Editing Compound Poisson distribution

{{Short description|Aspect of probability theory}}
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 distribution|Poisson-distributed]] variable. The result can be either a [[continuous distribution|continuous]] or a [[discrete distribution]].

==Definition==

Suppose that

:<math>N\sim\operatorname{Poisson}(\lambda),</math>

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

:<math>X_1, X_2, X_3, \dots</math>

are identically distributed random variables that are mutually independent and also independent of ''N''. Then the probability distribution of the sum of <math>N</math> i.i.d. random variables

:<math>Y = \sum_{n=1}^N X_n</math>

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''&nbsp;=&nbsp;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''&nbsp;|&nbsp;''N'' with the marginal distribution of ''N''.

==Properties==

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

:<math>\operatorname{E}(Y)= \operatorname{E}\left[\operatorname{E}(Y \mid N)\right]= \operatorname{E}\left[N \operatorname{E}(X)\right]= \operatorname{E}(N) \operatorname{E}(X)  ,</math>

: <math>
\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}
</math>

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

:<math>\operatorname{E}(Y)= \operatorname{E}(N)\operatorname{E}(X) = \lambda\operatorname{E}(X) ,</math>
:<math>\operatorname{Var}(Y) = \operatorname{E}(N)(\operatorname{Var}(X) + (\operatorname{E}(X))^2)= \operatorname{E}(N){\operatorname{E}(X^2)} = \lambda{\operatorname{E}(X^2)}.</math>

The probability distribution of ''Y'' can be determined in terms of [[Characteristic function (probability theory)|characteristic function]]s:

:<math>\varphi_Y(t) = \operatorname{E}(e^{itY})= \operatorname{E} \left( \left(\operatorname{E} (e^{itX}\mid N) \right)^N \right)= \operatorname{E} \left((\varphi_X(t))^N\right),  \,</math>
and hence, using the [[probability-generating function]] of the Poisson distribution, we have
:<math>\varphi_Y(t) = \textrm{e}^{\lambda(\varphi_X(t) - 1)}.\,</math>

An alternative approach is via [[cumulant generating function]]s:
:<math>K_Y(t)=\ln \operatorname{E}[e^{tY}]=\ln \operatorname E[\operatorname E[e^{tY}\mid N]]=\ln \operatorname E[e^{NK_X(t)}]=K_N(K_X(t)) . \,</math>

Via the [[law of total cumulance]] it can be shown that, if the mean of the Poisson distribution ''λ''&nbsp;=&nbsp;1, the [[cumulant]]s of ''Y'' are the same as the [[moment (mathematics)|moments]] of ''X''<sub>1</sub>.{{Citation needed|date=October 2010}}

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

==Discrete compound Poisson distribution==

When <math>X_1, X_2, X_3, \dots</math> are positive integer-valued i.i.d random variables with <math>P(X_1 = k) = \alpha_k,\ (k =1,2, \ldots )</math>, then this compound Poisson distribution is named  '''discrete compound Poisson distribution'''<ref name=libro>Johnson, N.L., [[Adrienne W. Kemp|Kemp, A.W.]], and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, {{ISBN|978-0-471-27246-5}}.</ref><ref name=zhang>{{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}}</ref><ref name=zhang2>{{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 }}</ref> (or stuttering-Poisson distribution<ref name=kemp>{{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 }}</ref>) . We say that the discrete random variable <math>Y</math> satisfying [[probability generating function]] characterization

:<math> 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| \le 1)</math>

has a discrete compound Poisson(DCP) distribution with parameters <math>(\alpha_1 \lambda,\alpha_2 \lambda, \ldots ) \in \mathbb{R}^\infty</math> (where <math display="inline">\sum_{i = 1}^\infty \alpha_i = 1</math>, with <math display="inline">\alpha_i \ge 0,\lambda  > 0</math>), which is denoted by

:<math>X \sim {\text{DCP}}(\lambda {\alpha _1},\lambda {\alpha _2}, \ldots )</math>

Moreover, if <math>X \sim {\operatorname{DCP}}(\lambda {\alpha _1}, \ldots ,\lambda {\alpha _r})</math>, we say <math>X</math> has a discrete compound Poisson distribution of order <math>r</math> . When  <math>r = 1,2</math>, DCP becomes [[Poisson distribution]] and [[Hermite distribution]], respectively. When  <math>r = 3,4</math>, DCP becomes triple stuttering-Poisson distribution and quadruple stuttering-Poisson distribution, respectively.<ref>Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering-Poisson distributions. Technometrics, 18(1), 67-73.</ref> Other special cases include: shift [[geometric distribution]], [[negative binomial distribution]], [[Geometric Poisson distribution]], [[Neyman Type A distribution|Neyman type A distribution]], Luria–Delbrück distribution in [[Luria–Delbrück experiment]]. For more special case of DCP, see the reviews paper<ref>Wimmer, G., Altmann, G. (1996). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical journal, 38(8), 995-1011.</ref> and references therein.

Feller's characterization of the compound Poisson distribution states that a non-negative integer valued r.v. <math>X</math> is [[infinite divisibility (probability)|infinitely divisible]] if and only if its distribution is a discrete compound Poisson distribution.<ref>{{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 }}</ref> The [[negative binomial distribution]] is discrete [[Infinite divisibility (probability)|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 ''X''<sub>1</sub>,&nbsp;...,&nbsp;''X''<sub>''n''</sub> 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]]<ref name=kemp/><ref>{{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 }}</ref>). The discrete compound Poisson distribution is also widely used in [[actuarial science]] for modelling the distribution of the total claim amount.<ref name=zhang/>

When some <math>\alpha_k</math> are negative, it is the discrete pseudo compound Poisson distribution.<ref name=zhang/> We define that any discrete random variable <math>Y</math> satisfying [[probability generating function]] characterization

:<math> G_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| \le 1)</math>

has a discrete pseudo compound Poisson distribution with parameters <math>(\lambda_1 ,\lambda_2, \ldots )=:(\alpha_1 \lambda,\alpha_2 \lambda, \ldots ) \in \mathbb{R}^\infty</math> where <math display="inline">\sum_{i = 1}^\infty  {\alpha_i} = 1</math> and <math display="inline">\sum_{i = 1}^\infty  {\left| {{\alpha _i}} \right|} < \infty</math>, with <math>{\alpha_i} \in \mathbb{R},\lambda  > 0 </math>.

==Compound Poisson Gamma distribution==

If ''X'' has a [[gamma distribution]], of which the [[exponential distribution]] is a special case, then the conditional distribution of ''Y''&nbsp;|&nbsp;''N'' is again a gamma distribution. The marginal distribution of ''Y'' is a [[Tweedie distribution]] with variance power 1&nbsp;<&nbsp;''p''&nbsp;<&nbsp;2 (proof via comparison of [[characteristic function (probability theory)|characteristic function]]).<ref 
name="Jørgensen-1997">{{cite book
      | author = Jørgensen, Bent 
      | year = 1997
      | title = The theory of dispersion models
      | publisher = Chapman & Hall
      | isbn = 978-0412997112
}}</ref> To be more explicit, if

:<math> N \sim\operatorname{Poisson}(\lambda) ,</math>

and

:<math> X_i \sim \operatorname{\Gamma}(\alpha, \beta) </math>

i.i.d., then the distribution of

:<math> Y = \sum_{i=1}^N X_i </math>

is a reproductive [[exponential dispersion model]] <math>ED(\mu, \sigma^2)</math> with

:<math>
\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}
</math>

The mapping of parameters Tweedie parameter <math>\mu, \sigma^2, p</math> to the Poisson and Gamma parameters <math>\lambda, \alpha, \beta</math> is the following:

:<math>
\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}
</math>

==Compound Poisson processes==

{{Main|Compound Poisson process}}

A [[compound Poisson process]] with rate <math>\lambda>0</math> and jump size distribution ''G'' is a continuous-time [[stochastic process]] <math>\{\,Y(t) : t \geq 0 \,\}</math> given by

:<math>Y(t) = \sum_{i=1}^{N(t)} D_i,</math>

where the sum is by convention equal to zero as long as ''N''(''t'')&nbsp;=&nbsp;0. Here, <math> \{\,N(t) : t \geq 0\,\}</math> is a [[Poisson process]] with rate <math>\lambda</math>, and <math> \{\,D_i : i \geq 1\,\}</math> are independent and identically distributed random variables, with distribution function ''G'', which are also independent of <math> \{\,N(t) : t \geq 0\,\}.\,</math><ref>{{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}}</ref>

For the discrete version of compound Poisson process, it can be used in [[survival analysis]] for the frailty models.<ref>{{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 }}</ref>

==Applications==
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.<ref name=Revf>{{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 }}</ref> Thompson applied the same model to monthly total rainfalls.<ref name=Thom>{{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 }}</ref>

There have been applications to [[Insurance|insurance claims]]<ref>{{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}}</ref><ref>{{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 }}</ref> and [[CT scan|x-ray computed tomography]].<ref>{{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 }}</ref><ref>{{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 }}</ref><ref>{{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 }}</ref>

== See also ==
{{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}}

==References==
{{Reflist}}

{{ProbDistributions|families}}

{{DEFAULTSORT:Compound Poisson Distribution}}
[[Category:Discrete distributions]]
[[Category:Poisson distribution]]
[[Category:Infinitely divisible probability distributions]]
[[Category:Compound probability distributions]]