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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 (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

[EAS-1] 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.

[UDD-2] 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-3] 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.

[DP-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 (in French). 227: 87–102.

[Luders-5] 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.

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Properties

References

Further reading

External links