Compound Poisson process

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A compound Poisson process is a continuous-time (random) stochastic process with jumps. The jumps arrive randomly according to a Poisson process and the size of the jumps is also random, with a specified probability distribution. A compound Poisson process, parameterised by a rate \lambda > 0 and jump size distribution G, is a process \{\,Y(t) : t \geq 0 \,\} given by

Y(t) = \sum_{i=1}^{N(t)} D_i

where,  \{\,N(t) : t \geq 0\,\} is a Poisson process with rate \lambda, 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\,\}.\,

When  D_i are non-negative integer-valued random variable, then this compound Poisson process is named stuttering Poisson process which has the feature that two or more events occur in a very short time .

Properties of the compound Poisson process[edit]

Using conditional expectation, the expected value of a compound Poisson process can be calculated using a result known as Wald's equation as:

\,E(Y(t)) = E(E(Y(t)|N(t))) = E(N(t)E(D)) = E(N(t))E(D) = \lambda t E(D).

Making similar use of the law of total variance, the variance can be calculated as:


\begin{align}
\operatorname{var}(Y(t)) &= E(\operatorname{var}(Y(t)|N(t))) + \operatorname{var}(E(Y(t)|N(t))) \\
&= E(N(t)\operatorname{var}(D)) + \operatorname{var}(N(t)E(D)) \\
&= \operatorname{var}(D)E(N(t)) + E(D)^2 \operatorname{var}(N(t)) \\
&= \operatorname{var}(D)\lambda t + E(D)^2\lambda t \\
&= \lambda t(\operatorname{var}(D) + E(D)^2) \\
&= \lambda t E(D^2).
\end{align}

Lastly, using the law of total probability, the moment generating function can be given as follows:

\,\Pr(Y(t)=i) = \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n)

\begin{align}
E(e^{sY}) & = \sum_i e^{si} \Pr(Y(t)=i) \\
& = \sum_i e^{si} \sum_{n} \Pr(Y(t)=i|N(t)=n)\Pr(N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si} \Pr(Y(t)=i|N(t)=n) \\
& = \sum_n \Pr(N(t)=n) \sum_i e^{si}\Pr(D_1 + D_2 + \cdots + D_n=i) \\
& = \sum_n \Pr(N(t)=n) M_D(s)^n \\
& = \sum_n \Pr(N(t)=n) e^{n\ln(M_D(s))} \\
& = M_{N(t)}(\ln(M_D(s))) \\
& = e^{\lambda t \left ( M_D(s) - 1\right ) }.
\end{align}

Exponentiation of measures[edit]

Let N, Y, and D be as above. Let μ be the probability measure according to which D is distributed, i.e.

\mu(A) = \Pr(D \in A).\,

Let δ0 be the trivial probability distribution putting all of the mass at zero. Then the probability distribution of Y(t) is the measure

\exp(\lambda t(\mu - \delta_0))\,

where the exponential exp(ν) of a finite measure ν on Borel subsets of the real line is defined by

\exp(\nu) = \sum_{n=0}^\infty {\nu^{*n} \over n!}

and

 \nu^{*n} = \underbrace{\nu * \cdots *\nu}_{n \text{ factors}}

is a convolution of measures, and the series converges weakly.

Fitting a compound Poisson process[edit]

The parameters for independent observations of a compound Poisson process can be chosen using a maximum likelihood estimator using Simar's algorithm,[1] which has been shown to converge.[2]

See also[edit]

References[edit]

  1. ^ Simar, L. (1976). "Maximum Likelihood Estimation of a Compound Poisson Process". The Annals of Statistics 4 (6): 1200. doi:10.1214/aos/1176343651. JSTOR 2958588.  edit
  2. ^ Böhning, D. (1982). "Convergence of Simar's Algorithm for Finding the Maximum Likelihood Estimate of a Compound Poisson Process". The Annals of Statistics 10 (3): 1006. doi:10.1214/aos/1176345890. JSTOR 2240923.  edit