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: 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: where K is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have. 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 where v = \alpha is the shape factor, 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, 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 where and is the Whittaker function. The n-th moments of K-distribution is given by So the mean and variance are given by

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.

Sources