Compound Poisson distribution

In probability theory, a compound Poisson distribution is the probability distribution of the sum of a number of independent identically-distributed random variables, where the number of terms to be added is itself a Poisson-distributed variable. In the simplest cases, the result can be either a continuous or a discrete distribution.

Definition

Suppose that

N\sim \operatorname {Poisson}(\lambda ),

i.e., N is a random variable whose distribution is a Poisson distribution with expected value λ, and that

X_{1},X_{2},X_{3},\dots

are identically distributed random variables that are mutually independent and also independent of N. Then the probability distribution of the sum of $N$ i.i.d. random variables conditioned on the number of these variables ( $N$ ):

Y\mid N=\sum _{{n=1}}^{N}X_{n}

has a well-defined distribution. In the case N = 0, then the value of Y is 0, so that then Y | N = 0 has a degenerate distribution.

The compound Poisson distribution is obtained by marginalising the joint distribution of (Y,N) over N, where this joint distribution is obtained by combining the conditional distribution Y | N with the marginal distribution of N.

Properties

Mean and variance of the compound distribution derive in a simple way from law of total expectation and the law of total variance. Thus

\operatorname {E}_{Y}(Y)=\operatorname {E}_{N}\left[\operatorname {E}_{{Y\mid N}}(Y)\right]=\operatorname {E}_{N}\left[N\operatorname {E}_{X}(X)\right]=\operatorname {E}_{N}(N)\operatorname {E}_{X}(X),

\operatorname {Var}_{Y}(Y)=E_{N}\left[\operatorname {Var}_{{Y\mid N}}(Y)\right]+\operatorname {Var}_{N}\left[E_{{Y\mid N}}(Y)\right]=\operatorname {E}_{N}\left[N\operatorname {Var}_{X}(X)\right]+\operatorname {Var}_{N}\left[N\operatorname {E}_{X}(X)\right],

giving

\operatorname {Var}_{Y}(Y)=\operatorname {E}_{N}(N)\operatorname {Var}_{X}(X)+\left(\operatorname {E}_{X}(X)\right)^{2}\operatorname {Var}_{N}(N).

Then, since E(N)=Var(N) if N is Poisson, and dropping the unnecessary subscripts, these formulae can be reduced to

\operatorname {E}(Y)=\operatorname {E}(N)\operatorname {E}(X),

\operatorname {Var}(Y)=E(N)(\operatorname {Var}(X)+{E(X)}^{2})=E(N){E(X^{2})}.

The probability distribution of Y can be determined in terms of characteristic functions:

\varphi _{Y}(t)=\operatorname {E}(e^{{itY}})=\operatorname {E}_{N}(\left(\operatorname {E}(e^{{itX}}))^{N}\right)=\operatorname {E}_{N}((\varphi _{X}(t))^{N}),\,

and hence, using the probability-generating function of the Poisson distribution, we have

\varphi _{Y}(t)={\textrm {e}}^{{\lambda (\varphi _{X}(t)-1)}}.\,

An alternative approach is via cumulant generating functions:

K_{Y}(t)=\ln E[e^{{tY}}]=\ln E[E[e^{{tY}}\mid N]]=\ln E[e^{{NK_{X}(t)}}]=K_{N}(K_{X}(t)).\,

Via the law of total cumulance it can be shown that, if the mean of the Poisson distribution λ = 1, the cumulants of Y are the same as the moments of X₁.

It can be shown that every infinitely divisible probability distribution is a limit of compound Poisson distributions.^[1] And compound Poisson distributions is infinitely divisible by the definition.

Discrete compound Poisson distribution

When $X_{1},X_{2},X_{3},\dots$ are non-negative integer-valued i.i.d random variables with $P(X_{1}=k)=\alpha _{k},\ (k=1,2,\ldots )$ , then this compound Poisson distribution is named discrete compound Poisson distribution^[2]^[3]^[4] (or stuttering-Poisson distribution^[5]) . We say that the discrete random variable $Y$ satisfying probability generating function characterization

P_{Y}(z)=\sum \limits _{{i=0}}^{\infty }P(Y=i)z^{i}=\exp \left(\sum \limits _{{k=1}}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)

has a discrete compound Poisson(DCP) distribution with parameters $(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in {\mathbb {R}}^{\infty }\left(\sum _{{i=1}}^{\infty }\alpha _{i}=1,\alpha _{i}\geq 0,\lambda >0\right)$ , which is denoted by

X\sim {\text{DCP}}(\lambda {\alpha _{1}},\lambda {\alpha _{r}},\ldots )

Moreover, if $X\sim {\operatorname {DCP} }(\lambda {\alpha _{1}},\ldots ,\lambda {\alpha _{r}})$ , we say $X$ has a discrete compound Poisson distribution of order $r$ . When $r=1,2$ , DCP becomes Poisson distribution and Hermite distribution, respectively. When $r=3,4$ , DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively.^[6] Other special cases include: shiftgeometric distribution, negative binomial distribution, Geometric Poisson distribution, Neyman type A distribution, Luria–Delbrück distribution in Luria–Delbrück experiment. For more special case of DCP, see the reviews paper^[7] and references therein.

Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v. $X$ is infinitely divisible if and only if its distribution is a discrete compound Poisson distribution.^[8] It can be shown that the negative binomial distribution is discrete infinitely divisible, i.e., if X has a negative binomial distribution, then for any positive integer n, there exist discrete i.i.d. random variables X₁, ..., X_n whose sum has the same distribution that X has. The shift geometric distribution is discrete compound Poisson distribution since it is a trivial case of negative binomial distribution.

This distribution can model batch arrivals (such as in a bulk queue ^[5]^[9]). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.^[3]

When some $\alpha _{k}$ are non-negative, it is the discrete pseudo compound Poisson distribution.^[3] We define that any discrete random variable $Y$ satisfying probability generating function characterization

G_{Y}(z)=\sum \limits _{{n=0}}^{\infty }P(Y=n)z^{n}=\exp \left(\sum \limits _{{k=1}}^{\infty }\alpha _{k}\lambda (z^{k}-1)\right),\quad (|z|\leq 1)

has a discrete pseudo compound Poisson distribution with parameters $(\lambda _{1},\lambda _{2},\ldots )=:(\alpha _{1}\lambda ,\alpha _{2}\lambda ,\ldots )\in \mathbb {R} ^{\infty }\left({\sum \limits _{k=1}^{\infty }{\alpha _{k}}=1,\sum \limits _{k=1}^{\infty }{\left|{\alpha _{k}}\right|}<\infty ,{\alpha _{k}}\in {\mathbb {R} },\lambda >0}\right)$ .

Other special cases

If the distribution of X is either an exponential distribution or a gamma distribution, then the conditional distributions of Y | N are gamma distributions in which the shape parameters are proportional to N. This shows that the formulation of the "compound Poisson distribution" outlined above is essentially the same as the more general class of compound probability distributions. However, the properties outlined above do depend on its formulation as the sum of a Poisson-distributed number of random variables. The distribution of Y in the case of the compound Poisson distribution with exponentially-distributed summands can be written in an form.^[10]^[11]

Compound Poisson processes

A compound Poisson process with rate $\lambda >0$ and jump size distribution G is a continuous-time stochastic process $\{\,Y(t):t\geq 0\,\}$ given by

Y(t)=\sum _{{i=1}}^{{N(t)}}D_{i},

where the sum is by convention equal to zero as long as N(t)=0. Here, $\{\,N(t):t\geq 0\,\}$ is a Poisson process with rate $\lambda$ , and $\{\,D_{i}:i\geq 1\,\}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\{\,N(t):t\geq 0\,\}.\,$ ^[12]

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.^[13]

Applications

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim^[10] to model the distribution of the total rainfall in a day, where each day contains a Poisson-distributed number of events each of which provides an amount of rainfall which has an exponential distribution. Thompson^[11] applied the same model to monthly total rainfalls.

References

↑ Lukacs, E. (1970). Characteristic functions. London: Griffin.
↑ Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, ISBN 978-0-471-27246-5.
1 2 3 Huiming, Zhang; Yunxiao Liu; Bo Li (2014). "Notes on discrete compound Poisson model with applications to risk theory". Insurance: Mathematics and Economics. 59: 325–336. doi:10.1016/j.insmatheco.2014.09.012.
↑ Huiming, Zhang; Bo Li (2016). "Characterizations of discrete compound Poisson distributions". Communications in Statistics - Theory and Methods. 45: 6789–6802. doi:10.1080/03610926.2014.901375.
1 2 Kemp, C. D. (1967). ""Stuttering – Poisson" distributions". Journal of the Statistical and Social Enquiry of Ireland. 21 (5): 151–157.
↑ Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering-Poisson distributions. Technometrics, 18(1), 67-73.
↑ Wimmer, G., Altmann, G. (1996). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical journal, 38(8), 995-1011.
↑ Feller, W. (1968).An introduction to probability theory and its applications,Vol. I. 3rd., Wiley, New York.
↑ Adelson, R. M. (1966). Compound poisson distributions. OR, 73–75.
1 2 Revfeim, K.J.A. (1984) An initial model of the relationship between rainfall events and daily rainfalls. Journal of Hydrology, 75, 357–364.
1 2 Thompson, C.S. (1984) Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model. J. Climatology, 4, 609 – 619.
↑ S. M. Ross (2007). Introduction to Probability Models (ninth ed.). Boston: Academic Press. ISBN 978-0-12-598062-3.
↑ Ata, N., & Özel, G. (2013). Survival functions for the frailty models based on the discrete compound Poisson process. Journal of Statistical Computation and Simulation, 83(11), 2105–2116.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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