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. Suppose that a random variable has 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 : 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: 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 where is the shape factor, is the scale factor, and is the modified Bessel function of second kind. The above two parameter formalization can also be obtained by setting , , and , albeit with different physical interpretation of and 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.