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 $X\!$ has a gamma distribution with shape parameter $r$ and rate parameter $\lambda$ 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 $X\sim \Gamma (r,\lambda )\!.$

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

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

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]

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

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

where

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

and

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

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

(1)

and

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

(2)

where

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

(3)

Alternative expressions are available in the literature on generalized chi-squared distribution, which is a field wherecomputer algorithms have been available for some years.

Generalization

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

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

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

Properties

The probability density function of $Z\!$ 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}}{}_{1}F_{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)}}}{}_{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

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

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

Computer modules

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

1 2 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.
↑ Coelho, C. A. (1998). The Generalized Integer Gamma distribution – a basis for distributions in Multivariate Statistics. Journal of Multivariate Analysis, 64, 86-102.
↑ 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.
1 2 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]
↑ Bilodeau, M., Brenner, D. (1999) "Theory of Multivariate Statistics". Springer, New York [Ch. 11, sec. 11.4]
↑ Das, S., Dey, D. K. (2010) "On Bayesian inference for generalized multivariate gamma distribution". Statistics and Probability Letters, 80, 1492-1499.
↑ 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.
↑ Paolella, M. S. (2007) "Intermediate Probability - A Computational Approach". J. Wiley & Sons, New York [Ch. 2, sec. 2.2]
↑ Timm, N. H. (2002) "Applied Multivariate Analysis". Springer, New York [Ch. 3, sec. 3.5]
↑ 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]
↑ 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]
↑ 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.
↑ 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
↑ Kaiser, T., Zheng, F. (2010) "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) "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) "Outage performance of cooperative cognitive relay networks". MsC Thesis, School of Engineering and Science, Victoria University, Melbourne, Australia [Ch. 3, sec. 3.4].

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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