Compound Poisson distribution

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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[edit]

Suppose that

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

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

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[edit]

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

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

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

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

An alternative approach is via cumulant generating functions:

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]

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[edit]

When are non-negative integer-valued i.i.d random variables with , 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 satisfying probability generating function characterization

has a discrete compound Poisson(DCP) distribution with parameters , which is denoted by

Moreover, if , we say has a discrete compound Poisson distribution of order . When , DCP becomes Poisson distribution and Hermite distribution, respectively. When , 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. 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, ..., Xn 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 are non-negative, it is the discrete pseudo compound Poisson distribution.[3] We define that any discrete random variable satisfying probability generating function characterization

has a discrete pseudo compound Poisson distribution with parameters .

Compound Poisson Gamma distribution[edit]

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 can be shown to be a Tweedie distribution[10] with variance power 1<p<2 (proof via comparison of characteristic function (probability theory)). To be more explicit, if

and

i.i.d., then the distribution of

is a reproductive exponential dispersion model with

The mapping of parameters Tweedie parameter to the Poisson and Gamma parameters is the following:

Compound Poisson processes[edit]

A compound Poisson process with rate and jump size distribution G is a continuous-time stochastic process given by

where the sum is by convention equal to zero as long as N(t)=0. Here, is a Poisson process with rate , and are independent and identically distributed random variables, with distribution function G, which are also independent of [11]

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

Applications[edit]

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]

See also[edit]

References[edit]

  1. ^ Lukacs, E. (1970). Characteristic functions. London: Griffin.
  2. ^ Johnson, N.L., Kemp, A.W., and Kotz, S. (2005) Univariate Discrete Distributions, 3rd Edition, Wiley, ISBN 978-0-471-27246-5.
  3. ^ a b c 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.
  4. ^ 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.
  5. ^ a b Kemp, C. D. (1967). ""Stuttering – Poisson" distributions". Journal of the Statistical and Social Enquiry of Ireland. 21 (5): 151–157.
  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. ^ Feller, W. (1968). An Introduction to Probability Theory and its Applications. Vol. I (3rd ed.). New York: Wiley.
  9. ^ Adelson, R. M. (1966). "Compound Poisson Distributions". OR. 17 (1): 73–75. doi:10.1057/jors.1966.8.
  10. ^ Jørgensen, Bent (1997). The theory of dispersion models. Chapman & Hall. ISBN 978-0412997112.
  11. ^ S. M. Ross (2007). Introduction to Probability Models (ninth ed.). Boston: Academic Press. ISBN 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. doi:10.1080/00949655.2012.679943.
  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. doi:10.1016/0022-1694(84)90059-3.
  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. doi:10.1002/joc.3370040605.