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 {\displaystyle X} has gamma distribution with mean σ {\displaystyle \sigma } and shape parameter α {\displaystyle \alpha } , with σ {\displaystyle \sigma } being treated as a random variable having another gamma distribution, this time with mean μ {\displaystyle \mu } and shape parameter β {\displaystyle \beta } . The result is that X {\displaystyle X} has the following probability density function (pdf) for x > 0 {\displaystyle x>0} :
f X ( x ; μ , α , β ) = 2 Γ ( α ) Γ ( β ) ( α β μ ) α + β 2 x α + β 2 − 1 K α − β ( 2 α β x μ ) , {\displaystyle f_{X}(x;\mu ,\alpha ,\beta )={\frac {2}{\Gamma (\alpha )\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 {\displaystyle K} is a modified Bessel function of the second kind. Note that for the modified Bessel function of the second kind, we have K ν = K − ν {\displaystyle 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 α {\displaystyle \alpha } , the second having a gamma distribution with mean μ {\displaystyle \mu } and shape parameter β {\displaystyle \beta } . A simpler two parameter formalization of the K-distribution can be obtained by setting β = 1 {\displaystyle \beta =1} as
f X ( x ; b , v ) = 2 b Γ ( v ) ( b x ) v − 1 K v − 1 ( 2 b x ) , {\displaystyle f_{X}(x;b,v)={\frac {2b}{\Gamma (v)}}\left({\sqrt {bx}}\right)^{v-1}K_{v-1}(2{\sqrt {bx}}),}
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