Delaporte distribution

Delaporte
Probability mass function Plot of the PMF for various Delaporte distributions. When $\alpha$ {\displaystyle \alpha } and $\beta$ {\displaystyle \beta } are 0, the distribution is the Poisson. When $\lambda$ {\displaystyle \lambda } is 0, the distribution is the negative binomial.
Cumulative distribution function Plot of the PMF for various Delaporte distributions. When $\alpha$ {\displaystyle \alpha } and $\beta$ {\displaystyle \beta } are 0, the distribution is the Poisson. When $\lambda$ {\displaystyle \lambda } is 0, the distribution is the negative binomial.
Parameters	$\lambda >0$ {\displaystyle \lambda >0} (fixed mean) $\alpha ,\beta >0$ {\displaystyle \alpha ,\beta >0} (parameters of variable mean)
Support	$k\in \{0,1,2,\ldots \}$ {\displaystyle k\in \{0,1,2,\ldots \}}
PMF	$\sum _{i=0}^{k}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{k-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(k-i)!}}$ {\displaystyle \sum _{i=0}^{k}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{k-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(k-i)!}}}
CDF	$\sum _{j=0}^{k}\sum _{i=0}^{j}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{j-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(j-i)!}}$ {\displaystyle \sum _{j=0}^{k}\sum _{i=0}^{j}{\frac {\Gamma (\alpha +i)\beta ^{i}\lambda ^{j-i}e^{-\lambda }}{\Gamma (\alpha )i!(1+\beta )^{\alpha +i}(j-i)!}}}
Mean	$\lambda +\alpha \beta$ {\displaystyle \lambda +\alpha \beta }
Mode	${\begin{cases}z,z+1&\{z\in \mathbb {Z} \}:\;z=(\alpha -1)\beta +\lambda \\\lfloor z\rfloor &{\textrm {otherwise}}\end{cases}}$ {\displaystyle {\begin{cases}z,z+1&\{z\in \mathbb {Z} \}:\;z=(\alpha -1)\beta +\lambda \\\lfloor z\rfloor &{\textrm {otherwise}}\end{cases}}}
Variance	$\lambda +\alpha \beta (1+\beta )$ {\displaystyle \lambda +\alpha \beta (1+\beta )}
Skewness	See #Properties
Excess kurtosis	See #Properties
MGF	${\frac {e^{\lambda (e^{t}-1)}}{(1-\beta (e^{t}-1))^{\alpha }}}$ {\displaystyle {\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.^[1]^[2] It can be defined using the convolution of a negative binomial distribution with a Poisson distribution.^[2] 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 $\lambda$ {\displaystyle \lambda } parameter, and a gamma-distributed variable component, which has the $\alpha$ {\displaystyle \alpha } and $\beta$ {\displaystyle \beta } parameters.^[3] The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959,^[4] although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders,^[5] where it was called the Formel II distribution.^[2]

Properties

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The skewness of the Delaporte distribution is:

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

The excess kurtosis of the distribution is:

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

References

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^ Panjer, Harry H. (2006). "Discrete Parametric Distributions". In Teugels, Jozef L.; Sundt, Bjørn (eds.). Encyclopedia of Actuarial Science. John Wiley & Sons. doi:10.1002/9780470012505.tad027. ISBN 978-0-470-01250-5.
^ ^a ^b ^c Johnson, Norman Lloyd; Kemp, Adrienne W.; Kotz, Samuel (2005). Univariate discrete distributions (Third ed.). John Wiley & Sons. pp. 241–242. ISBN 978-0-471-27246-5.
^ Vose, David (2008). Risk analysis: a quantitative guide (Third, illustrated ed.). John Wiley & Sons. pp. 618–619. ISBN 978-0-470-51284-5. LCCN 2007041696.
^ Delaporte, Pierre J. (1960). "Quelques problèmes de statistiques mathématiques poses par l'Assurance Automobile et le Bonus pour non sinistre" [Some problems of mathematical statistics as related to automobile insurance and no-claims bonus]. Bulletin Trimestriel de l'Institut des Actuaires Français (in French). 227: 87–102.
^ von Lüders, Rolf (1934). "Die Statistik der seltenen Ereignisse" [The statistics of rare events]. Biometrika (in German). 26 (1–2): 108–128. doi:10.1093/biomet/26.1-2.108. JSTOR 2332055.

External links

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Probability distributions (list)

Discrete
univariate

with finite support	Benford Bernoulli Beta-binomial Binomial Categorical Hypergeometric Negative Poisson binomial Rademacher Soliton Discrete uniform Zipf Zipf–Mandelbrot
with infinite support	Beta negative binomial Borel Conway–Maxwell–Poisson Discrete phase-type Delaporte Extended negative binomial Flory–Schulz Gauss–Kuzmin Geometric Logarithmic Mixed Poisson Negative binomial Panjer Parabolic fractal Poisson Skellam Yule–Simon Zeta

Continuous
univariate

supported on a bounded interval	Arcsine ARGUS Balding–Nichols Bates Beta Generalized Beta rectangular Continuous Bernoulli Irwin–Hall Kumaraswamy Logit-normal Noncentral beta PERT Raised cosine Reciprocal Triangular U-quadratic Uniform Wigner semicircle
supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind Beta prime Burr Chi Chi-squared Noncentral Inverse Scaled Dagum Davis Erlang Hyper Exponential Hyperexponential Hypoexponential Logarithmic F Noncentral Folded normal Fréchet Gamma Generalized Inverse gamma/Gompertz Gompertz Shifted Half-logistic Half-normal Hotelling's T-squared Inverse Gaussian Generalized Kolmogorov Lévy Log-Cauchy Log-Laplace Log-logistic Log-normal Log-t Lomax Matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami Pareto Phase-type Poly-Weibull Rayleigh Relativistic Breit–Wigner Rice Truncated normal type-2 Gumbel Weibull Discrete Wilks's lambda
supported on the whole real line	Cauchy Exponential power Fisher's z Kaniadakis κ-Gaussian Gaussian q Generalized normal Generalized hyperbolic Geometric stable Gumbel Holtsmark Hyperbolic secant Johnson's S_U Landau Laplace Asymmetric Logistic Noncentral t Normal (Gaussian) Normal-inverse Gaussian Skew normal Slash Stable Student's t Tracy–Widom Variance-gamma Voigt
with support whose type varies	Generalized chi-squared Generalized extreme value Generalized Pareto Marchenko–Pastur Kaniadakis κ-exponential Kaniadakis κ-Gamma Kaniadakis κ-Weibull Kaniadakis κ-Logistic Kaniadakis κ-Erlang q-exponential q-Gaussian q-Weibull Shifted log-logistic Tukey lambda