Generalized integer gamma distribution

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 ${\displaystyle X\!}$ has a gamma distribution with shape parameter ${\displaystyle r}$ and rate parameter ${\displaystyle \lambda }$ if its probability density function is

${\displaystyle f_{X}^{}(x)={\frac {\lambda ^{r}}{\Gamma (r)}}\,e^{-\lambda x}x^{r-1}~~~~~~(x>0;\,\lambda ,r>0)}$

and this fact is denoted by ${\displaystyle X\sim \Gamma (r,\lambda )\!.}$

Let ${\displaystyle X_{j}\sim \Gamma (r_{j},\lambda _{j})\!}$, where ${\displaystyle (j=1,\dots ,p),}$ be ${\displaystyle p}$ independent random variables, with all ${\displaystyle r_{j}}$ being positive integers and all ${\displaystyle \lambda _{j}\!}$ 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 ${\displaystyle \lambda _{j}}$ 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

${\displaystyle Y=\sum _{j=1}^{p}X_{j}}$

has a GIG (generalized integer gamma) distribution of depth ${\displaystyle p}$ with shape parameters ${\displaystyle r_{j}\!}$ and rate parameters ${\displaystyle \lambda _{j}\!}$${\displaystyle (j=1,\dots ,p)}$. This fact is denoted by

${\displaystyle Y\sim GIG(r_{j},\lambda _{j};p)\!.}$

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]}

${\displaystyle f_{Y}^{\text{GIG}}(y|r_{1},\dots ,r_{p};\lambda _{1},\dots ,\lambda _{p})\,=\,K\sum _{j=1}^{p}P_{j}(y)\,e^{-\lambda _{j}\,y}\,,~~~~(y>0)}$

and

${\displaystyle F_{Y}^{\text{GIG}}(y|r_{1},\dots ,r_{p};\lambda _{1},\dots ,\lambda _{p})\,=\,1-K\sum _{j=1}^{p}P_{j}^{*}(y)\,e^{-\lambda _{j}\,y}\,,~~~~(y>0)}$

where

${\displaystyle K=\prod _{j=1}^{p}\lambda _{j}^{r_{j}}~,~~~~~P_{j}(y)=\sum _{k=1}^{r_{j}}c_{j,k}\,y^{k-1}}$

and

${\displaystyle P_{j}^{*}(y)=\sum _{k=1}^{r_{j}}c_{j,k}\,(k-1)!\sum _{i=0}^{k-1}{\frac {y^{i}}{i!\,\lambda _{j}^{k-i}}}}$

with

${\displaystyle c_{j,r_{j}}={\frac {1}{(r_{j}-1)!}}\,\mathop {\prod _{i=1}^{p}} _{i\neq j}(\lambda _{i}-\lambda _{j})^{-r_{i}}~,~~~~~~j=1,\ldots ,p\,,}$

(1)

and

${\displaystyle c_{j,r_{j}-k}={\frac {1}{k}}\sum _{i=1}^{k}{\frac {(r_{j}-k+i-1)!}{(r_{j}-k-1)!}}\,R(i,j,p)\,c_{j,r_{j}-(k-i)}\,,~~~~~~(k=1,\ldots ,r_{j}-1;\,j=1,\ldots ,p)}$

(2)

where

${\displaystyle R(i,j,p)=\mathop {\sum _{k=1}^{p}} _{k\neq j}r_{k}\left(\lambda _{j}-\lambda _{k}\right)^{-i}~~~(i=1,\ldots ,r_{j}-1)\,.}$

(3)

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? ]}

Generalization

The GNIG (generalized near-integer gamma) distribution of depth ${\displaystyle p+1}$ is the distribution of the random variable ^[4]

${\displaystyle Z=Y_{1}+Y_{2}\!,}$

where ${\displaystyle Y_{1}\sim GIG(r_{j},\lambda _{j};p)\!}$ and ${\displaystyle Y_{2}\sim \Gamma (r,\lambda )\!}$ are two independent random variables, where ${\displaystyle r}$ is a positive non-integer real and where ${\displaystyle \lambda \neq \lambda _{j}}$${\displaystyle (j=1,\dots ,p)}$.

Properties

The probability density function of ${\displaystyle Z\!}$ is given by

${\displaystyle {\begin{array}{l}\displaystyle f_{Z}^{\text{GNIG}}(z|r_{1},\dots ,r_{p},r;\,\lambda _{1},\dots ,\lambda _{p},\lambda )=\\[5pt]\displaystyle \quad \quad \quad K\lambda ^{r}\sum \limits _{j=1}^{p}{e^{-\lambda _{j}z}}\sum \limits _{k=1}^{r_{j}}{\left\{{c_{j,k}{\frac {\Gamma (k)}{\Gamma (k+r)}}z^{k+r-1}{}_{1}F_{1}(r,k+r,-(\lambda -\lambda _{j})z)}\right\}}{\rm {,}}~~~~(z>0)\end{array}}}$

and the cumulative distribution function is given by

${\displaystyle {\begin{array}{l}\displaystyle F_{Z}^{\text{GNIG}}(z|r_{1},\ldots ,r_{p},r;\,\lambda _{1},\ldots ,\lambda _{p},\lambda )={\frac {\lambda ^{r}\,{z^{r}}}{\Gamma (r+1)}}{}_{1}F_{1}(r,r+1,-\lambda z)\\[12pt]\quad \quad \displaystyle -K\lambda ^{r}\sum \limits _{j=1}^{p}{e^{-\lambda _{j}z}}\sum \limits _{k=1}^{r_{j}}{c_{j,k}^{*}}\sum \limits _{i=0}^{k-1}{\frac {z^{r+i}\lambda _{j}^{i}}{\Gamma (r+1+i)}}{}_{1}F_{1}(r,r+1+i,-(\lambda -\lambda _{j})z)~~~~(z>0)\end{array}}}$

where

${\displaystyle c_{j,k}^{*}={\frac {c_{j,k}}{\lambda _{j}^{k}}}\Gamma (k)}$

with ${\displaystyle c_{j,k}}$ given by ( 1 )-( 3 ) above. In the above expressions ${\displaystyle _{1}F_{1}(a,b;z)}$ 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. ^{[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

1. Amari S.V. and Misra R.B. (1997). Closed-From Expressions for Distribution of Sum of Exponential Random Variables ^{[ permanent dead link ]}. 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. 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. MR 2063631 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. MR 2351709
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. MR 2245158 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].