Top Qs
Timeline
Chat
Perspective

Generalized integer gamma distribution

From Wikipedia, the free encyclopedia

Remove ads

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).

Remove ads

Definition

Summarize
Perspective

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[1][2][3]

and

where

and

with

and

where

Alternative expressions are available in the literature on generalized chi-squared distribution, which is a field where computer algorithms have been available for some years.[when?]

Remove ads

Generalization

Summarize
Perspective

The GNIG (generalized near-integer gamma) distribution of depth is the distribution of the random variable[4]

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 (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.

Remove ads

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. [5][6][7][8][9] 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. [4][10][11]

The GIG distribution is also the basis for a number of wrapped distributions in the wrapped gamma family. [12]

As being a special case of the generalized chi-squared distribution, there are many other applications; for example, in renewal theory[1] and in multi-antenna wireless communications.[13][14][15][16]

References

Loading related searches...

Wikiwand - on

Seamless Wikipedia browsing. On steroids.

Remove ads