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Delaporte distribution
Probability distribution in actuarial science
Probability distribution in actuarial science
| Field | Value | ||
|---|---|---|---|
| name | Delaporte | ||
| type | discrete | ||
| pdf_image | [[File:DelaportePMF.svg | 325px | Plot of the PMF for various Delaporte distributions.]] |
| When \alpha and \beta are 0, the distribution is the Poisson. | |||
| When \lambda is 0, the distribution is the negative binomial. | |||
| cdf_image | [[File:DelaporteCDF.svg | 325px | Plot of the PMF for various Delaporte distributions.]] |
| 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) | ||
| support | k \in \{0, 1, 2, \ldots\} | ||
| \sum_{i=0}^k\frac{\Gamma(\alpha + i)\beta^i\lambda^{k-i}e^{-\lambda |
When \alpha and \beta are 0, the distribution is the Poisson. When \lambda is 0, the distribution is the negative binomial. When \alpha and \beta are 0, the distribution is the Poisson. When \lambda is 0, the distribution is the negative binomial. \alpha, \beta 0 (parameters of variable mean) The Delaporte distribution is a discrete probability distribution that has received attention in actuarial science.{{cite encyclopedia | author-link = Harry Panjer | editor1-last = Teugels | editor1-first = Jozef L. | editor2-first = Bjørn | editor2-last = Sundt It can be defined using the convolution of a negative binomial distribution with a Poisson distribution.{{cite book | author1-link = Norman Lloyd Johnson | author3-link = Samuel Kotz 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.{{cite book The distribution is named for Pierre Delaporte, who analyzed it in relation to automobile accident claim counts in 1959,{{cite journal |trans-title= Some problems of mathematical statistics as related to automobile insurance and no-claims bonus although it appeared in a different form as early as 1934 in a paper by Rolf von Lüders,{{cite journal |trans-title= The statistics of rare events where it was called the Formel II distribution.
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
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