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).
Contents
Definition
The random variable
and this fact is denoted by
Let
Then the random variable Y defined by
has a GIG (generalized integer gamma) distribution of depth
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
where
Properties
The probability density function of
and the cumulative distribution function is given by
where
with
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 for the corresponding likelihood ratio test statistics themselves.
The GIG distribution is also the basis for a number of wrapped distributions in the wrapped gamma family.
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.
Computer modules
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.