compound Poisson distribution

Suppose that

:$N\backslash sim\backslash operatorname\{Poisson\}(\backslash lambda),$

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

:$X\_1,\; X\_2,\; X\_3,\; \backslash dots$

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

:$Y\; =\; \backslash sum\_\{n=1\}^N\; X\_n$

is a compound Poisson distribution.

In the case N = 0, then this is a sum of 0 terms, so the value of Y is 0. Hence the conditional distribution of Y given that N = 0 is a degenerate distribution.

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

The expected value and the variance of the compound distribution can be derived in a simple way from law of total expectation and the law of total variance. Thus

:$\backslash operatorname\{E\}(Y)=\; \backslash operatorname\{E\}\backslash left[\backslash operatorname\{E\}(Y\; \backslash mid\; N)\backslash right]=\; \backslash operatorname\{E\}\backslash left[N\; \backslash operatorname\{E\}(X)\backslash right]=\; \backslash operatorname\{E\}(N)\; \backslash operatorname\{E\}(X)\; ,$

: $$

\begin{align}

\operatorname{Var}(Y) & = \operatorname{E}\left[\operatorname{Var}(Y\mid N)\right] + \operatorname{Var}\left[\operatorname{E}(Y \mid N)\right] =\operatorname{E} \left[N\operatorname{Var}(X)\right] + \operatorname{Var}\left[N\operatorname{E}(X)\right] , \\[6pt]

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

\end{align}

Then, since E(N) = Var(N) if N is Poisson-distributed, these formulae can be reduced to

:$\backslash operatorname\{E\}(Y)=\; \backslash operatorname\{E\}(N)\backslash operatorname\{E\}(X)\; =\; \backslash lambda\backslash operatorname\{E\}(X)\; ,$

:$\backslash operatorname\{Var\}(Y)\; =\; \backslash operatorname\{E\}(N)(\backslash operatorname\{Var\}(X)\; +\; (\backslash operatorname\{E\}(X))^2)=\; \backslash operatorname\{E\}(N)\{\backslash operatorname\{E\}(X^2)\}\; =\; \backslash lambda\{\backslash operatorname\{E\}(X^2)\}.$

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

:$\backslash varphi\_Y(t)\; =\; \backslash operatorname\{E\}(e^\{itY\})=\; \backslash operatorname\{E\}\; \backslash left(\; \backslash left(\backslash operatorname\{E\}\; (e^\{itX\}\backslash mid\; N)\; \backslash right)^N\; \backslash right)=\; \backslash operatorname\{E\}\; \backslash left((\backslash varphi\_X(t))^N\backslash right),\; \backslash ,$

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

:$\backslash varphi\_Y(t)\; =\; \backslash textrm\{e\}^\{\backslash lambda(\backslash varphi\_X(t)\; -\; 1)\}.\backslash ,$

An alternative approach is via cumulant generating functions:

:$K\_Y(t)=\backslash ln\; \backslash operatorname\{E\}[e^\{tY\}]=\backslash ln\; \backslash operatorname\; E[\backslash operatorname\; E[e^\{tY\}\backslash mid\; N]]=\backslash ln\; \backslash operatorname\; E[e^\{NK\_X(t)\}]=K\_N(K\_X(t))\; .\; \backslash ,$

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.{{Citation needed|date=October 2010}}

Every infinitely divisible probability distribution is a limit of compound Poisson distributions.{{cite book |last=Lukacs |first=E. |year=1970 |title=Characteristic functions |location=London |publisher=Griffin |isbn=0-85264-170-2 }} And compound Poisson distributions is infinitely divisible by the definition.

When $X\_1,\; X\_2,\; X\_3,\; \backslash dots$ are positive integer-valued i.i.d random variables with $P(X\_1\; =\; k)\; =\; \backslash alpha\_k,\backslash \; (k\; =1,2,\; \backslash ldots\; )$, then this compound Poisson distribution is named discrete compound Poisson distributionJohnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, {{ISBN|978-0-471-27246-5}}.{{Cite journal |first =Zhang | last = Huiming |author2=Yunxiao Liu |author3=Bo Li |title=Notes on discrete compound Poisson model with applications to risk theory |journal=Insurance: Mathematics and Economics |volume=59 |year=2014|pages=325–336 |doi=10.1016/j.insmatheco.2014.09.012}}{{Cite journal |first =Zhang | last = Huiming |author2=Bo Li |title=Characterizations of discrete compound Poisson distributions |journal=Communications in Statistics - Theory and Methods |volume=45 | issue = 22 |year=2016|pages=6789–6802 |doi=10.1080/03610926.2014.901375| s2cid = 125475756 }} (or stuttering-Poisson distribution{{cite journal | title = "Stuttering – Poisson" distributions | first = C. D. | last = Kemp | journal = Journal of the Statistical and Social Enquiry of Ireland | year = 1967 | volume = 21 | issue = 5 | pages = 151–157 | hdl = 2262/6987 }}) . We say that the discrete random variable $Y$ satisfying probability generating function characterization

:$P\_Y(z)\; =\; \backslash sum\backslash limits\_\{i\; =\; 0\}^\backslash infty\; P(Y\; =\; i)z^i\; =\; \backslash exp\backslash left(\backslash sum\backslash limits\_\{k\; =\; 1\}^\backslash infty\; \backslash alpha\_k\; \backslash lambda\; (z^k\; -\; 1)\backslash right),\; \backslash quad\; (|z|\; \backslash le\; 1)$

has a discrete compound Poisson(DCP) distribution with parameters $(\backslash alpha\_1\; \backslash lambda,\backslash alpha\_2\; \backslash lambda,\; \backslash ldots\; )\; \backslash in\; \backslash mathbb\{R\}^\backslash infty$ (where $\backslash sum\_\{i\; =\; 1\}^\backslash infty\; \backslash alpha\_i\; =\; 1$, with $\backslash alpha\_i\; \backslash ge\; 0,\backslash lambda\; >\; 0$), which is denoted by

:$X\; \backslash sim\; \{\backslash text\{DCP\}\}(\backslash lambda\; \{\backslash alpha\; \_1\},\backslash lambda\; \{\backslash alpha\; \_2\},\; \backslash ldots\; )$

Moreover, if $X\; \backslash sim\; \{\backslash operatorname\{DCP\}\}(\backslash lambda\; \{\backslash alpha\; \_1\},\; \backslash ldots\; ,\backslash lambda\; \{\backslash alpha\; \_r\})$, 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.Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering-Poisson distributions. Technometrics, 18(1), 67-73. Other special cases include: shift geometric 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 paperWimmer, G., Altmann, G. (1996). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical journal, 38(8), 995-1011. 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.{{cite book |last=Feller |first=W. |year=1968 |title=An Introduction to Probability Theory and its Applications |volume=I |edition=3rd |publisher=Wiley |location=New York }} 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{{cite journal |last=Adelson |first=R. M. |year=1966 |title=Compound Poisson Distributions |journal= Journal of the Operational Research Society|volume=17 |issue=1 |pages=73–75 |doi=10.1057/jors.1966.8 }}). The discrete compound Poisson distribution is also widely used in actuarial science for modelling the distribution of the total claim amount.

When some $\backslash alpha\_k$ are negative, it is the discrete pseudo compound Poisson distribution. We define that any discrete random variable $Y$ satisfying probability generating function characterization

:$G\_Y(z)\; =\; \backslash sum\backslash limits\_\{i\; =\; 0\}^\backslash infty\; P(Y\; =\; i)z^i\; =\; \backslash exp\backslash left(\backslash sum\backslash limits\_\{k\; =\; 1\}^\backslash infty\; \backslash alpha\_k\; \backslash lambda\; (z^k\; -\; 1)\backslash right),\; \backslash quad\; (|z|\; \backslash le\; 1)$

has a discrete pseudo compound Poisson distribution with parameters $(\backslash lambda\_1\; ,\backslash lambda\_2,\; \backslash ldots\; )=:(\backslash alpha\_1\; \backslash lambda,\backslash alpha\_2\; \backslash lambda,\; \backslash ldots\; )\; \backslash in\; \backslash mathbb\{R\}^\backslash infty$ where $\backslash sum\_\{i\; =\; 1\}^\backslash infty\; \{\backslash alpha\_i\}\; =\; 1$ and , with $\{\backslash alpha\_i\}\; \backslash in\; \backslash mathbb\{R\},\backslash lambda\; >\; 0$.

If X has a gamma distribution, of which the exponential distribution is a special case, then the conditional distribution of Y | N is again a gamma distribution. The marginal distribution of Y is a Tweedie distribution with variance power 1 < p < 2 (proof via comparison of characteristic function).

name="Jørgensen-1997">{{cite book

| author = Jørgensen, Bent

| year = 1997

| title = The theory of dispersion models

| publisher = Chapman & Hall

| isbn = 978-0412997112

}} To be more explicit, if

:$N\; \backslash sim\backslash operatorname\{Poisson\}(\backslash lambda)\; ,$

and

:$X\_i\; \backslash sim\; \backslash operatorname\{\backslash Gamma\}(\backslash alpha,\; \backslash beta)$

i.i.d., then the distribution of

:$Y\; =\; \backslash sum\_\{i=1\}^N\; X\_i$

is a reproductive exponential dispersion model $ED(\backslash mu,\; \backslash sigma^2)$ with

:$$

\begin{align}

\operatorname{E}[Y] & = \lambda \frac{\alpha}{\beta} =: \mu , \\[4pt]

\operatorname{Var}[Y]& = \lambda \frac{\alpha(1+\alpha)}{\beta^2}=: \sigma^2 \mu^p .

\end{align}

The mapping of parameters Tweedie parameter $\backslash mu,\; \backslash sigma^2,\; p$ to the Poisson and Gamma parameters $\backslash lambda,\; \backslash alpha,\; \backslash beta$ is the following:

:$$

\begin{align}

\lambda &= \frac{\mu^{2-p}}{(2-p)\sigma^2} ,

\\[4pt]

\alpha &= \frac{2-p}{p-1} ,

\\[4pt]

\beta &= \frac{\mu^{1-p}}{(p-1)\sigma^2} .

\end{align}

{{Main|Compound Poisson process}}

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

:$Y(t)\; =\; \backslash sum\_\{i=1\}^\{N(t)\}\; D\_i,$

where the sum is by convention equal to zero as long as N(t) = 0. Here, $\backslash \{\backslash ,N(t)\; :\; t\; \backslash geq\; 0\backslash ,\backslash \}$ is a Poisson process with rate $\backslash lambda$, and $\backslash \{\backslash ,D\_i\; :\; i\; \backslash geq\; 1\backslash ,\backslash \}$ are independent and identically distributed random variables, with distribution function G, which are also independent of $\backslash \{\backslash ,N(t)\; :\; t\; \backslash geq\; 0\backslash ,\backslash \}.\backslash ,${{cite book|author=S. M. Ross|title=Introduction to Probability Models|edition=ninth|publisher=Academic Press|location=Boston|year=2007|isbn=978-0-12-598062-3}}

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.{{cite journal |last1=Ata |first1=N. |last2=Özel |first2=G. |year=2013 |title=Survival functions for the frailty models based on the discrete compound Poisson process |journal=Journal of Statistical Computation and Simulation |volume=83 |issue=11 |pages=2105–2116 |doi=10.1080/00949655.2012.679943 |s2cid=119851120 }}

A compound Poisson distribution, in which the summands have an exponential distribution, was used by Revfeim 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.{{cite journal |last=Revfeim |first=K. J. A. |year=1984 |title=An initial model of the relationship between rainfall events and daily rainfalls |journal=Journal of Hydrology |volume=75 |issue=1–4 |pages=357–364 |doi=10.1016/0022-1694(84)90059-3 |bibcode=1984JHyd...75..357R }} Thompson applied the same model to monthly total rainfalls.{{cite journal |last=Thompson |first=C. S. |year=1984 |title=Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model |journal=Journal of Climatology |volume=4 |issue=6 |pages=609–619 |doi=10.1002/joc.3370040605 |bibcode=1984IJCli...4..609T }}

There have been applications to insurance claims{{cite journal |last1=Jørgensen |first1=Bent |last2=Paes De Souza |first2=Marta C. |title=Fitting Tweedie's compound poisson model to insurance claims data |journal=Scandinavian Actuarial Journal |date=January 1994 |volume=1994 |issue=1 |pages=69–93 |doi=10.1080/03461238.1994.10413930}}{{cite journal |last1=Smyth |first1=Gordon K. |last2=Jørgensen |first2=Bent |title=Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling |journal=ASTIN Bulletin |date=29 August 2014 |volume=32 |issue=1 |pages=143–157 |doi=10.2143/AST.32.1.1020|doi-access=free }} and x-ray computed tomography.{{cite journal |last1=Whiting |first1=Bruce R. |editor-first1=Larry E. |editor-first2=Martin J. |editor-last1=Antonuk |editor-last2=Yaffe |title=Signal statistics in x-ray computed tomography |journal=Medical Imaging 2002: Physics of Medical Imaging |date=3 May 2002 |volume=4682 |pages=53–60 |doi=10.1117/12.465601 |publisher=International Society for Optics and Photonics|bibcode=2002SPIE.4682...53W |s2cid=116487704 }}{{cite journal |last1=Elbakri |first1=Idris A. |last2=Fessler |first2=Jeffrey A. |editor2-first=J. Michael |editor2-last=Fitzpatrick |editor1-first=Milan |editor1-last=Sonka |title=Efficient and accurate likelihood for iterative image reconstruction in x-ray computed tomography |journal=Medical Imaging 2003: Image Processing |date=16 May 2003 |volume=5032 |pages=1839–1850 |doi=10.1117/12.480302 |publisher=SPIE|bibcode=2003SPIE.5032.1839E |s2cid=12215253 |citeseerx=10.1.1.419.3752 }}{{cite journal |last1=Whiting |first1=Bruce R. |last2=Massoumzadeh |first2=Parinaz |last3=Earl |first3=Orville A. |last4=O'Sullivan |first4=Joseph A. |last5=Snyder |first5=Donald L. |last6=Williamson |first6=Jeffrey F. |title=Properties of preprocessed sinogram data in x-ray computed tomography |journal=Medical Physics |date=24 August 2006 |volume=33 |issue=9 |pages=3290–3303 |doi=10.1118/1.2230762|pmid=17022224 |bibcode=2006MedPh..33.3290W }}

{{div col|colwidth=30em}}

• Compound Poisson process
• Hermite distribution
• Negative binomial distribution
• Geometric distribution
• Geometric Poisson distribution
• Gamma distribution
• Poisson distribution
• Zero-inflated model

{{div col end}}

{{Reflist}}

{{ProbDistributions|families}}

{{DEFAULTSORT:Compound Poisson Distribution}}

Category:Discrete distributions

Category:Poisson distribution

Category:Infinitely divisible probability distributions

Category:Compound probability distributions