View source for Generalized integer gamma distribution

In [[Probability|probability]] and [[Statistics|statistics]], the '''generalized integer gamma distribution''' (GIG) is the distribution of the sum of independent 
[[Gamma distribution|gamma distributed random variables]], all with integer shape parameters and different rate parameters. This is a special case of the [[Generalized chi-squared distribution|generalized chi-squared distribution]]. A related concept is the '''generalized near-integer gamma distribution''' (GNIG).

==Definition==

The [[Random variable|random variable]]  <math>X\!</math> has a [[Gamma distribution|gamma distribution]] with [[Shape parameter|shape parameter]] <math>r</math> and [[Scale parameter#Rate parameter|rate parameter]] <math>\lambda</math> if its [[Physics:Probability density function|probability density function]] is
:<math>
f^{}_X(x)=\frac{\lambda^r}{\Gamma(r)}\,e^{-\lambda x} x^{r-1}~~~~~~(x>0;\,\lambda,r>0)
</math>
and this fact is denoted by <math>X\sim\Gamma(r,\lambda)\!.</math>

Let <math>X_j\sim\Gamma(r_j,\lambda_j)\!</math>, where <math>(j=1,\dots,p),</math> be <math>p</math> independent random variables, with all <math>r_j</math> being positive integers and all <math>\lambda_j\!</math> 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 <math>\lambda_j</math> 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
:<math>
Y=\sum^p_{j=1} X_j
</math>
has a GIG (generalized integer gamma) distribution of depth <math>p</math> with [[Shape parameter|shape parameter]]s  <math>r_j\!</math> and [[Scale parameter#Rate parameter|rate parameters]] <math>\lambda_j\!</math> <math>(j=1,\dots,p)</math>.  This fact is denoted by
:<math>Y\sim GIG(r_j,\lambda_j;p)\! .</math>
It is also a special case of the [[Generalized chi-squared distribution|generalized chi-squared distribution]].

===Properties===
The probability density function and the [[Cumulative distribution function|cumulative distribution function]] of ''Y'' are respectively given by<ref name=amari>Amari 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.</ref><ref> 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.</ref><ref> 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.</ref>

:<math>
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)
</math>
and
:<math>
F_Y^{\text{GIG}}(y|r_1,\dots,r_j;\lambda_1,\dots,\lambda_p)\,=\,1-K\sum^p_{j=1}P^*_j(y)\,e^{-\lambda_j\,y}\,,~~~~(y>0)
</math>
where
:<math>
K=\prod^p_{j=1}\lambda_j^{r_j}~,~~~~~P_j(y)=\sum^{r_j}_{k=1} c_{j,k}\,y^{k-1}
</math>
and
:<math>
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}}
</math>
with
{{NumBlk|:|<math>
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\,,
</math>|{{EquationRef|1}}}}
and
{{NumBlk|:|<math>
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)
</math>|{{EquationRef|2}}}}
where
{{NumBlk|:|<math>
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)\,.
</math>|{{EquationRef|3}}}}

Alternative expressions are available in the literature on [[Generalized chi-squared distribution|generalized chi-squared distribution]], which is a field where computer algorithms have been available for some years.{{when|date=November 2023}}

==Generalization==
The GNIG (generalized near-integer gamma) distribution of depth <math>p+1</math> is the distribution of the random variable<ref name="Coe04">Coelho, 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]</ref>
:<math>Z=Y_1+Y_2\!,</math>
where <math>Y_1\sim GIG(r_j,\lambda_j;p)\!</math> and <math>Y_2\sim\Gamma(r,\lambda)\!</math> are  two independent random variables, where <math>r</math> is a positive non-integer real and where <math>\lambda\neq\lambda_j</math> <math>(j=1,\dots,p)</math>.

===Properties===
The probability density function of <math>Z\!</math> is given by
:<math>
\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}
</math>
and the cumulative distribution function is given by
:<math>
\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}
</math>
where 
:<math>
c_{j,k}^*  = \frac{{c_{j,k} }}{{\lambda _j^k }}\Gamma (k)
</math>
with <math>c_{j,k}</math> given by ({{EquationNote|1}})-({{EquationNote|3}}) above.  In the above expressions <math>_1F_1(a,b;z)</math> 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|multivariate analysis]]. <ref name="Bil99">Bilodeau, M., Brenner, D. (1999) [https://books.google.com/books?id=1pa0VpPC8gsC "Theory of Multivariate Statistics"]. Springer, New York [Ch. 11, sec. 11.4]</ref><ref name="Das10">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.</ref><ref name="Kara06">Karagiannidis, K., Sagias, N. C., Tsiftsis, T. A. (2006) [http://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.</ref><ref name="Pao07">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]</ref><ref name="Timm02">Timm, N. H. (2002) [https://books.google.com/books?id=vtiyg6fnnskC "Applied Multivariate Analysis"]. Springer, New York [Ch. 3, sec. 3.5]</ref> 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. <ref name=Coe04/><ref name="Coe06">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}}</ref><ref name="CoeAlbGri06">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}}</ref>

The GIG distribution is also the basis for a number of [[Wrapped distribution|wrapped distribution]]s in the wrapped gamma family.
<ref name="Coe07">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.</ref>

As being a special case of the [[Generalized chi-squared distribution|generalized chi-squared distribution]], there are many other applications; for example, in renewal theory<ref name=amari /> and in multi-antenna wireless communications.<ref>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, 4027-4041</ref><ref name="Kai10">Kaiser, 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]</ref><ref name="Sur08">Suraweera, H. A., Smith, P. J., Surobhi, N. A. (2008) [http://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)}}.</ref><ref name="Sur10">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].</ref>

==References==
{{reflist}}
{{ProbDistributions|continuous-semi-infinite}}

[[Category:Continuous distributions]]
[[Category:Factorial and binomial topics]]

{{Sourceattribution|Generalized integer gamma distribution}}