K-distribution

Parameters	${\displaystyle \mu \in (0,+\infty )}$, ${\displaystyle \alpha \in [0,+\infty )}$, ${\displaystyle \beta \in [0,+\infty )}$
Support	${\displaystyle x\in [0,+\infty )\;}$
PDF	${\displaystyle {\frac {2}{\Gamma (\alpha )\Gamma (\beta )))\,\left({\frac {\alpha \beta }{\mu ))\right)^{\frac {\alpha +\beta }{2))\,x^((\frac {\alpha +\beta }{2))-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu ))}\right),}$
Mean	${\displaystyle \mu }$
Variance	${\displaystyle \mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta ))}$
MGF	${\displaystyle \left({\frac {\xi }{s))\right)^{\beta /2}\exp \left({\frac {\xi }{2s))\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s))\right)}$

In probability and statistics, the generalized K-distribution is a three-parameter family of continuous probability distributions. The distribution arises by compounding two gamma distributions. In each case, a re-parametrization of the usual form of the family of gamma distributions is used, such that the parameters are:

• the mean of the distribution,
• the usual shape parameter.

K-distribution is a special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution. A simpler special case of the generalized K-distribution is often referred as the K-distribution.

Density

Suppose that a random variable ${\displaystyle X}$ has gamma distribution with mean ${\displaystyle \sigma }$ and shape parameter ${\displaystyle \alpha }$, with ${\displaystyle \sigma }$ being treated as a random variable having another gamma distribution, this time with mean ${\displaystyle \mu }$ and shape parameter ${\displaystyle \beta }$. The result is that ${\displaystyle X}$ has the following probability density function (pdf) for ${\displaystyle x>0}$:^[1]

${\displaystyle f_{X}(x;\mu ,\alpha ,\beta )={\frac {2}{\Gamma (\alpha )\Gamma (\beta )))\,\left({\frac {\alpha \beta }{\mu ))\right)^{\frac {\alpha +\beta }{2))\,x^((\frac {\alpha +\beta }{2))-1}K_{\alpha -\beta }\left(2{\sqrt {\frac {\alpha \beta x}{\mu ))}\right),}$

where ${\displaystyle K}$ is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have ${\displaystyle K_{\nu }=K_{-\nu ))$. In this derivation, the K-distribution is a compound probability distribution. It is also a product distribution:^[1] it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter ${\displaystyle \alpha }$, the second having a gamma distribution with mean ${\displaystyle \mu }$ and shape parameter ${\displaystyle \beta }$.

A simpler two parameter formalization of the K-distribution can be obtained by setting ${\displaystyle \beta =1}$ as^[2][3]

${\displaystyle f_{X}(x;b,v)={\frac {2b}{\Gamma (v)))\left({\sqrt {bx))\right)^{v-1}K_{v-1}(2{\sqrt {bx))),}$

where ${\displaystyle v=\alpha }$ is the shape factor, ${\displaystyle b=\alpha /\mu }$ is the scale factor, and ${\displaystyle K}$ is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting ${\displaystyle \alpha =1}$, ${\displaystyle v=\beta }$, and ${\displaystyle b=\beta /\mu }$, albeit with different physical interpretation of ${\displaystyle b}$ and ${\displaystyle v}$ parameters. This two parameter formalization is often referred to as the K-distribution, while the three parameter formalization is referred to as the generalized K-distribution.

This distribution derives from a paper by Eric Jakeman and Peter Pusey (1978) who used it to model microwave sea echo.^[4] Jakeman and Tough (1987) derived the distribution from a biased random walk model.^[5] Keith D. Ward (1981) derived the distribution from the product for two random variables, z = a y, where a has a chi distribution and y a complex Gaussian distribution. The modulus of z, |z|, then has K-distribution.^[6]

Moments

The moment generating function is given by^[7]

${\displaystyle M_{X}(s)=\left({\frac {\xi }{s))\right)^{\beta /2}\exp \left({\frac {\xi }{2s))\right)W_{-\delta /2,\gamma /2}\left({\frac {\xi }{s))\right),}$

where ${\displaystyle \gamma =\beta -\alpha ,}$ ${\displaystyle \delta =\alpha +\beta -1,}$ ${\displaystyle \xi =\alpha \beta /\mu ,}$ and ${\displaystyle W_{-\delta /2,\gamma /2}(\cdot )}$ is the Whittaker function.

The n-th moments of K-distribution is given by^[1]

${\displaystyle \mu _{n}=\xi ^{-n}{\frac {\Gamma (\alpha +n)\Gamma (\beta +n)}{\Gamma (\alpha )\Gamma (\beta ))).}$

So the mean and variance are given by^[1]

${\displaystyle \operatorname {E} (X)=\mu }$

${\displaystyle \operatorname {var} (X)=\mu ^{2}{\frac {\alpha +\beta +1}{\alpha \beta )).}$

Other properties

All the properties of the distribution are symmetric in ${\displaystyle \alpha }$ and ${\displaystyle \beta .}$^[1]

Applications

K-distribution arises as the consequence of a statistical or probabilistic model used in synthetic-aperture radar (SAR) imagery. The K-distribution is formed by compounding two separate probability distributions, one representing the radar cross-section, and the other representing speckle that is a characteristic of coherent imaging. It is also used in wireless communication to model composite fast fading and shadowing effects.