Delaporte distribution

Delaporte

Probability mass function When $\alpha$ and $\beta$ are 0, the distribution is the Poisson. When $\lambda$ is 0, the distribution is the negative binomial.
Cumulative distribution function When $\alpha$ and $\beta$ are 0, the distribution is the Poisson. When $\lambda$ is 0, the distribution is the negative binomial.
Parameters	$\lambda >0$ (fixed mean) $\alpha ,\beta >0$ (parameters of variable mean)
Support	$k\in \{0,1,2,\dots \}$
pmf	$\sum _{i=0}^k\frac{\mathrm{\Gamma }(\alpha +i){\beta }^i{\lambda }^{k-i}{e}^{-\lambda }}{\mathrm{\Gamma }(\alpha )i!(1+\beta {)}^{\alpha +i}(k-i)!}$
CDF	$\sum _{j=0}^k\sum _{i=0}^j\frac{\mathrm{\Gamma }(\alpha +i){\beta }^i{\lambda }^{j-i}{e}^{-\lambda }}{\mathrm{\Gamma }(\alpha )i!(1+\beta {)}^{\alpha +i}(j-i)!}$
Mean	$\lambda +\alpha \beta$
Mode	$\{\begin{array}{ll}z,z+1& \{z\in \mathbb{\mathbb{Z}}\}:z=(\alpha -1)\beta +\lambda \\ \lfloor z\rfloor & \text{<mrow data-mjx-texclass="ORD">otherwise</mrow>}\end{array}$
Variance	$\lambda +\alpha \beta (1+\beta )$
Skewness	See #Properties
Kurtosis	See #Properties
MGF	$\frac{e^{\lambda (e^t-1)}}{(1-\beta (e^t-1){)}^{\alpha }}$

The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science. It can be defined using the convolution of a negative binomial distribution with a Poisson distribution. Just as the negative binomial distribution can be viewed as a Poisson distribution where the mean parameter is itself a random variable with a gamma distribution, the Delaporte distribution can be viewed as a compound distribution based on a Poisson distribution, where there are two components to the mean parameter: a fixed component, which has the $\lambda$ parameter, and a gamma-distributed variable component, which has the $\alpha$ and $\beta$ parameters. The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959, although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders, where it was called the Formel II distribution.

Properties

The skewness of the Delaporte distribution is:

$\frac{\lambda +\alpha \beta (1+3\beta +2{\beta }^2)}{{(\lambda +\alpha \beta (1+\beta ))}^{\frac{3}{2}}}$

The excess kurtosis of the distribution is:

$\frac{\lambda +3{\lambda }^2+\alpha \beta (1+6\lambda +6\lambda \beta +7\beta +12{\beta }^2+6{\beta }^3+3\alpha \beta +6\alpha {\beta }^2+3\alpha {\beta }^3)}{{(\lambda +\alpha \beta (1+\beta ))}^2}$

Further reading

• Murat, M.; Szynal, D. (1998). "On moments of counting distributions satisfying the k'th-order recursion and their compound distributions". Journal of Mathematical Sciences 92 (4): 4038–4043. doi:10.1007/BF02432340.