K-distribution

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

$f_X(x;\mu ,\alpha ,\beta )=\frac{2}{\mathrm{\Gamma }(\alpha )\mathrm{\Gamma }(\beta )}{\left(\frac{\alpha \beta }{\mu }\right)}^{\frac{\alpha +\beta }{2}}{x}^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\frac{\alpha \beta x}{\mu }}\right),$

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

A simpler two parameter formalization of the K-distribution can be obtained by setting $\beta =1$ as

$f_X(x;b,v)=\frac{2b}{\mathrm{\Gamma }(v)}{\left(\sqrt{bx}\right)}^{v-1}{K}_{v-1}(2\sqrt{bx}),$

where $v=\alpha$ is the shape factor, $b=\alpha /\mu$ is the scale factor, and $K$ is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting $\alpha =1$, $v=\beta$, and $b=\beta /\mu$, albeit with different physical interpretation of $b$ and $v$ 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. Jakeman and Tough (1987) derived the distribution from a biased random walk model. 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.

Moments

The moment generating function is given by

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

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

The n-th moments of K-distribution is given by

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

So the mean and variance are given by

$\mathrm{E}(X)=\mu$

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

Other properties

All the properties of the distribution are symmetric in $\alpha$ and $\beta .$

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.

Further reading

• Ward, Keith D.; Tough, Robert J. A; Watts, Simon (2006) Sea Clutter: Scattering, the K Distribution and Radar Performance, Institution of Engineering and Technology. Template:ISBN.