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	$k \in \{0, 1, 2, \ldots\}$
pmf	$\sum_{i=0}^k\frac{\Gamma(\alpha + i)\beta^i\lambda^{k-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(k-i)!}$
CDF	$\sum_{j=0}^k\sum_{i=0}^j\frac{\Gamma(\alpha + i)\beta^i\lambda^{j-i}e^{-\lambda}}{\Gamma(\alpha)i!(1+\beta)^{\alpha+i}(j-i)!}$
Mean	$\lambda + \alpha\beta$
Mode	$\begin{cases}z, z+1 & \{z \in \mathbb{Z}\}:\; z = (\alpha-1)\beta+\lambda\\ \lfloor z \rfloor & \textrm{otherwise}\end{cases}$
Variance	$\lambda + \alpha\beta(1+\beta)$
Skewness	See #Properties
Ex. kurtosis	See #Properties

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 $\lambda$ parameter, and a gamma-distributed variable component, which has the $\alpha$ and $\beta$ 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

The skewness of the Delaporte distribution is:

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

The excess kurtosis of the distribution is:

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

References

↑ Panjer, Harry H. (2006). "Discrete Parametric Distributions". In Teugels, Jozef L.; Sundt, Bjørn. Encyclopedia of Actuarial Science. John Wiley & Sons. doi:10.1002/9780470012505.tad027. ISBN 978-0-470-01250-5.
1 2 3 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: 108–128. doi:10.1093/biomet/26.1-2.108. JSTOR 2332055.

External links

Delaporte distribution at Vose Software. Details of derivation.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

Some common univariate probability distributions

Continuous	beta Cauchy chi-squared exponential F gamma Laplace log-normal normal Pareto Student's t uniform Weibull

Discrete	Bernoulli binomial discrete uniform geometric hypergeometric negative binomial Poisson

List of probability distributions

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Delaporte distribution

Properties

References

Further reading

External links