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).

Definition[edit]

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[edit]

The probability density function and the cumulative distribution function of Y are respectively given by[1][2][3]

and

where

and

with

 

 

 

 

(1)

and

 

 

 

 

(2)

where

 

 

 

 

(3)

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[edit]

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[edit]

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.

Applications[edit]

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]

Computer modules[edit]

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.

References[edit]

  1. ^ a b Amari S.V. and Misra R.B. (1997). Closed-From Expressions for Distribution of Sum of Exponential Random Variables. IEEE Transactions on Reliability, vol. 46, no. 4, 519-522.
  2. ^ Coelho, C. A. (1998). The Generalized Integer Gamma distribution – a basis for distributions in Multivariate Statistics. Journal of Multivariate Analysis, 64, 86-102.
  3. ^ Coelho, C. A. (1999). Addendum to the paper ’The Generalized IntegerGamma distribution - a basis for distributions in MultivariateAnalysis’. Journal of Multivariate Analysis, 69, 281-285.
  4. ^ a b Coelho, C. A. (2004). "The Generalized Near-Integer Gamma distribution – a basis for ’near-exact’ approximations to the distributions of statistics which are the product of an odd number of particular independent Beta random variables". Journal of Multivariate Analysis, 89 (2), 191-218. [MR2063631 (2005d:62024)] [Zbl 1047.62014] [WOS: 000221483200001]
  5. ^ Bilodeau, M., Brenner, D. (1999) "Theory of Multivariate Statistics". Springer, New York [Ch. 11, sec. 11.4]
  6. ^ Das, S., Dey, D. K. (2010) "On Bayesian inference for generalized multivariate gamma distribution". Statistics and Probability Letters, 80, 1492-1499.
  7. ^ Karagiannidis, K., Sagias, N. C., Tsiftsis, T. A. (2006) "Closed-form statistics for the sum of squared Nakagami-m variates and its applications". Transactions on Communications, 54, 1353-1359.
  8. ^ Paolella, M. S. (2007) "Intermediate Probability - A Computational Approach". J. Wiley & Sons, New York [Ch. 2, sec. 2.2]
  9. ^ Timm, N. H. (2002) "Applied Multivariate Analysis". Springer, New York [Ch. 3, sec. 3.5]
  10. ^ Coelho, C. A. (2006) "The exact and near-exact distributions of the product of independent Beta random variables whose second parameter is rational". Journal of Combinatorics, Information & System Sciences, 31 (1-4), 21-44. [MR2351709]
  11. ^ Coelho, C. A., Alberto, R. P. and Grilo, L. M. (2006) "A mixture of Generalized Integer Gamma distributions as the exact distribution of the product of an odd number of independent Beta random variables.Applications". Journal of Interdisciplinary Mathematics, 9, 2, 229-248. [MR2245158] [Zbl 1117.62017]
  12. ^ Coelho, C. A. (2007) "The wrapped Gamma distribution and wrapped sums and linear combinations of independent Gamma and Laplace distributions". Journal of Statistical Theory and Practice, 1 (1), 1-29.
  13. ^ E. Björnson, D. Hammarwall, B. Ottersten (2009) "Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems", IEEE Transactions on Signal Processing, 57, 4027-4041
  14. ^ Kaiser, T., Zheng, F. (2010) "Ultra Wideband Systems with MIMO". J. Wiley & Sons, Chichester, U.K. [Ch. 6, sec. 6.6]
  15. ^ Suraweera, H. A., Smith, P. J., Surobhi, N. A. (2008) "Exact outage probability of cooperative diversity with opportunistic spectrum access". IEEE International Conference on Communications, 2008, ICC Workshops '08, 79-86 (ISBN 978-1-4244-2052-0 - doi: 10.1109/ICCW.2008.20).
  16. ^ Surobhi, N. A. (2010) "Outage performance of cooperative cognitive relay networks". MsC Thesis, School of Engineering and Science, Victoria University, Melbourne, Australia [Ch. 3, sec. 3.4].