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

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

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.

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.