Mixed Poisson distribution

A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that the conditional distribution of a random variable, given the value of the rate parameter, is a Poisson distribution, and that the rate parameter itself is considered as a random variable. Hence it is a special case of a compound probability distribution. Mixed Poisson distributions can be found in actuarial mathematics as a general approach for the distribution of the number of claims and is also examined as an epidemiological model.[1] It should not be confused with compound Poisson distribution or compound Poisson process.[2]

mixed Poisson distribution
Notation
Parameters
Support
PMF
Mean
Variance
Skewness
MGF , with the MGF of π
CF
PGF

Definition

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A random variable X satisfies the mixed Poisson distribution with density π(λ) if it has the probability distribution[3]

 

If we denote the probabilities of the Poisson distribution by qλ(k), then

 

Properties

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In the following let   be the expected value of the density   and   be the variance of the density.

Expected value

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The expected value of the mixed Poisson distribution is

 

Variance

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For the variance one gets[3]

 

Skewness

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The skewness can be represented as

 

Characteristic function

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The characteristic function has the form

 

Where   is the moment generating function of the density.

Probability generating function

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For the probability generating function, one obtains[3]

 

Moment-generating function

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The moment-generating function of the mixed Poisson distribution is

 

Examples

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Theorem — Compounding a Poisson distribution with rate parameter distributed according to a gamma distribution yields a negative binomial distribution.[3]

Proof

Let   be a density of a   distributed random variable.

 

Therefore we get  

Theorem — Compounding a Poisson distribution with rate parameter distributed according to a exponential distribution yields a geometric distribution.

Proof

Let   be a density of a   distributed random variable. Using integration by parts n times yields:   Therefore we get  

Table of mixed Poisson distributions

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mixing distribution mixed Poisson distribution[4]
gamma negative binomial
exponential geometric
inverse Gaussian Sichel
Poisson Neyman
generalized inverse Gaussian Poisson-generalized inverse Gaussian
generalized gamma Poisson-generalized gamma
generalized Pareto Poisson-generalized Pareto
inverse-gamma Poisson-inverse gamma
log-normal Poisson-log-normal
Lomax Poisson–Lomax
Pareto Poisson–Pareto
Pearson’s family of distributions Poisson–Pearson family
truncated normal Poisson-truncated normal
uniform Poisson-uniform
shifted gamma Delaporte
beta with specific parameter values Yule

Literature

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  • Jan Grandell: Mixed Poisson Processes. Chapman & Hall, London 1997, ISBN 0-412-78700-8 .
  • Tom Britton: Stochastic Epidemic Models with Inference. Springer, 2019, doi:10.1007/978-3-030-30900-8

References

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  1. ^ Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, Lecture Notes in Statistics, vol. 156, New York, NY: Springer New York, pp. 37–49, doi:10.1007/978-1-4613-0111-0_3, ISBN 978-0-387-95135-5, retrieved 2022-07-08
  2. ^ Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59–S79. doi:10.1017/S051503610001165X. ISSN 0515-0361.
  3. ^ a b c d Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Astin Bulletin. 16: 5–7. doi:10.1017/S051503610001165X. S2CID 17737506.
  4. ^ Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review. 73 (1): 35–58. doi:10.1111/j.1751-5823.2005.tb00250.x. ISSN 0306-7734. JSTOR 25472639. S2CID 53637483.