Generalized integer gamma distribution
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In probability and statistics, the generalized integer gamma distribution (GIG) is the distribution of the sum of independent gamma distributed random variables, all with integer shape parameters and different rate parameters. This is a special case of the generalized chi-squared distribution. A related concept is the generalized near-integer gamma distribution (GNIG).
and this fact is denoted by
Let , where be independent random variables, with all being positive integers and all different. In other words, each variable has the Erlang distribution with different shape parameters. The uniqueness of each shape parameter comes without loss of generality, because any case where some of the are equal would be treated by first adding the corresponding variables: this sum would have a gamma distribution with the same rate parameter and a shape parameter which is equal to the sum of the shape parameters in the original distributions.
Then the random variable Y defined by
It is also a special case of the generalized chi-squared distribution.
Alternative expressions are available in the literature on generalized chi-squared distribution, which is a field wherecomputer algorithms have been available for some years.
The GNIG (generalized near-integer gamma) distribution of depth is the distribution of the random variable
where and are two independent random variables, where is a positive non-integer real and where .
The probability density function of is given by
and the cumulative distribution function is given by
with given by (1)-(3) above. In the above expressions is the Kummer confluent hypergeometric function. This function has usually very good convergence properties and is nowadays easily handled by a number of software packages.
The GIG and GNIG distributions are the basis for the exact and near-exact distributions of a large number of likelihood ratio test statistics and related statistics used in multivariate analysis.  More precisely, this application is usually for the exact and near-exact distributions of the negative logarithm of such statistics. If necessary, it is then easy, through a simple transformation, to obtain the corresponding exact or near-exact distributions for the corresponding likelihood ratio test statistics themselves. 
As being a special case of the generalized chi-squared distribution, there are many other applications; for example, in renewal theory and in multi-antenna wireless communications.   
Modules for the computation of the p.d.f. and c.d.f. of both the GIG and the GNIG distributions are made available at this web-page on near-exact distributions.
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