generalized integer gamma distribution

The random variable $X\backslash !$ has a gamma distribution with shape parameter $r$ and rate parameter $\backslash lambda$ if its probability density function is

:$$

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 $X\backslash sim\backslash Gamma(r,\backslash lambda)\backslash !.$

Let $X\_j\backslash sim\backslash Gamma(r\_j,\backslash lambda\_j)\backslash !$, where $(j=1,\backslash dots,p),$ be $p$ independent random variables, with all $r\_j$ being positive integers and all $\backslash lambda\_j\backslash !$ 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 $\backslash lambda\_j$ 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

:$$

Y=\sum^p_{j=1} X_j

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

:$Y\backslash sim\; GIG(r\_j,\backslash lambda\_j;p)\backslash !\; .$

It is also a special case of the generalized chi-squared distribution.

The probability density function and the cumulative distribution function of Y are respectively given byAmari S.V. and Misra R.B. (1997). [http://www.dei.unipd.it/~tomasin/RelatedWork/sum-exp-rvs.pdf Closed-From Expressions for Distribution of Sum of Exponential Random Variables]{{Dead link|date=December 2019 |bot=InternetArchiveBot |fix-attempted=yes }}. IEEE Transactions on Reliability, vol. 46, no. 4, 519-522. Coelho, C. A. (1998). [http://www.sciencedirect.com/science/article/B6WK9-45J4Y1R-19/2/b439ff01637b5e9d2c682459a5b9c135 The Generalized Integer Gamma distribution – a basis for distributions in Multivariate Statistics]. Journal of Multivariate Analysis, 64, 86-102. Coelho, C. A. (1999). [http://www.sciencedirect.com/science/article/pii/S0047259X9891805X Addendum to the paper ’The Generalized IntegerGamma distribution - a basis for distributions in MultivariateAnalysis’]. Journal of Multivariate Analysis, 69, 281-285.

:$$

f_Y^{\text{GIG}}(y|r_1,\dots,r_p;\lambda_1,\dots,\lambda_p)\,=\,K\sum^p_{j=1}P_j(y)\,e^{-\lambda_j\,y}\,,~~~~(y>0)

and

:$$

F_Y^{\text{GIG}}(y|r_1,\dots,r_p;\lambda_1,\dots,\lambda_p)\,=\,1-K\sum^p_{j=1}P^*_j(y)\,e^{-\lambda_j\,y}\,,~~~~(y>0)

where

:$$

K=\prod^p_{j=1}\lambda_j^{r_j}~,~~~~~P_j(y)=\sum^{r_j}_{k=1} c_{j,k}\,y^{k-1}

and

:$$

P^*_j(y)=\sum^{r_j}_{k=1}c_{j,k}\,(k-1)!\sum^{k-1}_{i=0}\frac{y^i}{i!\,\lambda_j^{k-i}}

with

{{NumBlk|:|$$

c_{j,r_j}

=\frac{1}{(r_j-1)!}\,\mathop{\prod^p_{i=1}}_{i\neq j}(\lambda_i-\lambda_j)^{-r_i}~,~~~~~~

j=1,\ldots,p\,,

|{{EquationRef|1}}}}

and

{{NumBlk|:|$$

c_{j,r_j-k}=\frac{1}{k}\sum^k_{i=1}\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)

|{{EquationRef|2}}}}

where

{{NumBlk|:|$$

R(i,j,p)=\mathop{\sum^p_{k=1}}_{k\neq j}r_k\left(\lambda_j-\lambda_k\right)^{-i}~~~(i=1,\ldots,r_j-1)\,.

|{{EquationRef|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|date=November 2023}}

The GNIG (generalized near-integer gamma) distribution of depth $p+1$ is the distribution of the random variableCoelho, C. A. (2004). [http://www.sciencedirect.com/science/article/pii/S0047259X03002112 "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]

:$Z=Y\_1+Y\_2\backslash !,$

where $Y\_1\backslash sim\; GIG(r\_j,\backslash lambda\_j;p)\backslash !$ and $Y\_2\backslash sim\backslash Gamma(r,\backslash lambda)\backslash !$ are two independent random variables, where $r$ is a positive non-integer real and where $\backslash lambda\backslash neq\backslash lambda\_j$ $(j=1,\backslash dots,p)$.

The probability density function of $Z\backslash !$ is given by

:$$

\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} {}_1F_1 (r,k+r, - (\lambda-\lambda _j )z)} \right\}} {\rm , } ~~~~(z > 0)

\end{array}

and the cumulative distribution function is given by

:$$

\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)}}{}_1F_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)}}} {}_1F_1 (r,r+1+i, - (\lambda - \lambda _j )z) ~~~~ (z>0)

\end{array}

where

:$$

c_{j,k}^* = \frac{{c_{j,k} }}{{\lambda _j^k }}\Gamma (k)

with $c\_\{j,k\}$ given by ({{EquationNote|1}})-({{EquationNote|3}}) above. In the above expressions $\_1F\_1(a,b;z)$ 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.

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. Bilodeau, M., Brenner, D. (1999) [https://books.google.com/books?id=1pa0VpPC8gsC "Theory of Multivariate Statistics"]. Springer, New York [Ch. 11, sec. 11.4]Das, S., Dey, D. K. (2010) [http://www.sciencedirect.com/science/article/pii/S0167715210001616 "On Bayesian inference for generalized multivariate gamma distribution"]. Statistics and Probability Letters, 80, 1492-1499.Karagiannidis, K., Sagias, N. C., Tsiftsis, T. A. (2006) [https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1673666&tag=1 "Closed-form statistics for the sum of squared Nakagami-m variates and its applications"]. Transactions on Communications, 54, 1353-1359.Paolella, M. S. (2007) [https://books.google.com/books?id=9SHARfvyiR4C&dq=Intermediate+Probability+-+A+Computational+Approach&pg=PP2 "Intermediate Probability - A Computational Approach"]. J. Wiley & Sons, New York [Ch. 2, sec. 2.2]Timm, N. H. (2002) [https://books.google.com/books?id=vtiyg6fnnskC "Applied Multivariate Analysis"]. Springer, New York [Ch. 3, sec. 3.5] 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. Coelho, C. A. (2006) [https://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Coelho%2C%20Carlos%20A%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=12&mx-pid=2351709 "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}}Coelho, C. A., Alberto, R. P. and Grilo, L. M. (2006) [https://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=TI&pg6=PC&pg7=ALLF&pg8=ET&review_format=html&s4=Coelho%2C%20Carlos%20A%2A&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=12&mx-pid=2351709 "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}}

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

Coelho, C. A. (2007) [http://www.tandfonline.com/doi/abs/10.1080/15598608.2007.10411821 "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.

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.E. Björnson, D. Hammarwall, B. Ottersten (2009) [http://www.ee.kth.se/php/modules/publications/reports/2009/IR-EE-SB_2009_010.pdf "Exploiting Quantized Channel Norm Feedback through Conditional Statistics in Arbitrarily Correlated MIMO Systems"], IEEE Transactions on Signal Processing, 57, 4027-4041Kaiser, T., Zheng, F. (2010) [https://books.google.com/books?id=D15M_FLNPGwC&dq=Kaiser+Ultra+Wideband+Systems+With+MIMO&pg=PA247 "Ultra Wideband Systems with MIMO"]. J. Wiley & Sons, Chichester, U.K. [Ch. 6, sec. 6.6]Suraweera, H. A., Smith, P. J., Surobhi, N. A. (2008) [https://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=4531868&tag=1 "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)}}.Surobhi, N. A. (2010) [http://vuir.vu.edu.au/15509/1/nusrat2009.pdf "Outage performance of cooperative cognitive relay networks"]. MsC Thesis, School of Engineering and Science, Victoria University, Melbourne, Australia [Ch. 3, sec. 3.4].

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