Compound Poisson distribution explained

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. The result can be either a continuous or a discrete distribution.

Definition

Suppose that

N\sim\operatorname{Poisson}( λ ),

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,} ...

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 variablesY = \sum ^{N} _{n =1} 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.

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

\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) , \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] , \\[ 6 pt] & = \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

\operatorname{E}(Y) = \operatorname{E}(N)\operatorname{E}(X) = λ \operatorname{E}(X) , \operatorname{Var}(Y) = \operatorname{E}(N)(\operatorname{Var}(X) + (\operatorname{E}(X)) ^{2)=} \operatorname{E}(N){\operatorname{E}(X ^{2)}} = λ {\operatorname{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} \left( \left(\operatorname{E} (e ^{itX} \mid N) \right) ^{N} \right) = \operatorname{E} \left((\varphi ^{N\right),} _{X(t))} and hence, using the probability-generating function of the Poisson distribution, we have\varphi _{Y(t)} = rm{e} ^{λ (\varphi _{X(t)} -1 )} .

An alternative approach is via cumulant generating functions:

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))} .

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.

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,} ... are positive 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 characterizationP _{Y(z)} = \sum\limits ^{inf ty} _{i =0} P(Y = i)z ^{i} = \exp\left(\sum\limits ^{inf ty} _{k =1} \alpha _{k} λ (z ^{k} - 1 )\right), (|z| \le 1 )

has a discrete compound Poisson(DCP) distribution with parameters

(\alpha _{1} λ ,\alpha _{2} λ , \ldots ) \in R ^{infty} (where $\backslash sum\_^\backslash infty\; \backslash alpha\_i\; =\; 1$, with $\backslash alpha\_i\; \backslash ge\; 0,\backslash lambda\; >\; 0$), which is denoted byX \sim { DCP }(\lambda,\lambda, \ldots)

Moreover, if

X \sim {\operatorname{DCP}}( λ {\alpha _{1},} \ldots , λ {\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: 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 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] 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 negative, it is the discrete pseudo compound Poisson distribution.^[3] We define that any discrete random variable Y satisfying probability generating function characterizationG _{Y(z)} = \sum\limits ^{inf ty} _{i =0} P(Y = i)z ^{i} = \exp\left(\sum\limits ^{inf ty} _{k =1} \alpha _{k} λ (z ^{k} - 1 )\right), (|z| \le 1 )

has a discrete pseudo compound Poisson distribution with parameters

( λ _{1} , λ _{2,} \ldots ) = :(\alpha _{1} λ ,\alpha _{2} λ , \ldots ) \in R ^{infty} where $\backslash sum\_^\backslash infty\; =\; 1$ and , with {\alpha _{i}} \in R , λ > 0 .

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 | N is again a gamma distribution. The marginal distribution of Y is a Tweedie distribution^[10] with variance power 1 < p < 2 (proof via comparison of characteristic function (probability theory)). To be more explicit, if

N \sim\operatorname{Poisson}( λ ) ,

and

X _{i} \sim \operatorname{\Gamma}(\alpha, \beta)

i.i.d., then the distribution of

Y = \sum ^{N} _{i =1} X _{i} ED(\mu, \sigma ^{2)} with\begin{align} \operatorname{E}[Y] & = λ (\alpha)/(\beta) = : \mu , \\[ 4 pt] \operatorname{Var}[Y]& = λ (\alpha( 1+ \alpha))/(\beta ^{2}) = : \sigma ^{2} \mu ^{p} . \end{align}

The mapping of parameters Tweedie parameter

\mu, \sigma ^{2,} p to the Poisson and Gamma parameters λ , \alpha, \beta is the following:\begin{align} λ & = (\mu ^{2-p})/(( 2- p)\sigma ^{2}) , \\[ 4 pt] \alpha & = (2- p)/(p -1) , \\[ 4 pt] \beta & = (\mu ^{1-p})/((p -1 )\sigma ^{2}) . \end{align}

Compound Poisson processes

See main article: Compound Poisson process.

A compound Poisson process with rate

λ > 0 and jump size distribution G is a continuous-time stochastic process \{Y(t) : t \geq 0 \} given byY(t) = \sum ^{N(t)} _{i =1} 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 λ , 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 \}. ^[11]

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.^[12]

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.^[13] Thompson applied the same model to monthly total rainfalls.^[14]

There have been applications to insurance claims^{[15] [16]} and x-ray computed tomography.^{[17] [18] [19]}

See also

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

Notes and References

1. Book: Lukacs, E. . 1970 . Characteristic functions . London . Griffin . 0-85264-170-2 .
2. Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, .
3. Zhang . Huiming . Yunxiao Liu . Bo Li . Notes on discrete compound Poisson model with applications to risk theory . Insurance: Mathematics and Economics . 59 . 2014. 325–336 . 10.1016/j.insmatheco.2014.09.012.
4. Zhang . Huiming . Bo Li . Characterizations of discrete compound Poisson distributions . Communications in Statistics - Theory and Methods . 45 . 22 . 2016. 6789–6802 . 10.1080/03610926.2014.901375. 125475756 .
5. "Stuttering – Poisson" distributions . C. D. . Kemp . Journal of the Statistical and Social Enquiry of Ireland . 1967 . 21 . 5 . 151–157 . 2262/6987 .
6. Patel, Y. C. (1976). Estimation of the parameters of the triple and quadruple stuttering-Poisson distributions. Technometrics, 18(1), 67-73.
7. Wimmer, G., Altmann, G. (1996). The multiple Poisson distribution, its characteristics and a variety of forms. Biometrical journal, 38(8), 995-1011.
8. Book: Feller, W. . 1968 . An Introduction to Probability Theory and its Applications . I . 3rd . Wiley . New York .
9. Adelson . R. M. . 1966 . Compound Poisson Distributions . Journal of the Operational Research Society. 17 . 1 . 73–75 . 10.1057/jors.1966.8 .
10. Book: Jørgensen, Bent . 1997 . The theory of dispersion models . Chapman & Hall . 978-0412997112.
11. Book: S. M. Ross. Introduction to Probability Models. ninth. Academic Press. Boston. 2007. 978-0-12-598062-3.
12. 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 . 10.1080/00949655.2012.679943 . 119851120 .
13. Revfeim . K. J. A. . 1984 . An initial model of the relationship between rainfall events and daily rainfalls . Journal of Hydrology . 75 . 1–4 . 357–364 . 10.1016/0022-1694(84)90059-3 . 1984JHyd...75..357R .
14. Thompson . C. S. . 1984 . Homogeneity analysis of a rainfall series: an application of the use of a realistic rainfall model . J. Climatology . 4 . 6 . 609–619 . 10.1002/joc.3370040605 . 1984IJCli...4..609T .
15. Jørgensen . Bent . Paes De Souza . Marta C. . Fitting Tweedie's compound poisson model to insurance claims data . Scandinavian Actuarial Journal . January 1994 . 1994 . 1 . 69–93 . 10.1080/03461238.1994.10413930.
16. Smyth . Gordon K. . Jørgensen . Bent . Fitting Tweedie's Compound Poisson Model to Insurance Claims Data: Dispersion Modelling . ASTIN Bulletin . 29 August 2014 . 32 . 1 . 143–157 . 10.2143/AST.32.1.1020. free .
17. Whiting . Bruce R. . Larry E. . Martin J. . Antonuk . Yaffe . Signal statistics in x-ray computed tomography . Medical Imaging 2002: Physics of Medical Imaging . 3 May 2002 . 4682 . 53–60 . 10.1117/12.465601 . International Society for Optics and Photonics. 2002SPIE.4682...53W . 116487704 .
18. Elbakri . Idris A. . Fessler . Jeffrey A. . J. Michael . Fitzpatrick . Milan . Sonka . Efficient and accurate likelihood for iterative image reconstruction in x-ray computed tomography . Medical Imaging 2003: Image Processing . 16 May 2003 . 5032 . 1839–1850 . 10.1117/12.480302 . SPIE. 2003SPIE.5032.1839E . 12215253 . 10.1.1.419.3752 .
19. Whiting . Bruce R. . Massoumzadeh . Parinaz . Earl . Orville A. . O'Sullivan . Joseph A. . Snyder . Donald L. . Williamson . Jeffrey F. . Properties of preprocessed sinogram data in x-ray computed tomography . Medical Physics . 24 August 2006 . 33 . 9 . 3290–3303 . 10.1118/1.2230762. 17022224 . 2006MedPh..33.3290W .