In probability and statistics, the 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.

Density

The model is that random variable has a gamma distribution with mean and shape parameter , with being treated as a random variable having another gamma distribution, this time with mean and shape parameter . The result is that has the following probability density function (pdf) for :[1]

where 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:[1] it is the distribution of the product of two independent random variables, one having a gamma distribution with mean 1 and shape parameter , the second having a gamma distribution with mean and shape parameter .

A simpler two parameter formalization of the K-distribution can be obtained by setting as[2]

where is the shape factor, is the scale factor, and is the modified Bessel function of second kind.

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. 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[3]

where and is the Whittaker function.

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

So the mean and variance are given[1] by

Other properties

All the properties of the distribution are symmetric in and [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.

Notes

  1. ^ a b c d e Redding (1999).
  2. ^ Long (2001)
  3. ^ Bithas (2006).

Sources

Further reading