What are the gamma-Pareto convolutions and how have they been used?

Question

The Pareto distributions, i.e., density functions (pdf), are types I through IV and the type II variant; the Lomax distribution. This makes for a number of possible gamma-Pareto convolutions (GPC; also density functions). Of these the GPC type I and GPC Lomax have been used in the literature, at least in so far as I know. Let us start by defining the parameterizations used for GPC type I.

The gamma distribution (GD) can be given as

\begin{equation}\label{eq:GD} \text{GD}(a,b;t) = \,\dfrac{1}{t}\;\dfrac{e^{-b \, t}(b \, t)^{\,a} }{\Gamma (a)} \;\theta(t)\;, \end{equation}

where the gamma function satisfies $\Gamma (a)=\int _0^{\infty } e^{-t} t^{a-1}\,dt$, and $\theta(t)$ is the unit step function, i.e., 0 for $t<0$ and 1 for $t\geq0$. The GD rate parameter is $b$, whereas $\frac{1}{b}$ is the scale parameter. The shape parameter for the GD is $a$. The shape parameter aids in fitting data and data shapes. There is no location parameter.

Next, we define a type I Pareto distribution ($\text{PD}_{\text{I}}$ a pdf),

\begin{equation} \text{PD}_{\text{I}}(\alpha, \beta;t)= \dfrac{\alpha}{t} \left(\dfrac{\beta}{t}\right) ^{\alpha } \;\theta(t-\beta)\;. \end{equation}

Note the shift to the right of the unit step function; $\theta(t-\beta)$, i.e., the PD function is zero for $t<\beta$. $\beta$ is the scale parameter. The shape parameter is $\alpha$. There is no location parameter.

The PD type II pdf is

$$\text{PD}_{\text{II}}(\alpha ,\beta ,\mu;t)=\frac{\beta }{\alpha }\left(\frac{\alpha -\mu +t}{\alpha }\right)^{-\beta -1}\theta (t-\mu ). $$

The PD type II Lomax pdf is

$$\text{PD}_{\text{Lomax}}(\alpha ,\beta ,0;t)=\frac{\beta }{\alpha }\left(\frac{\alpha +t}{\alpha }\right)^{-\beta -1}\theta (t ) .$$

The PD type III pdf is

$$\text{PD}_{\text{III}}(\alpha ,1,\gamma ,\mu;t)=\frac{\alpha ^{-1/\gamma } (t-\mu )^{\frac{1}{\gamma }-1}}{\gamma \left[\left(\frac{\alpha }{t-\mu }\right)^{-1/\gamma }+1\right]^2}\theta (t-\mu ) .$$

The PD type IV pdf is

$$\text{PD}_{\text{IV}}(\alpha ,\beta,\gamma ,\mu;t)=\frac{\beta}{\gamma } \alpha ^{-1/\gamma } (t-\mu )^{\frac{1}{\gamma }-1} \left[\left(\frac{\alpha }{t-\mu }\right)^{-1/\gamma }+1\right]^{-\beta -1}\theta (t-\mu ) .$$

The infinite series describing the GPC type I convolution is given in the answer below. There is no closed form solution. Such functions are computer implemented as truncated infinite series.

I will award a bounty to anyone who shows a solution$^\star$ for a GPC distribution type whose solution is not referenced here, i.e., GPC type II (non-Lomax), GPC type III or GPC type IV.

$^\star$EDIT: Any exact and calculable reformatting of any convolution integral of the form: $$\begin{align}\text{GPC}_j\left(\begin{array}{cc} a & b \\ \alpha &\beta \end{array} \Big|\,t\right)&=\text{GD}(a,b;x)\ast\text{PD}_j(\alpha,\beta;x)\,(t)\\&=\int_{-\infty}^{\infty} \text{GD}(a,b;x-t)\,\text{PD}_{j}(\alpha,\beta,\,,\,;t) \;dt,\end{align}$$ where $j=\text{II, III, or IV}$. I do not consider limiting discrete convolutions for $\Delta t\to0$, i.e., a limiting or Riemann sum of the convolution integral, to be exact solutions unless the limits sign can be removed in such a fashion as to provide a calculation of any desired accuracy.

Answer 1

The GPC type I has been open access published recently and used to model metformin drug serum concentrations following intravenous bolus injection. Derived from series expansion of the GD exponential above, followed by convolution with a PD, as above and summation of the expanded parts. The GPC type I model inherits two shape parameters, $a$ and $\alpha$, and two scale related parameters, $b\; (T^{-1})$ and $\beta\;(T)$ from its convolved pdf's,

\begin{equation} \begin{split} \text{GPC}_{\text{I}}(t;a,b,\alpha,\beta) &=\text{GD}( x;a,b)\ast \text{PD}_{\text{I}}(x;\alpha , \beta) \;(t)\\ &=\theta (t-\beta) \frac{\alpha b^a\beta^\alpha}{\Gamma (a)}t^{a-\alpha-1}\sum _{n=0}^{\infty } \frac{(-b\,t)^n}{n!} B_{1-\frac{\beta}{t}}(a+n,-\alpha) \end{split}\;\;, \end{equation}

where the incomplete beta function is $B_z(A,B)=\int _0^z u^{A-1} (1-u)^{B-1}d u$. Note that this is exact but is calculable by removing the limiting $\infty$ for $n$ as follows. Because the series is alternating, for $n>m$, where $m$ is some positive integer, no matter how large, the series sum thereafter has decreasing absolute error, as it then satisfies the Leibniz test, where the absolute error is less than the absolute magnitude of the $n>m^{\text{th}}$ term.

Proof: First note that $\lim_{t\to\infty}B_{1-\frac{\beta}{t}}(a+n,-\alpha)=B(a+n,-\alpha)$, where $B(\cdot,\cdot)$ is the beta function. Next, $\lim_{n\to\infty}B(a+n,-\alpha)=0$. Then $\lim_{n\to\infty}t^{a-\alpha-1} \frac{(-b\,t)^n}{n!}=0$, no matter how large $t$. Finally, $0\cdot0=0$, thus the $n^{\text{th}}$ term absolute value can be made smaller than $\epsilon$, no matter how small for $n$ sufficiently large.||

An alternative formulation is used for $t$-long, because the alternating term magnitude inflates as $t$ increases, requiring very high precision for use of the primary definition above. The alternate formula is \begin{equation} \begin{split} \text{GPC}_{\text{I}}(t;a,b,\alpha,\beta)=&-\theta (t-\beta) \frac{ \alpha b^a }{\Gamma (a)}t^{a-1}\sum _{k=1}^{\infty }\left(\frac{\beta }{t}\right)^k \frac{(1-a)_k}{k! (k-\alpha )} \, _1F_1(a,a-k;-b t)\\ &+\theta (t-\beta )\left[ \frac{b^a }{\Gamma (a)} e^{-b t} t^{a-1}-\pi \csc (\pi \alpha )\frac{b^a \beta ^{\alpha } }{\Gamma (\alpha )} t^{a-\alpha -1} \, _1\tilde{F}_1(a,a-\alpha ;-b t)\right] \end{split}\;\;\;. \end{equation}

The above is shown as Theorem 1 of this paper.

The GPC Lomax distribution: On the convolution of Pareto and gamma distributions was published in 2007 and is proposed for usage to determine inter arrival times between on-traffic (off-traffic) random variates. It is derived explicitly in that text in a fashion analogous to the GPC type I above, however, it was parameterized differently and to see it either contact the authors or download it from a library service.

Closed form expressions for GPC types I to IV may not be possible. However, by combining certain parameters and reducing their number by one parameter, closed form expressions for simplified GPC types II through IV can be written.