Density and distribution evaluation for convolution of independent gamma variables

Abstract

Convolutions of independent gamma variables are encountered in many applications such as insurance, reliability, and network engineering. Accurate and fast evaluations of their density and distribution functions are critical for such applications, but no open source, user-friendly software implementation has been available. We review several numerical evaluations of the density and distribution of convolution of independent gamma variables and compare them with respect to their accuracy and speed. The methods that are based on the Kummer confluent hypergeometric function are computationally most efficient. The benefit of employing the confluent hypergeometric function is further demonstrated by a renewal process application, where the gamma variables in the convolution are Erlang variables. We derive a new computationally efficient formula for the probability mass function of the number of renewals by a given time. An R package coga provides efficient C++ based implementations of the discussed methods and are available in CRAN.

Appendices

Proofs

1.1 Proof of Proposition 1

First, note that

$$\begin{aligned}&\Pr (N(t) = n) = \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n+1} U_k> t \right) \\&\quad = p \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_1> t\right) \\&\qquad +(1-p) \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_2 > t\right) , \end{aligned}$$

where $E_1$ and $E_2$ are independent of $\{ U_k \}_{k \ge 1}$ with $\mathcal {E}xp(\beta _1)$ and $\mathcal {E}xp(\beta _2)$ distributions, respectively. Next, using mixture randomization we get

$$\begin{aligned} \begin{aligned} \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_1 > t \right)&= \Pr \left( \sum _{k=1}^{n} U_k \le t\right) - \Pr \left( \sum _{k=1}^{n} U_k + E_1 \le t\right) \\&= \sum _{k=0}^{n} H(t; (k, \beta _1), (n-k, \beta _2)) {n\atopwithdelims ()k} p^k (1-p)^{n-k}. \end{aligned} \end{aligned}$$

Similarly,

$$\begin{aligned} \begin{aligned} \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_2 > t\right)&= \Pr \left( \sum _{k=1}^{n} U_k \le t\right) - \Pr \left( \sum _{k=1}^{n} U_k + E_2 \le t\right) \\&= \sum _{k=0}^{n} H(t; (n-k, \beta _2), (k, \beta _1)) {n\atopwithdelims ()k} p^k (1-p)^{n-k}. \end{aligned} \end{aligned}$$

That is

$$\begin{aligned} \begin{aligned}&\Pr (N(t) = n)\\&\quad = \sum _{k=0}^{n} \left[ p H(t; (k, \beta _1), (n-k, \beta _2)) + (1-p) H(t; (n-k, \beta _2), (k, \beta _1))\right] {n\atopwithdelims ()k} p^k (1-p)^{n-k} \end{aligned} \end{aligned}$$

(7)

Equation (6) of Lemma 1 completes the proof. $\square$

1.2 Proof of Lemma 1

Equation (4) gives us that

$$\begin{aligned} H(y; (\alpha _1, \beta _1),&(\alpha _2, \beta _2)) \\&\quad = \left( \frac{\beta _1}{\beta _2}\right) ^{\alpha _2} \sum _{k = 0}^{\infty } \left( {\begin{array}{c}\alpha _2 + k - 1\\ k\end{array}}\right) \left( 1-\beta _1/\beta _2\right) ^k G(y/\beta _1; k + \alpha _1 + \alpha _2)\\&\qquad \quad - \left( \frac{\beta _1}{\beta _2}\right) ^{\alpha _2} \sum _{k = 0}^{\infty } \left( {\begin{array}{c}\alpha _2 + k - 1\\ k\end{array}}\right) \left( 1-\beta _1/\beta _2\right) ^k G(y/\beta _1; k + \alpha _1 + \alpha _2 + 1)\\&\quad = \left( \frac{\beta _1}{\beta _2}\right) ^{\alpha _2} \sum _{k = 0}^{\infty } \left( {\begin{array}{c}\alpha _2 + k - 1\\ k\end{array}}\right) (1-\beta _1/\beta _2)^k \frac{(y/\beta _1)^{k+\alpha _1+\alpha _2} e^{-y/\beta _1}}{\varGamma (k + \alpha _1 + \alpha _2 + 1)}\\&\quad = \left( \frac{\beta _1}{\beta _2}\right) ^{\alpha _2} \left( \frac{y}{\beta _1}\right) ^{\alpha _1 + \alpha _2} e^{-y/\beta _1} \sum _{k=0}^{\infty } \frac{\left( {\begin{array}{c}\alpha _2 + k - 1\\ k\end{array}}\right) \left[ y(1/\beta _1 - 1/\beta _2)\right] ^k}{\varGamma (k + \alpha _1 + \alpha _2 + 1)}, \end{aligned}$$

where the second equation follows from

$$\begin{aligned} G(y; \alpha ) - G(y; \alpha + 1) = \frac{y^\alpha e^{-y}}{\varGamma (\alpha + 1)}. \end{aligned}$$

Because of the identities

$$\begin{aligned} \left( {\begin{array}{c}\alpha _2 + k + 1\\ k\end{array}}\right) = \frac{(\alpha _2)_k}{k!} \end{aligned}$$

and

$$\begin{aligned} \varGamma (k + \alpha _1 + \alpha _2 + 1) = (\alpha _1 + \alpha _2 + 1)_k \varGamma (\alpha _1 + \alpha _2 + 1), \end{aligned}$$

we obtain that

$$\begin{aligned} \begin{aligned} H(y; (\alpha _1, \beta _1),&(\alpha _2, \beta _2)) \\ =&\frac{y^{\alpha _1 + \alpha _2} e^{-y/\beta _1}}{\beta _1^{\alpha _1} \beta _2^{\alpha _2} \varGamma (\alpha _1 + \alpha _2 + 1)} \sum _{k=0}^{\infty } \frac{(\alpha _2)_k [y(1/\beta _1 - 1/\beta _2)]^k}{(\alpha _1 + \alpha _2 + 1)_k k!}\\ =&\frac{y^{\alpha _1 + \alpha _2} e^{-y/\beta _1}}{\beta _1^{\alpha _1} \beta _2^{\alpha _2} \varGamma (\alpha _1 + \alpha _2 + 1)} {}_1F_1(\alpha _2; \alpha _1 + \alpha _2 + 1; y(1/\beta _1 - 1/\beta _2)). \end{aligned} \end{aligned}$$

$\square$

1.3 Proof of Corollary 1

Note first that

$$\begin{aligned} \begin{aligned} \Pr (N(t) = n) =&\Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n+1} U_k> t \right) \\ =&\sum _{s=1}^S p_s \Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_s > t\right) , \end{aligned} \end{aligned}$$

where $E_s$ are independent of $\{ U_k \}_{k \ge 1}$ with an $\mathcal {E}xp(\beta _s)$ distribution. Then we get that

$$\begin{aligned}&\Pr \left( \sum _{k=1}^{n} U_k \le t, \sum _{k=1}^{n} U_k + E_s > t\right) \\&\quad =\Pr \left( \sum _{k=1}^{n} U_k \le t \right) - \Pr \left( \sum _{k=1}^{n} U_k + E_s \le t \right) \\&\qquad = \sum _{k_1=0}^{n} \sum _{k_2=0}^{n-k_1} \dots \sum _{k_{S-1}=0}^{n-k_1- \dots -k_{S-2}} H_s(t, (k_1, \beta _1), \dots , (k_S, \beta _S)) \frac{n!}{k_1!k_2! \dots k_S!} p_1^{k_1} \dots p_S^{k_S}. \end{aligned}$$

$\square$

An alternative expression for confluent hypergeometric function

For completeness, we illustrate the derivation given by Di Salvo (2008) here. Equation (2) can be expressed as a multiple integral over $n-1$-dimensional standard simplex (Srivastava and Karlsson 1985), that is,

$$\begin{aligned} \begin{aligned} \phi&=\frac{\varGamma (\gamma )}{\prod _{i=1}^n\varGamma (\alpha _i)}\int _{E_{n-1}} \exp \left\{ x\sum _{i=2}^n t_i\left( \frac{1}{\beta _1}-\frac{1}{\beta _i}\right) \right\} \\&\qquad \times \prod _{i=2}^n t_i^{\alpha _i-1}(1-t_2-\dots -t_n)^{\alpha _1-1} \mathrm {d}t_2 \dots \mathrm {d}t_n, \end{aligned} \end{aligned}$$

(8)

where $E_{n-1}=\{(t_2,\dots ,t_n):t_2>0, \dots , t_n>0, t_2+\dots +t_n\le 1\}$. Di Salvo (2008) demonstrated that, with transformation

$$\begin{aligned} \begin{aligned}&t_2=u_2 \\&t_3=u_3(1-u_2) \\&t_4=u_4(1-u_2)(1-u_3) \\&\dots \\&t_n=u_n(1-u_2)(1-u_3)\dots (1-u_{n-1}), \end{aligned} \end{aligned}$$

Equation (8) can be written as a multiple integral over the standard hypercube, that is,

$$\begin{aligned} \begin{aligned} \phi&=\frac{\varGamma (\gamma )}{\prod _{i=1}^n\varGamma (\alpha _i)}\int _{[0,1]^{n-1}} \exp \{x\sum _{i=2}^n u_i(\frac{1}{\beta _1}-\frac{1}{\beta _i})h_i(\mathbf {u})\}\\&\quad \times \prod _{i=2}^n u_i^{\alpha _i - 1}(1-u_i)^{\bar{\alpha _i}} \mathrm {d}u_2 \dots \mathrm {d}u_n, \end{aligned} \end{aligned}$$

(9)

where $h_2 = 1$, $h_i(\mathbf {u})=\prod _{j=2}^{i-1}(1-u_j)$, $i=3,\dots , n$, and $\bar{\alpha _i}=\sum _{j=i+1}^n \alpha _j + \alpha _1 - 1$, $i=2, \dots , n-1$, $\bar{\alpha _n}=\alpha _1-1$. Because of good analytical properties of integrand in Eq. (9), the multiple integral can be efficiently computed.