality of the descriptors $x_t$.
$G^x_{\mu,i}$ is the D-dimensional gradient with respect to the mean
$\mu_{i}$ and standard deviation $\sigma_{i}$ of Gaussian $i$. Mathematical derivations
lead to <cit.>:

\begin{equation}\label{eqFisher1} G^x_{\mu,i} = \frac{1}{T \sqrt{\omega_{i}}} \sum\limits_{t=1}^T  { \gamma(i) \frac {x_{t}-\mu_{i}}{\sigma_{i}} } \end{equation}

\begin{equation}\label{eqFisher2} G^x_{\sigma,i} = \frac{1}{T \sqrt{2\omega_{i}}} \sum\limits_{t=1}^T  { \gamma(i) \left [{\frac {(x_{t}-\mu_{i})^2}{{\sigma_{i}}^2}} - 1  \right  ] } \end{equation}

where the division between vectors is a term-by-term operation. The final
gradient vector $G^x$ is the concatenation of the $G^x_{\mu,i}$ and $G^x_{\sigma,i}$
vectors, for $i = 1...K$. The final feature vector becomes a $2KD$ dimensional vector.

Most of content based systems involve two main steps: feature extraction and
document classification. The first step mainly consists of computing one feature per
document that needs to capture as many relevant characteristic for that document category
as possible. For  video documents, most of the approaches compute a feature for each frame, and than
aggregate all the features in one descriptor by computing the mean, dispersion or other statistics
over all the frames. But, by aggregating these statistics in these ways, all notion of time is lost.
Alternatively, we can represent a video by multiple vectors and to compute the distance between two sets of points
using, for example, the Earth Mover distance <cit.>. However, these metrics involve a huge computational cost
for large databases.

By using the Fisher kernel representation, we obtain a natural solution to the problem above. The Fisher Kernel
was originally designed to map multiple vectors into a fixed length representation,
and this approach is exactly what we need for this problem. It takes advantage of the expressive power
of generative models to map sequences of features of variable length, such as video sequences,
into a fixed length representation.

For cutting up the video into temporal fragments, one approach is to divide the video document into frames
and to compute a visual descriptor for each video frame. However, for large multimedia databases the
number of frames is huge (25 frames per second and thousands of hours of video footage) and this
approach can create computational problems. In order to efficiently browse through the significant
video content, summarization is required <cit.>. So, we extract a small collection of
salient images (keyframes) that best represent the underlying content <cit.>.

We train the Gaussian Mixture model on the features of the top $n$ videos.
Once the generative model is trained, for every training
sequence of feature vectors $X_i = \{ x_1; x_2; .. ; x_{T} \}$, composed
of $T$ feature vectors we transform it into a vector of fixed
dimension. The only difference between the previous approach and this one regards on what data the GMM has learned.
Instead of using one global aggregated video feature,  we will use more features per document.
The resulting Fisher kernel representation will have the same number of dimensions.
Experiments from Subsection <ref> will show the performance of temporal Fisher Kernels on the Relevance feedback problem.

§ EXPERIMENTAL SETUP

§.§ Dataset

The validation of the proposed content descriptors is carried out in the context
of the MediaEval 2012 Benchmarking Initiative for Multimedia Evaluation,
the Video Genre Tagging Task <cit.>. This task addresses the automatic categorization
of web media genres used with the blip.tv platform (see http://blip.tv/).
For testing, we use the MediaEval 2012 Video Genre Tagging dataset consisting of 15000 sequences (up to 2000 hours
of video footage), labeled according to 26 video genre categories, namely:
art, autos and vehicles, business, citizen journalism, comedy, conferences and other
events, documentary, educational, food and drink, gaming, health, literature,
movies and television, music and entertainment, personal of auto-biographical, politics, religion,school
and education, sports, technology, environment, mainstream media, travel, videoblogging,
web development and "default" category (accounts for movies which
cannot be assigned to neither one of the previous categories). The videos are predominantly English
with non-English content divided over a range of languages including French, Spanish and Dutch.
For more details on the genre categories see <cit.>.

The main challenge of this task is the diversity of videos that contain high level concepts for videos genres,
each genre category has a high variety of video materials. Figure <ref> illustrates image
examples from the dataset.

The dataset is divided in two main parts: 5196 videos for training and 9300 test documents.
We didn't add any aditional training data to MediaEval dataset.

Image examples for several video genres. The bottom images
correspond to videos from the same genre category, i.e. "politics"
(source blip.tv).

To assess the retrieval performance, we use the global measure of performance,
Mean Average Precision (MAP), which provides a single-figure measure of quality across recall levels.
Among evaluation measures, MAP has been shown to have especially good discrimination and stability. For a single information need, Average Precision
is the average of the precision value obtained for the set of top $k$ documents existing after each relevant document is retrieved,
and this value is then averaged over information needs (see also trec eval scoring tool at http://trec.nist.gov/trec_eval/ <cit.>).

§.§ Content descriptors

For content description we have used two types of descriptors: visual and audio.

Visual descriptors

HoG (81 values) <cit.> from this category, we compute global Histogram of oriented Gradients (HoG) over all frames.
The essential thought behind the Histogram of Oriented Gradient descriptors is that local object appearance and shape within an image can be described by the
distribution of intensity gradients or edge directions. The implementation of these descriptors can be achieved by dividing the image into small connected regions,
called cells, and for each cell compiling a histogram of gradient directions or edge orientations for the pixels within the cell. The combination of these histograms
then represents the descriptor.

Color Naming histogram (11 dimensions) to describe global color contents, we use the Color Naming Histogram proposed
in [22]. It maps colors to 11 universal color names: “black”, “blue”,“brown”, “grey”, “green”, “orange”, “pink”, “purple”, “red”, “white”, and “yellow”.
Audio features

Audio descriptors
Standard audio features (196 values) <cit.> <cit.>
- we used a set of general-purpose audio descriptors: Linear Predictive Coefficients (LPCs), Line Spectral Pairs
(LSPs), MFCCs, Zero-Crossing Rate (ZCR), spectral centroid, flux, rolloff and kurtosis, augmented with the variance of
each feature over a certain window (we used the common setup for capturing enough local context that is equal to 1.28 s).

§ RESULTS

In the following subsections, we present our experiments.
The first experiment (Section <ref>) studies the influence of Fisher Kernel parameters on system's accuracy,
and in Section <ref> we compare our work with oficial MediaEval 2012 competition results.

§.§ RF using Fisher Kernels

In this experiment we study the influence of Fisher Kernel parameters on the system's performance.
We present the following experiments: the influience of each descriptor (HoG, CN and Audio), the influence of centroids number,
the influience of frame number, the Fisher Kernels normalization strategies and performance of each FK component.

In the first experiment we analyze the influence of the number of Gaussian centroids.
Figure <ref> presents the variation of MAP using
a different number of Gaussian centroids. It can be observed
that the results are increasing using a higher number of Gaussian centroids.

The influence of GMM centroids number on system performance (using HoG, CN and standard audio features)

The second experiment presents the influence of Fisher normalization strategies on system performance.
In <cit.>, it was demonstrated that some normalization strategies can improve the performance
of Fisher Kernels. We have tested four normalization algorithms and some combination of them: L1 normalization, L2 normalization,
power normalization ($f(x) = sign(x) \sqrt(\alpha|x|)$ ) and logarithmic normalization ($f(x) = sign(x) log(1+\alpha|x|)$).
The results are presented in Table <ref>. It can be observed that using the combination of L2 normalization -
alpha normalization we obtained the best results.

The influience of different normalization algorithms on system's performance (mean average precision values)
NormalizationHOG     CN      HOG + CN
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%
xx.xx%  	      xx.xx% 	         xx.xx%

In order to compare our algorithm with other state-of-the-art approaches, we have
selected the settings that provide the greatest improvement in performance: 800 GMM centroids,
L2 normalization with alpha normalization. We used the linear SVM classifier.

§.§ Improving Metadata Features using Visual FK

The most efficient retrieval approach remains the inclusion of metadata information,
as it provides a higher semantic level of description than audio-visual information.
The high level metadata descriptors contains information such as document title, tags,keywords,
user's description and comments from the blip.tv blog. Metadata features are fast to compute,
but manual video annotation is time-consuming, laborious and expensive, and in a real scenario
many videos don't have any metadata.

In the following we compare our results to the more high level metadata features, and we try to show that we can improve the metadata's performance using
our fast visual descriptors.

We use the standard Term Frequency-Inverse Document Frequency approach. We reduce
further the term space by keeping only those terms that discriminate best between genres according to the χ2-test. We
generate a global list by retaining for each genre class, the m terms (e.g. 20 for each genre ) with
the highest χ2 values that occur more frequently than in complement classes. This results in a vector representation
for each document that is subsequently cosine normalized.

Comparison between metadata features and Visual FK
FeatureGenre     1     2     3     4     5     6     7     8     9     10     11     12     13     14     15     16     17     18     19     20     21     22     23     24     25     26     global MAP
33.16%     91.14%     47.10%     38.24%     37.81%     23.21%     28.11%     18.43%     45.23%     56.71%     39.33%     65.27%     59.27%     63.26%     38.54 %     16.83%     30.61 %    56.40%     52.12%     53.56%     56.45%     39.75%     43.44%     15.21%     29.26%     28.21%     42.44%
43.32%     84.67%     34.04%     39.49%     39.69%     13.72%     39.63%     16.15%     36.00%     74.90%     88.14%     66.83%     75.29%     50.41%     51.22 %     14.79%     50.06%     68.56%     66.43%     49.75%     54.73%     60.07%     54.43%     24.58%     35.43%     20.82%     48.19%
43.32%     84.67%     34.04%     39.49%     39.69%     13.72%     39.63%     16.15%     36.00%     74.90%     88.14%     66.83%     75.29%     50.41%     51.22 %     14.79%     50.06%     68.56%     66.43%     49.75%     54.73%     60.07%     54.43%     24.58%     35.43%     20.82%     48.19%
43.32%     84.67%     34.04%     39.49%     39.69%     13.72%     39.63%     16.15%     36.00%     74.90%     88.14%     66.83%     75.29%     50.41%     51.22 %     14.79%     50.06%     68.56%     66.43%     49.75%     54.73%     60.07%     54.43%     24.58%     35.43%     20.82%     48.19%

§.§ Computational complexity

In order to analyze the computational complexity of entire framework, we analyse the time for calculating each step,
from feature extraction to video classification. All the video frames are rescaled to resolution of 320x400 pixels.
The run-time is obtained on a Dell with a 2.5 GHz quad-core Xeon CPU and 8GB RAM. We do not parallelize our code
and only use a single core of the CPU.

Figure 12 analyzes the percentage of time spent on each step of computing FK with HOG and CN features, with 600 GMM
centroids and L2 with square-root normalization. The computation of features consumes most of the time: HOG features
with 62% and CN with 31%. Once the features is computed, the creation of Fisher Kernels is relatively inexpensive,
it only takes 3% of the total computation time. Classification is also fast, we consume only 3% percent for SVM with
linear kernel.

The percentages of the time spent on the major steps of computing Fisher Kernels with HOG and CN features

!!!Maybe we can show other charts with the influience of the GMM centroids number or of the number of frames on time complexity!!!.

§.§ Comparison to MediaEval 2012 Tagging Task

A summary of the best team runs at MediaEval 2012 Video Genre Tagging Task is presented
in Table <ref> (results are presented by decreasing MAP values).
The most efficient retrieval approach remains the inclusion of textual information,
as it provides a higher semantic level of description than audio-visual
information. The best MAP value is equal to 52.55% and it is achieved by
including high level metadata descriptors, such as document title, tags and user's description.

In this work we tried to prove that, notwithstanding the
superiority of metadata descriptors, visual information also has great potential in classification tasks,
and it can archieve similar results to other more complicated algorithms.

Each team best result for MediaEval 2012 Video Genre Tagging Task (MAP values).

Team     Method                                      MAP
BoW text metadata                                   52.55%
TF-IDF text metadata & ASR                         37.93%
TF on Visual word, ASR & metadata               36.75%
Visuals descriptors(Color, Texture, rgbSIFT)        34.99%
TF-IDF ASR                                          25.00%
Visual BOW                                 21.12%
3|p8.5cm|Notations: SIFT - Scale
Invariant Features Transform, TF-IDF - Term Frequency-Inverse Document Frequency,
BoW - Bag-of-Words, TF - Term Frequency, ASR - text extracted with automatic speech recognition.

§ CONCLUSIONS

This paragraph will end the body of this sample document.
Remember that you might still have Acknowledgments or
Appendices; brief samples of these
follow.  There is still the Bibliography to deal with; and
we will make a disclaimer about that here: with the exception
of the reference to the  book, the citations in
this paper are to articles which have nothing to
do with the present subject and are used as
examples only.

§ ACKNOWLEDGMENTS

This section is optional; it is a location for you
to acknowledge grants, funding, editing assistance and
what have you.  In the present case, for example, the
authors would like to thank Gerald Murray of ACM for
his help in codifying this Author's Guide
and the .cls and .tex files that it describes.

§ HEADINGS IN APPENDICES

The rules about hierarchical headings discussed above for
the body of the article are different in the appendices.
In the appendix environment, the command
section is used to
indicate the start of each Appendix, with alphabetic order
designation (i.e. the first is A, the second B, etc.) and
a title (if you include one).  So, if you need
hierarchical structure
within an Appendix, start with subsection as the
highest level. Here is an outline of the body of this
document in Appendix-appropriate form:

§.§ Introduction

§.§ The Body of the Paper

§.§.§ Type Changes and  Special Characters

§.§.§ Math Equations

Inline (In-text) Equations

Display Equations

§.§.§ Citations

§.§.§ Tables

§.§.§ Figures

§.§.§ Theorem-like Constructs

§.§.§ A Caveat for the  Expert

§.§ Conclusions

§.§ Acknowledgments

§.§ Additional Authors

This section is inserted by ; you do not insert it.
You just add the names and information in the
command at the start
of the document.

§.§ References

Generated by bibtex from your  .bib file.  Run latex,
then bibtex, then latex twice (to resolve references)
to create the  .bbl file.  Insert that  .bbl file into
the .tex source file and comment out
the command .

§ MORE HELP FOR THE HARDY

The sig-alternate.cls file itself is chock-full of succinct
and helpful comments.  If you consider yourself a moderately
experienced to expert user of , you may find reading
it useful but please remember not to change it.