Delaporte distribution
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Probability mass function When α {\displaystyle \alpha } and β {\displaystyle \beta } are 0, the distribution is the Poisson. When λ {\displaystyle \lambda } is 0, the distribution is the negative binomial. | |||
Cumulative distribution function When α {\displaystyle \alpha } and β {\displaystyle \beta } are 0, the distribution is the Poisson. When λ {\displaystyle \lambda } is 0, the distribution is the negative binomial. | |||
| Parameters |
λ > 0 {\displaystyle \lambda >0} (fixed mean) α , β > 0 {\displaystyle \alpha ,\beta >0} (parameters of variable mean) | ||
|---|---|---|---|
| Support | k ∈ { 0 , 1 , 2 , … } {\displaystyle k\in \{0,1,2,\ldots \}} | ||
| PMF | ∑ i = 0 k Γ ( α + i ) β i λ k − i e − λ Γ ( α ) i ! ( 1 + β ) α + 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 | ∑ j = 0 k ∑ i = 0 j Γ ( α + i ) β i λ j − i e − λ Γ ( α ) i ! ( 1 + β ) α + 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 | λ + α β {\displaystyle \lambda +\alpha \beta } | ||
| Mode | { z , z + 1 { z ∈ Z } : z = ( α − 1 ) β + λ ⌊ z ⌋ otherwise {\displaystyle {\begin{cases}z,z+1&\{z\in \mathbb {Z} \}:\;z=(\alpha -1)\beta +\lambda \\\lfloor z\rfloor &{\textrm {otherwise}}\end{cases}}} | ||
| Variance | λ + α β ( 1 + β ) {\displaystyle \lambda +\alpha \beta (1+\beta )} | ||
| Skewness | See #Properties | ||
| Excess kurtosis | See #Properties | ||
| MGF | e λ ( e t − 1 ) ( 1 − β ( e t − 1 ) ) α {\displaystyle {\frac {e^{\lambda (e^{t}-1)}}{(1-\beta (e^{t}-1))^{\alpha }}}} | ||
and
β
{\displaystyle \beta }
are 0, the distribution is the Poisson.
is 0, the distribution is the negative binomial.
(fixed mean)
(parameters of variable mean)
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 λ {\displaystyle \lambda } parameter, and a gamma-distributed variable component, which has the α {\displaystyle \alpha } and β {\displaystyle \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:
λ + α β ( 1 + 3 β + 2 β 2 ) ( λ + α β ( 1 + β ) ) 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:
λ + 3 λ 2 + α β ( 1 + 6 λ + 6 λ β + 7 β + 12 β 2 + 6 β 3 + 3 α β + 6 α β 2 + 3 α β 3 ) ( λ + α β ( 1 + β ) ) 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
- ^ 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" . Bulletin Trimestriel de l'Institut des Actuaires Français (in French). 227: 87–102.
- ^ von Lüders, Rolf (1934). "Die Statistik der seltenen Ereignisse" . Biometrika (in German). 26 (1–2): 108–128. doi:10.1093/biomet/26.1-2.108. JSTOR 2332055.
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. S2CID 122625458.
