K-distribution

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: 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 an special case of variance-gamma distribution, which in turn is a special case of generalised hyperbolic distribution. The model is that random variable X {displaystyle X} has a gamma distribution with mean σ {displaystyle sigma } and shape parameter L {displaystyle L} , with σ {displaystyle sigma } being treated as a random variable having another gamma distribution, this time with mean μ {displaystyle mu } and shape parameter ν {displaystyle u } . The result is that X {displaystyle X} has the following probability density function (pdf) for x > 0 {displaystyle x>0} : where α = ν − L , {displaystyle alpha = u -L,} β = L + ν − 1 , {displaystyle eta =L+ u -1,} ξ = L ν / μ , {displaystyle xi =L u /mu ,} and K {displaystyle K} is a modified Bessel function of the second kind. 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 L {displaystyle L} , the second having a gamma distribution with mean μ {displaystyle mu } and shape parameter ν {displaystyle u } . A simpler two parameter formalization of the K-distribution is where v is the shape factor, b is the scale factor, and K 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. The moment generating function is given by where W − β / 2 , α / 2 ( ⋅ ) {displaystyle W_{-eta /2,alpha /2}(cdot )} is the Whittaker function.