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). Definition The random variable has a gamma distribution with shape parameter and rate parameter if its probability density function is 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 has a GIG (generalized integer gamma) distribution of depth with shape parameters and rate parameters . This fact is denoted by It is also a special case of the generalized chi-squared distribution. Properties The probability density function and the cumulative distribution function of Y are respectively given by and where and with and where Alternative expressions are available in the literature on generalized chi-squared distribution, which is a field wherecomputer algorithms have been available for some years. Generalization 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 . Properties The probability density function of is given by and the cumulative distribution function is given by where with given by ()-() 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. Applications 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