Compound Poisson Distribution (Pollaczek–Geiringer)

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What is a compound Poisson distribution?

The compound Poisson distribution is a sum of independent and identically distributed (iid) random variables that follow a Poisson distribution [1]. The compound Poisson is commonly used in a variety of fields, such as frequency of insurance claims and frequencies and amount of rainfall [2].

Compound Poisson distribution definition

Compound Poisson processes (called Pollaczek–Geiringer distributions) were devised in the 1930s as a tool to model the behavior of extreme events such as accidents, diseases, and suicides. 

Suppose that a random variable N is a Poisson distributed random variable with expected value λ, i.e. [1], N ~ Poisson(λ). Also suppose that X₁, X₂, X₃ are i.i.d. random variables that are mutually independent of N. A compound Poisson distribution is defined as the sum of N i.i.d variables:

An alternate definition stems from a relationship between mixing and compounding. If U(.) is the cumulative distribution function (CDF) of an infinitely divisible distribution, then the mixed Poisson distribution is also infinitely divisible. This implies that the following integral defines a compound Poisson distribution [3]:

For large values of λ, the compound Poisson distribution can be approximated by a normal distribution with mean λE(X) and variance λ E(X²) [4].

The probability density function (PDF) and cumulant generating function (CGF) of a compound Poisson distribution can be calculated by using the properties of the underlying Poisson process and the probability distribution of the random variables. Specifically:

• The PDF of a compound Poisson distribution is the sum of the random variables’ PDFs weighted by their corresponding Poisson probabilities.
• The CGF is the product of random variables’ CGFs and the CGF of the Poisson process.

Compound Poisson processes

A compound Poisson distribution can also be defined by an underlying compound Poisson process. The arrivals in a Poisson process are associated with real-valued random variables that represent the value of each arrival. These variables are i.i.d, which means they have the same PDF and are not affected by the Poisson process itself. Examples of random variables that can be used in compound Poisson processes include those drawn from insurance claims, product failures, or stock prices.

A Poisson-distributed random sum of random variables is the sum of the random variables for all arrivals up to time t. In a compound Poisson process, this random sum’s distribution is the compound Poisson distribution, a convolution of the underlying Poisson process and the probability distribution of the random variables. In probability, convolution (F * G) sums the products of two probability distributions F and G.

Relationship to other distributions

Many probability distributions are also compound Poisson distributions. For example:

• The compound Poisson distribution is an example of a stopped sum distribution, which can model two waves, a primary wave and a secondary wave [5]. Stopped sum distributions are so-named because they include a stopping rule, which specifies the point at which the sum should be stopped.
• The negative binomial distribution can be expressed as a compound Poisson distribution when the failure probability is distributed as a gamma distribution.
• The gamma distribution can be expressed as a compound Poisson distribution when the failure rate is a random variable.
• The stuttering Poisson distribution is a special case of compound Poisson distribution [3].
• When a random variable Xi has variational Cauchy distribution then the random variable Y has compound Poisson distribution as the sum of variational Cauchy distribution [1].
• The compound Poisson distribution motivates the Bernoulli process, a continuous-time stochastic process with jumps of randomly distributed size, which arrive according to a Poisson process [2].
• The Tweedie compound Poisson distribution is a mixture of a degenerate distribution at the origin and a continuous distribution on the positive real line [6].

Other names for the compound Poisson distribution

The compound Poisson distribution goes by many other names, including:

• clustered Poisson distribution,
• composed Poisson distribution,
• generalized Poisson distribution,
• multiple Poisson distribution,
• Poisson power series distribution,
• Poisson-stopped sum,
• stuttering Poisson distribution.

References

[1] Adelson, R. M. “Compound Poisson Distributions.” OR, vol. 17, no. 1, 1966, pp. 73–75. JSTOR, https://doi.org/10.2307/3007241. Accessed 17 Apr. 2023.

[2] Wolfram Research (2012), CompoundPoissonDistribution, Wolfram Language function, https://reference.wolfram.com/language/ref/CompoundPoissonDistribution.html.

[3] Willmot, G. (1986). Mixed compound Poisson distributions. Retrieved April 17, 2023 from: https://www.cambridge.org/core/services/aop-cambridge-core/content/view/EB000303D7A5230E79869B13CDEC04CE/S051503610001165Xa.pdf/mixed_compound_poisson_distributions.pdf

[4] Wolfgang Härdle, Rafał Weron (2005). Statistical Tools for Finance and Insurance. Springer.

[5] Neyman, J. (1939). On a new class of “contagious” distributions, applicable in entomology and bacteriology. The Annals of Mathematical Statistics, 10(1), 35–57. doi:10.1214/aoms/1177732245

[6] Zhang, Y (2013). Likelihood-based and Bayesian Methods for Tweedie Compound Poisson Linear Mixed Models, Statistics and Computing, 23, 743-757.