Mixed Poisson distribution: Difference between revisions
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A '''mixed Poisson distribution''' is a [[Univariate distribution|univariate]] discrete [[probability distribution]] in stochastics. It results from assuming that a random variable is [[Poisson distribution|Poisson distributed]], where the [[Scale parameter#Rate parameter|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 science|actuarial mathematics]] as a general approach for the distribution of the number of claims and is also examined as an [[Mathematical modelling of infectious disease|epidemiological model]]. It should not be confused with [[compound Poisson distribution]] or [[compound Poisson process]]. |
A '''mixed Poisson distribution''' is a [[Univariate distribution|univariate]] discrete [[probability distribution]] in stochastics. It results from assuming that a random variable is [[Poisson distribution|Poisson distributed]], where the [[Scale parameter#Rate parameter|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 science|actuarial mathematics]] as a general approach for the distribution of the number of claims and is also examined as an [[Mathematical modelling of infectious disease|epidemiological model]].<ref>{{Citation |last=Willmot |first=Gordon E. |title=Mixed Poisson distributions |date=2001 |url=http://link.springer.com/10.1007/978-1-4613-0111-0_3 |work=Lundberg Approximations for Compound Distributions with Insurance Applications |volume=156 |pages=37–49 |place=New York, NY |publisher=Springer New York |doi=10.1007/978-1-4613-0111-0_3 |isbn=978-0-387-95135-5 |access-date=2022-07-08 |last2=Lin |first2=X. Sheldon}}</ref> It should not be confused with [[compound Poisson distribution]] or [[compound Poisson process]].<ref>{{Cite journal |last=Willmot |first=Gord |date=1986 |title=Mixed Compound Poisson Distributions |url=https://www.cambridge.org/core/product/identifier/S051503610001165X/type/journal_article |journal=ASTIN Bulletin |language=en |volume=16 |issue=S1 |pages=S59–S79 |doi=10.1017/S051503610001165X |issn=0515-0361}}</ref> |
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== Definition == |
== Definition == |
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!mixed Poisson distribution<ref>{{Cite |
!mixed Poisson distribution<ref>{{Cite journal |last=Karlis |first=Dimitris |last2=Xekalaki |first2=Evdokia |date=2005 |title=Mixed Poisson Distributions |url=https://www.jstor.org/stable/25472639 |journal=International Statistical Review / Revue Internationale de Statistique |volume=73 |issue=1 |pages=35–58 |issn=0306-7734}}</ref> |
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|[[Gamma distribution|gamma]] |
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Revision as of 19:46, 8 July 2022
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PGF |
A mixed Poisson distribution is a univariate discrete probability distribution in stochastics. It results from assuming that a random variable is Poisson distributed, where 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]
Definition
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
- The variance is always bigger than the expected value. This property is called overdispersion. This is in contrast to the Poisson distribution where mean and variance are the same.
- In practice, almost only densities of gamma distributions, logarithmic normal distributions and inverse Gaussian distributions are used as densities π(λ). If we choose the density of the gamma distribution, we get the negative binomial distribution, which explains why this is also called the Poisson gamma distribution.
In the following be the expected value of the density and the variance of the density.
Expected value
The expected value of the Mixed Poisson Distribution is
- .
Variance
- .
Skewness
The skewness can be represented as
- .
Characteristic function
The characteristic function has the form
- .
Where is the moment generating function of the density.
Probability generating function
For the probability generating function, one obtains[3]
- .
Moment-generating function
The moment-generating function of the mixed Poisson distribution is
- .
Examples
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
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
- 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
- ^ Willmot, Gordon E.; Lin, X. Sheldon (2001), "Mixed Poisson distributions", Lundberg Approximations for Compound Distributions with Insurance Applications, 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
- ^ Willmot, Gord (1986). "Mixed Compound Poisson Distributions". ASTIN Bulletin. 16 (S1): S59–S79. doi:10.1017/S051503610001165X. ISSN 0515-0361.
- ^ a b c d Willmot, Gord (2014-08-29). "Mixed Compound Poisson Distributions". Cambridge. pp. 5–7. doi:10.1017/S051503610001165X.
{{cite web}}
: CS1 maint: url-status (link) - ^ Karlis, Dimitris; Xekalaki, Evdokia (2005). "Mixed Poisson Distributions". International Statistical Review / Revue Internationale de Statistique. 73 (1): 35–58. ISSN 0306-7734.