Delaporte distribution explained

Delaporte
Type:	discrete
Pdf Image:	When \alpha and \beta are 0, the distribution is the Poisson. When λ is 0, the distribution is the negative binomial.
Cdf Image:	When \alpha and \beta are 0, the distribution is the Poisson. When λ is 0, the distribution is the negative binomial.
Parameters:	λ > 0 (fixed mean)\alpha, \beta > 0 (parameters of variable mean)
Support:	k \in \{ 0 , 1 , 2 , \ldots\}
Pdf:	\sum ^{k (\Gamma(\alpha + i)\beta ^{iλ} ^{k-i} e ^{-λ})/(\Gamma(\alpha)i!( 1+ \beta) ^{\alpha+i} (k - i)!)} _{i =0}
Cdf:	\sum ^{j (\Gamma(\alpha + i)\beta ^{iλ} ^{j-i} e ^{-λ})/(\Gamma(\alpha)i!( 1+ \beta) ^{\alpha+i} (j - i)!)} _{i =0}
Mean:	λ + \alpha\beta
Mode:	\begin{cases}z, z +1 & \{z \in Z \}: z = (\alpha -1 )\beta +λ \ \lfloor z \rfloor & rm{otherwise}\end{cases}
Variance:	λ + \alpha\beta( 1+ \beta)
Skewness:	See Properties
Kurtosis:	See Properties
Mgf:	(e ^{λ (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] 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

λ 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.

Properties

The skewness of the Delaporte distribution is:

(λ+ \alpha\beta( 1+3 \beta +2 \beta ^{2)})/(\left( λ + \alpha\beta( 1+ \beta)\right) ^{(3)/(2)})

The excess kurtosis of the distribution is:

(λ+3λ ^{2+\alpha\beta(1+6λ+6λ\beta+7\beta+12\beta} ^{2+6\beta} ^{3+3\alpha\beta+6\alpha\beta} ^{2+3\alpha\beta} ^{3)})/(\left( λ+ \alpha\beta( 1+ \beta)\right) ^{2})

Further reading

• Murat . M.. Szynal . D.. On moments of counting distributions satisfying the k'th-order recursion and their compound distributions. Journal of Mathematical Sciences. 1998. 4038 - 4043. 92 . 4. 10.1007/BF02432340 . 122625458. free. none.

Notes and References

1. Encyclopedia: Panjer . Harry Panjer . Harry H. . Teugels . Jozef L. . Bjørn . Sundt . Encyclopedia of Actuarial Science . Discrete Parametric Distributions . 2006 . . 978-0-470-01250-5 . 10.1002/9780470012505.tad027.
2. Book: Johnson. Norman Lloyd. Norman Lloyd Johnson. Kemp. Adrienne W.. Kotz. Samuel. Samuel Kotz. Univariate discrete distributions. Third. 2005. John Wiley & Sons. 978-0-471-27246-5. 241–242.
3. Book: Vose. David. Risk analysis: a quantitative guide. Third, illustrated. 2008. John Wiley & Sons. 978-0-470-51284-5. 2007041696. 618–619.
4. 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. 227. 87–102. French.
5. von Lüders. Rolf . 1934. Die Statistik der seltenen Ereignisse. The statistics of rare events. Biometrika. 26. 1–2 . 108–128. German. 10.1093/biomet/26.1-2.108. 2332055.