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On the distribution of linear combinations of independent Gumbel random variables

Abstract

The distribution of linear combinations of independent Gumbel random variables is of great interest for modeling risk and extremes in the most different areas of application. In this paper we develop near-exact approximations for the distribution of linear combination of independent Gumbel random variables based on a shifted generalized near-integer gamma distribution and on the distribution of the difference of two independent generalized integer gamma distributions. These near-exact distributions are computationally appealing and numerical studies confirm their accuracy, as assessed by a proximity measure used in related studies. We illustrate the proposed approximations on applied problems in networks engineering, computational biology, and flood risk management.

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Acknowledgments

We thank the Editor, the Associate Editor, and two Reviewers for their careful reading and constructive comments. We also thank Anthony Davison, Barry Arnold, and Vanda de Carvalho for helpful discussions and recommendations on an earlier version of this paper. Part of this work was developed during an academic visit of F. Marques and C. Coelho to the Ecole Polytechnique Fédérale de Lausanne where M. de Carvalho was a Post-Doctoral Fellow. This work was partially supported by the Fundação para a Ciência e a Tecnologia (Portuguese Foundation for Science and Technology) through the project PEst-OE/MAT/UI0297/2014 (Centro de Matemática e Aplicações), and under the Fondecyt Project 11121186.

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Correspondence to Filipe J. Marques.

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Appendix

Appendix

Appendix 1: Results and definitions on distributions of interest

Part I: The GIG and GNIG distributions

Let \(X_j\overset{\text {ind.}}{\sim }\text {Gamma}(r_j,\lambda _j)\) with shape parameters \(r_j \in {\mathbb {N}}\) and rate parameters \(\lambda _j\in {{\mathbb {R}}_+^*}\), all different, for \(j=1,\ldots ,\ell \). The GIG distribution of depth \(\ell \in {\mathbb {N}}\), introduced by Coelho (1998), is defined as the distribution of \(Y = \sum _{j=1}^\ell X_j\), and we denote this by \(Y \sim \text {GIG}(\varvec{r},\varvec{\lambda },\ell )\), for \(\varvec{r} = (r_1,\ldots ,r_\ell )\) and \(\varvec{\lambda } = (\lambda _1,\ldots ,\lambda _\ell ).\) The density and distribution functions of \(Y\) are

$$\begin{aligned} f_{Y}(y; \varvec{r}, \varvec{\lambda }, \ell ) = K\mathop {\sum }\limits _{j=1}^\ell \pi _j(y) \exp \{-\lambda _j y\}, \end{aligned}$$

and

$$\begin{aligned} F_Y(y; \varvec{r}, \varvec{\lambda }, \ell ) = 1- K \mathop {\sum }\limits _{j=1}^\ell \varPi _j(y) \exp \{-\lambda _j y\}, \end{aligned}$$

where \(y > 0,\,K=\prod _{i=1}^p \lambda _i^{r_i}\),

$$\begin{aligned} {\left\{ \begin{array}{ll} \pi _j(y) = \mathop {\sum }\limits _{k=1}^{r_j} c_{j,k}y^{k-1}, \\ \varPi _j(y) = \mathop {\sum }\limits _{k=1}^{r_j} c_{j,k}(k-1)!\mathop {\sum }\limits _{i=0}^{k-1}\frac{y^{k}}{i!\lambda _j^{k-i}}, \end{array}\right. } \end{aligned}$$

and the \(c_{j,k}\) are given in (11)–(13) in Coelho (1998). The GNIG distribution of depth \((\ell +1) \in {\mathbb {N}}\), introduced by Coelho (2004), is defined as the distribution of \(Y^{\star } = X^{\star } + \sum _{j=1}^{\ell } X_j\), where \(X^{\star }\) is independent of \(\sum _{j=1}^{\ell } X_j\), and \(X^{\star } \sim \text {Gamma}(\rho \),  \(l\)), with \(\rho \in {{\mathbb {R}}_+^*} \backslash {\mathbb {N}}\). We denote this by \(Y^{\star } \sim \text {GNIG}(\varvec{r}^{\star }, \varvec{\lambda }^{\star }, \ell +1)\), where \(\varvec{r}^{\star }=(\varvec{r},\rho )\) and \(\varvec{\lambda }^{\star } = (\varvec{\lambda }, l)\), and the corresponding density and distribution functions are

$$\begin{aligned}&f_{Y^{\star }}(y; \varvec{r}^{\star }, \varvec{\lambda }^{\star }, \ell +1)\\&\quad = K l ^\rho \sum \limits _{j = 1}^\ell {\exp \{- \lambda _j y\} } \\&\quad \times \sum \limits _{k = 1}^{r_j } {\bigg \{ {c_{j,k} \tfrac{{\varGamma (k)}}{{\varGamma (k\!+\!\rho )}}y^{k + \rho - 1} {}_1F_1 (\rho ,k\!+\!\rho , - (l\!-\!\lambda _j )y)} \bigg \}}, \end{aligned}$$

and

$$\begin{aligned}&F_{Y^{\star }}(y; \varvec{r}^{\star },\varvec{\lambda }^{\star }, \ell +1)\nonumber \\&\quad = \tfrac{l ^\rho \,{y^\rho }}{{\varGamma (\rho \!+\!1)}}{}_1F_1 (\rho ,\rho \!+\!1, - l y) - K l ^\rho \sum \limits _{j = 1}^\ell {\exp \{- \lambda _j y\} }\nonumber \\&\quad \times \sum \limits _{k = 1}^{r_j } {c_{j,k}^* } \sum \limits _{i = 0}^{k - 1} {\tfrac{{y^{r + i} \lambda _j^i }}{{\varGamma (\rho \!+\!1\!+\!i)}}} {}_1F_1 (\rho ,\rho \!+\!1\!+\!i, - (l - \lambda _j )y), \nonumber \\ \end{aligned}$$
(23)

for \(y > 0\) and where \(c^*_{j,k}=(c_{j,k}\lambda _j^k)/\varGamma (k)\); in the above expressions \(_1F_1(\cdot )\) denotes the Kummer confluent hypergeometric function.

The random variable \(X^*=X+\theta \) is a shifted Gamma distribution with rate \(\lambda \in {{\mathbb {R}}_+^*}\), shape \(r \in {{\mathbb {R}}_+^*}\), and shift \(\theta \in {\mathbb {R}}\), if \(X\sim \text {Gamma}(r,\lambda )\), and we denote this by \(X^*\sim \text {SGamma}(r,\lambda ,\theta )\); the shifted GIG and GNIG distributions are analogously defined and denoted by \(\mathrm{SGIG}(\varvec{r},\varvec{\lambda },\ell ,\theta )\) and \(\mathrm{SGNIG}(\varvec{r}^{\star },\varvec{\lambda }^{\star },\ell +1,\theta )\).

Part II: The DGIG distribution and the sum (and the difference) of a DGIG random variable with an independent Gamma random variable

Let \(X_1\sim \text {GIG}(\varvec{r}_1,\varvec{\lambda }_1,p_1)\), with \(\varvec{r}_1=(r_{11},\dots ,r_{1p_1})\) and \(\varvec{\lambda }_1=(\lambda _{11},\dots ,\lambda _{1p_1})\), and \(X_2\sim \text {GIG}(\varvec{r}_2,\varvec{\lambda }_2,p_2)\), with \(\varvec{r}_2=(r_{21},\dots ,r_{2p_2})\) and \(\varvec{\lambda }_2=(\lambda _{21},\dots ,\lambda _{2p_2})\) be two independent random variables with GIG distributions. Let us then consider the random variable \(Y=X_1-X_2\). \(Y\) has a DGIG distribution whose density and distribution functions are given by (2.12) and (2.15) in Coelho and Mexia (2010), and we denote this by \(Y\sim \text {DGIG}(\varvec{r}_1,\varvec{r}_2,\varvec{\lambda }_1,\varvec{\lambda }_2,p_1,p_2)\). The shifted SDGIG distribution, with shift \(\theta \in {\mathbb {R}}\), is denoted by \(Y\sim \mathrm{SDGIG}(\varvec{r}_1,\varvec{r}_2,\varvec{\lambda }_1,\varvec{\lambda }_2,p_1,p_2,\theta )\). Next we obtain results on the distribution of the sum (and the difference) of a DGIG with an independent Gamma random variable; these results are relevant for our third near-exact distribution. One useful way to look at the distribution of \(Y\) is to see it as a particular mixture of integer Gamma or Erlang distributions. Indeed, after some rearrangements the density and distribution functions of \(Y\) may be respectively written as

$$\begin{aligned} f^{}_Y(y)=\left\{ \begin{array}{lcl} \displaystyle \mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p_{jki}\,f^{}_{Y_{jki}}(y), &{}&{} y\ge 0,\\ \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,f^{}_{Y^*_{jki}}(-y), &{}&{} y< 0,\\ \end{array} \right. \end{aligned}$$

and

$$\begin{aligned} F^{}_Y(y)=\left\{ \begin{array}{lcl} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\\ \displaystyle +\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p_{jki}\,F^{}_{Y_{jki}}(y), &{}&{} y\ge 0,\\ \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\\ \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,F^{}_{Y^*_{jki}}(-y), &{}&{} y< 0,\\ \end{array} \right. \end{aligned}$$

where, for \({j\!=\!1,\dots ,p_1;k\!=\!1,\dots ,r_{1j};i\!=\!0,\dots ,k-1}\),

$$\begin{aligned} p^{}_{jki}=\frac{K_1K_2}{\lambda _{1j}^{k-i}}\,c_{jk} \mathop {\sum }\limits _{\ell =1}^{p_2}\mathop {\sum }\limits _{h=1}^{r_{2\ell }} d_{\ell h}\,\frac{(k-1)!}{i!}\,\frac{(h+i-1)!}{(\lambda _{1j}+\lambda _{2\ell })^{h+i}} \end{aligned}$$

and, for \({j=1,\dots ,p_2;k=1,\dots ,r_{2j};i=0,\dots ,k-1}\),

$$\begin{aligned} p^{*}_{jki}=\frac{K_1K_2}{\lambda _{2j}^{k-i}}\,d_{jk}\mathop {\sum }\limits _{\ell =1}^{p_1}\mathop {\sum }\limits _{h=1}^{r_{1\ell }} c_{\ell h}\,\frac{(k-1)!}{i!}\,\frac{(h+i-1)!}{(\lambda _{1j}+\lambda _{2\ell })^{h+i}}, \end{aligned}$$

with

$$\begin{aligned} K_1=\prod _{j=1}^{p_1}\lambda _{1j}^{r_{1j}},\quad K_2=\prod _{j=1}^{p_2}\lambda _{2j}^{r_{2j}}, \end{aligned}$$

and \(c_{jk}\,{(j=1,\dots ,p_1;k=1,\dots ,r_{1j})}\) given by (2.9)–(2.11) in Coelho and Mexia (2010), with \(p\) replaced by \(p_1\) and \(r_j\) replaced by \(r_{1j}\) and \(d_{jk}\,(j\!=\!1,\dots ,p_2;k\!=\!1,\dots ,r_{2j})\) defined in a similar manner, replacing \(p_1\) by \(p_2\) and \(r_{1j}\) by \(r_{2j}\), and where, for \(y\ge 0\),

$$\begin{aligned} f^{}_{Y_{jki}}(y)=\frac{\lambda _{1j}^{k-i}}{\varGamma (k-i)} \,y^{k-i-1}e^{-\lambda _{1j}y}, \end{aligned}$$

and

$$\begin{aligned} F^{}_{Y_{jki}}(y)=1-\mathop {\sum }\limits _{t=0}^{k-i-1} \frac{\lambda _{1j}^{t}}{t!}\,y^t\,e^{-\lambda _{1j}y}, \end{aligned}$$
(24)

are respectively the density and distribution functions of \(Y_{jki}\sim \text {Gamma}(k-i,\lambda _{1j})\), while \(f^{}_{Y^*_{jki}}(\,\cdot \,)\) and \(F^{}_{Y^*_{jki}}(\,\cdot \,)\) are the density and distribution functions of \(Y^*_{jki}\sim \text {Gamma} (k-i,\lambda _{2j})\).

The weights \(p^{}_{jki}\) and \(p^*_{jki}\) verify the relation

$$\begin{aligned} \mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}+\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}=1\,. \end{aligned}$$

Let now \(W\sim \text {Gamma}(\rho ,\lambda )\), where \(\rho \) is a positive non-integer real, be independent of \(Y\). We will consider the random variables \(Z_1=Y+W\) and \(Z_2=Y-W\) and derive their distribution functions. The distribution function of \(Z_1\), will be given by

$$\begin{aligned} F^{}_{Z_1}(z)=\int _0^{+\infty } F^{}_Y(z-w)\,f^{}_W(w)\,\text {d}w, \end{aligned}$$

which, for \({z\ge 0}\), using the notation introduced above for the GNIG distribution function, with \(\varvec{r}^\star =(k-i,\rho )\) and \(\varvec{\lambda }_1^\star =(\lambda _{1j},\lambda )\), may be written as

$$\begin{aligned} \begin{array}{rcl} F^{}_{Z_1}(z) &{} = &{} \displaystyle \int _0^z F^{}_Y(\,\underbrace{z-w}_{\ge 0}\,)\,f^{}_W(w)\,\text {d}w\\ &{}&{} \displaystyle + \int _z^{+\infty } \!\!F^{}_Y(\,\underbrace{z-w}_{\le 0}\,)\,f^{}_W(w)\,\text {d}w \\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\int _0^z\,f^{}_W(w)\,\text {d}w\\ &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}\! \underbrace{\int _0^z \!\!\!F^{}_{Y_{jki}}(z\!-\!w)\,f^{}_W(w)\, \text {d}w}_{\begin{array}{c}\\ \hbox {distribution function of }\\ \scriptstyle G_1\sim \text {GNIG}(\varvec{r}^\star , \varvec{\lambda }_1^\star ,2) \end{array}} \end{array} \\ \begin{array}{rcl} &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\left( 1-\int _0^z f^{}_W(w)\,\text {d}w\right) \\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\!\underbrace{\int _z^{+\infty }\!\!\! F^{}_{Y^*_{jki}}(w\!-\!z)\,f^{}_W(w)\, \text {d}w}_{1-F^{}_{W-Y^*_{jki}}(z)}\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\\ &{}&{} \displaystyle + \mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}F^{}_{G_1}(z,\varvec{r}^\star , \varvec{\lambda }_1^\star ,2)\\ &{}&{} \displaystyle - \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki} \left( 1-F^{}_{W-Y^*_{jki}}(z)\right) \\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}} \mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}F^{}_{G_1}(z,\varvec{r}^\star , \varvec{\lambda }_1^\star ,2)\\ &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki} \,F^{}_{W-Y^*_{jki}}(z), \end{array} \end{aligned}$$

while for \(z< 0\) we have

$$\begin{aligned} \begin{array}{rcl} F^{}_{Z_1}(z) &{} = &{} \displaystyle \int _0^{+\infty } F^{}_Y(\,\underbrace{z-w}_{\le 0}\,)\,f^{}_W(w)\,\text {d}w\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\underbrace{\int _0^{+\infty }f^{}_W(w)\,\text {d}w}_{=1}\\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\!\underbrace{\int _0^{+\infty }\!\!\!F^{}_{Y^*_{jki}}(w\!-\!z)\,f^{}_W(w)\, \text {d}w}_{=1-F^{}_{W-Y^*_{jki}}(z)}\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,F^{}_{W-Y^*_{jki}}(z)\,. \end{array} \end{aligned}$$

We thus have

$$\begin{aligned} F^{}_{Z_1}(z)=\left\{ \begin{array}{lcl} \displaystyle \mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}F^{}_{G_1}(z;\varvec{r}^\star , \varvec{\lambda }_1^\star ,2)\\ \displaystyle +\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\, F^{}_{W-Y^*_{jki}}(z), &{}&{} z\ge 0, \\ \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\, F^{}_{W-Y^*_{jki}}(z), &{}&{} z< 0. \end{array}\right. \end{aligned}$$
(25)

Concerning \(Z_2=Y-W\) we have, for \(z<0\), using the notation introduced above for the GNIG distribution function, with \(\varvec{r}^\star =(k-i,\rho )\) and \(\varvec{\lambda }_2^\star =(\lambda _{2j},\lambda )\),

$$\begin{aligned} \begin{array}{rcl} F^{}_{Z_2}(z) &{} = &{} \displaystyle P(Y-W\le z)=P(Y\le W+z)\\ &{} = &{} \displaystyle \int _0^{-z}F^{}_Y(\,\underbrace{w+z}_{\le 0}\,)\,f^{}_W(w)\, \text {d}w\\ &{}&{} \displaystyle +\int _{-z}^{+\infty }\!\!F^{}_Y(\,\underbrace{w+z}_{\ge 0}\,) \,f^{}_W(w)\,\text {d}w\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\int _0^{-z} f^{}_W(w)\,\text {d}w\\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1}\! p^*_{jki}\!\underbrace{\int _0^{-z}\!\!\! F^{}_{Y^*_{jki}}(\!-w\!-\!z)\,f^{}_W(w)\,\text {d}w\!}_{ \begin{array}{c} \hbox {distribution function of}\\ {\scriptstyle G_2\sim \text {GNIG}(\varvec{r}^\star , \varvec{\lambda }_2^\star ,2)}\end{array}}\\ \end{array} \\ \begin{array}{rcl} &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki} \int _{-z}^{+\infty }f^{}_W(w)\,\text {d}w\\ &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki} \!\underbrace{\int _{-z}^{+\infty }\!\!\! F^{}_{Y^*_{jki}}(w\!+\!z)\,f^{}_W(w)\,\text {d}w\!}_{1-F^{}_{W-Y_{jki}}(-z)}\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki} +\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}\\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\, F^{}_{G_2}(-z;\varvec{r}^\star ,\varvec{\lambda }_2^\star ,2)\\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\, F^{}_{W-Y_{jki}}(-z)\\ &{} = &{} \displaystyle 1-\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,F^{}_{W-Y_{jki}}(-z)\\ &{}&{} \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\, F^{}_{G_2}(-z;\varvec{r}^\star ,\varvec{\lambda }_2^\star ,2) \end{array} \end{aligned}$$

while for \(z\ge 0\) we have

$$\begin{aligned} \begin{array}{rcl} F^{}_{Z_2}(z) &{} = &{} \displaystyle P(Y-W\le z)=P(Y\le W+z)\\ &{} = &{} \displaystyle \int _{0}^{+\infty }F^{}_Y(\, \underbrace{w+z}_{\ge 0}\,)\,f^{}_W(w)\,\text {d}w\\ &{} = &{} \displaystyle \mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki} \int _0^{+\infty } f^{}_W(w)\,\text {d}w\\ &{}&{} \displaystyle +\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki}\! \underbrace{\int _0^{+\infty }\!\!\!F^{}_{Y_{jki}}(w\!+\!z)\, f^{}_W(w)\,\text {d}w}_{1-F^{}_{W-Y_{jki}}(-z)}\\ &{} = &{} \displaystyle 1-\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki} F^{}_{W-Y_{jki}}(-z), \end{array} \end{aligned}$$

so that

$$\begin{aligned} \begin{array}{l} \displaystyle F^{}_{Z_2}(z)=\\ \left\{ \begin{array}{lcl} \displaystyle 1-\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^{}_{jki} F^{}_{W-Y_{jki}}(-z), &{}&{} z\ge 0,\\ \displaystyle 1-\mathop {\sum }\limits _{j=1}^{p_1}\mathop {\sum }\limits _{k=1}^{r_{1j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,F^{}_{W-Y_{jki}}(-z)\\ \displaystyle -\mathop {\sum }\limits _{j=1}^{p_2}\mathop {\sum }\limits _{k=1}^{r_{2j}}\mathop {\sum }\limits _{i=0}^{k-1} p^*_{jki}\,F^{}_{G_2}(-z;\varvec{r}^\star , \varvec{\lambda }_2^\star ,2),&{}&{} z<0\,. \end{array}\right. \end{array} \end{aligned}$$
(26)

It remains now to obtain the distribution function of random variables of the type of \(Z^*=W-Y^*\), where \(W\sim \text {Gamma}(\rho ,\lambda )\) and \(Y^*\sim \text {Gamma}(r,\lambda _1)\), where \(\rho ,\lambda _1\) and \(\lambda _2\) are positive reals and \(r\) is a positive integer. The distribution function of \(Z^*\) is given by

$$\begin{aligned} \begin{array}{rcl} F^{}_{Z^*}(z) &{} = &{} \displaystyle P(W-Y^*\le z)=P(-Y^*\le z-W)\\ &{} = &{} \displaystyle 1-P(Y^*\le W-z)\\ &{} = &{} \displaystyle 1-\int _0^{+\infty } F^{}_{Y^*}(w-z)\,f^{}_W(w)\,\text {d}w \end{array} \end{aligned}$$

which for \(z\ge 0\), using the expression in (24) for the distribution function of an integer Gamma or Erlang distribution, yields

$$\begin{aligned} F_{Z^*}(z)&= \displaystyle 1-\int _0^z \underbrace{F_{Y^*}(\underbrace{w-z}_{\le 0})}_{=0}\,f_W(w)\,\text {d}w\\&\displaystyle -\int _z^{+\infty } F_{Y^{*}}(w-z)\,f_W(w)\,\text {d}w\\&= \displaystyle 1-\int _z^{+\infty } \{1-P(Y^{*}>w-z)\}\,f_W(w)\,\text {d}w\\&= \displaystyle 1-\int _z^{+\infty } \!\!\!\!f_W(w)\,\text {d}w\\&\displaystyle +\int _z^{+\infty } \!\!\!\!P(Y^{*}>w-z)\,f_W(w)\,\text {d}w\\&= \displaystyle F_W(z)+\frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z} \left\{ \mathop {\sum }\limits _{t=0}^{r-1}\frac{\lambda _1^r}{t!}\right. \\&\displaystyle \left. \int _z^{+\infty }(w-z)^t w^{\rho -1}\,e^{-w(\lambda +\lambda _1)}\,\text {d}w\!\right\} \\&= \displaystyle F_W(z)+\frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z} \left\{ \mathop {\sum }\limits _{t=0}^{r-1}\frac{\lambda _1^r}{t!}\mathop {\sum }\limits _{k=0}^t\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )(-z)^k\right. \\&\displaystyle \left. \int _z^{+\infty } w^{t+\rho -k-1}e^{-w(\lambda +\lambda _1)} \text {d}w\!\right\} \\&= \displaystyle 1\!-\! \frac{\varGamma (\rho ,\lambda z)}{\varGamma (\rho )}\!+\! \frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z}\left\{ \mathop {\sum }\limits _{t=0}^{r-1} \frac{\lambda _1^r}{t!}\mathop {\sum }\limits _{k=0}^t{\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )}\right. \\&\displaystyle \left. (\!-z)^k (\lambda \!+\!\lambda _1)^{-t-\rho +k} \varGamma (t\!+\!\rho \!-\!k,(\lambda \!+\!\lambda _1)z)\right\} \end{aligned}$$

while for \(z<0\) it yields

$$\begin{aligned} F_{Z^*}(z)&= \displaystyle 1-\int _0^{+\infty }F_{Y^*}(w-z)\,f_W(w)\, \text {d}w\\&= \displaystyle 1-\int _0^{+\infty }\{1-P(Y^*>w-z)\}\,f_W(w)\,\text {d}w\\&= \displaystyle 1-\int _0^{+\infty } f_W(w)\,\text {d}w\\&\displaystyle +\int _0^{+\infty }P(Y^*>w-z)\,f_W(w)\,\text {d}w\\&= \displaystyle \frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z} \left\{ \mathop {\sum }\limits _{t=0}^{r-1}\frac{\lambda _1^r}{t!} \mathop {\sum }\limits _{k=0}^t{\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )}(-z)^k\right. \\&\displaystyle \left. \int _0^{+\infty }\! w^{t+\rho -k-1}e^{-w(\lambda +\lambda _1)} \text {d}w\right\} \\&= \displaystyle \frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z} \left\{ \mathop {\sum }\limits _{t=0}^{r-1}\frac{\lambda _1^r}{t!} \mathop {\sum }\limits _{k=0}^t{\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )}(-z)^k\right. \\&\displaystyle \left. (\lambda +\lambda _1)^{-t-\rho +k} \varGamma (t+\rho -k)\right\} \!, \end{aligned}$$

and as such

$$\begin{aligned} \begin{array}{l} F_{Z^*}(z)=\\ \left\{ \begin{array}{lcl} \displaystyle 1\!-\!\frac{\varGamma (\rho ,\lambda z)}{\varGamma (\rho )}\!+\! \frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z}\left\{ \mathop {\sum }\limits _{t=0}^{r-1} \frac{\lambda _1^r}{t!}\mathop {\sum }\limits _{k=0}^t{\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )}\right. \\ \displaystyle \left. (\!-z)^k (\lambda \!+\!\lambda _1)^{-t-\rho +k} \varGamma (t\!+\!\rho \!-\!k,(\lambda \!+\!\lambda _1)z)\!\right\} \!\!, &{}&{}\! z\ge 0,\\ \displaystyle \frac{\lambda ^\rho }{\varGamma (\rho )}\,e^{\lambda _1z}\left\{ \mathop {\sum }\limits _{t=0}^{r-1}\frac{\lambda _1^r}{t!}\mathop {\sum }\limits _{k=0}^t{\Big ({\begin{array}{ll}t \\ k\end{array}}\Big )}(-z)^k \right. \\ \displaystyle \left. (\lambda +\lambda _1)^{-t-\rho +k}\varGamma (t+\rho -k)\right\} \!\!, &{}&{}\! z\!<0. \end{array} \right. \end{array} \end{aligned}$$

Appendix 2: Representation of a logarithmized Gamma distribution as an infinite sum of shifted exponential distributions

If \(X\sim \mathrm{Gamma}(r,\lambda )\) its \(h\)th moment is given by

$$\begin{aligned} E\left( X^h\right) =\frac{\varGamma \left( r+h\right) }{\varGamma (r)}\,\lambda ^{-h}\,. \end{aligned}$$
(27)

Then, the random variable \(Y=-\log \, X\) has what we call a logarithmized Gamma distribution and its characteristic function may be obtained from (27) in the following way

$$\begin{aligned} \varPhi _{Y}(t)=E(Y^{-\mathrm {i}t})=\frac{\varGamma \left( r-\mathrm {i}t\right) }{\varGamma (r)}\,\lambda ^{\mathrm {i}t},\quad t\in {\mathbb {R}}, \end{aligned}$$

Using the equality

$$\begin{aligned} \varGamma (z)=\frac{1}{z}\prod _{n=1}^{\infty }\left[ \left( 1+\frac{1}{n}\right) ^z\left( 1+\frac{z}{n}\right) ^{-1}\right] ,\quad z\in \mathbb {C}, \end{aligned}$$

we have

$$\begin{aligned} \varPhi _{Y}(t)&= \frac{1}{\varGamma (r)}\frac{1}{r-\mathrm {i}t}\prod _{n=1}^{\infty }\left[ \left( 1+\frac{1}{n}\right) ^{r-\mathrm {i}t}\right. \\&\times \left. \left( 1+\frac{r-\mathrm {i}t}{n}\right) ^{-1} \right] \exp \{\log \lambda ^{\mathrm {i}t}\}\\&= \left\{ \frac{r}{r-\mathrm {i}t}\,\exp \{\log \lambda ^{\mathrm {i}t}\}\right\} \bigg [\prod _{n=1}^{\infty }\frac{n+r}{n+r-\mathrm {i}t}\, \\&\times \exp \left\{ \mathrm {i}t\left( -\log \left( 1+\frac{1}{n}\right) \right) \right\} \bigg ]. \end{aligned}$$

Hence \(\varPhi _Y\) is also the characteristic function of an infinite sum of independent shifted Exponential distributions. This shows that a logarithmized Gamma random variable may be represented as an infinite sum of independent shifted Exponential random variables.

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Marques, F.J., Coelho, C.A. & de Carvalho, M. On the distribution of linear combinations of independent Gumbel random variables. Stat Comput 25, 683–701 (2015). https://doi.org/10.1007/s11222-014-9453-5

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Keywords

  • Generalized integer gamma distribution
  • Generalized near-integer gamma distribution
  • Gumbel distribution
  • Near-exact distribution
  • Phase type distributions
  • Risk