Compound Poisson distribution

Template:Short description 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

<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 = 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

<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) = 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 functions:

<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 functions:

<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 = 1, the cumulants of Y are the same as the moments of X₁.Template:Citation needed

Every infinitely divisible probability distribution is a limit of compound Poisson distributions.<ref>Template:Cite book</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., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, Template:ISBN.</ref><ref name=zhang>Template:Cite journal</ref><ref name=zhang2>Template:Cite journal</ref> (or stuttering-Poisson distribution<ref name=kemp>Template:Cite journal</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, 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 infinitely divisible if and only if its distribution is a discrete compound Poisson distribution.<ref>Template:Cite book</ref> 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<ref name=kemp/><ref>Template:Cite journal</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| Template:\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 | 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).<ref name="Jørgensen-1997">Template:Cite book</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

{{#invoke:Labelled list hatnote|labelledList|Main article|Main articles|Main page|Main pages}}

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) = 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>Template:Cite book</ref>

For the discrete version of compound Poisson process, it can be used in survival analysis for the frailty models.<ref>Template:Cite journal</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>Template:Cite journal</ref> Thompson applied the same model to monthly total rainfalls.<ref name=Thom>Template:Cite journal</ref>

There have been applications to insurance claims<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref> and x-ray computed tomography.<ref>Template:Cite journal</ref><ref>Template:Cite journal</ref><ref>Template:Cite journal</ref>

See also

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