Skellam distribution - Definition and Overview

The Skellam distribution is the probability distribution of the difference N₁ − N₂ of two uncorrelated random variables N₁ and N₂ having Poisson distributions with different expected values μ₁ and μ₂.

Recall that probability mass function of a Poisson distribution with mean μ is given by

<math>

P_n(\mu)={\mu^n\over n!}e^{-\mu}.
<math>

(Skellam, 1946). The Skellam probability density function is:

<math>

 P_n(\mu_1,\mu_2)=\sum_{k=-\infty}^\infty
 P_{n+k}(\mu_1)P_k(\mu_2)
 <math>

<math>

 =e^{-(\mu_1+\mu_2)}\sum_{k=-\infty}^\infty
 {{\mu_1^{n+k}\mu_2^k}\over{k!(n+k)!}}
 <math>

<math>

 = e^{-(\mu_1+\mu_2)}
 \left({\mu_1\over\mu_2}\right)^{n/2}I_n(2\sqrt{\mu_1\mu_2})
 <math>

where I _n(z) is the modified Bessel function of the first kind. The above formulas have assumed that any term with a negative factorial is set to zero. The special case for μ₁ = μ₂ is given by (Irwin, 1937):

<math>

 P_n\left(\mu,\mu\right) = e^{-2\mu}I_n(2\mu)
<math>

Properties

The Skellam probability density function is of course normalized:

<math>

 \sum_{k=-\infty}^\infty P_k(\mu_1,\mu_2)=1
 <math>

We know that the generating function for a Poisson distribution is:

<math>

 G\left(t,\mu\right)= e^{\mu(t-1)}
 <math>

It follows that the generating function G(t,μ₁,μ₂) for a Skellam probability function will be:

<math>G(t,\mu_1,\mu_2) = \sum_{k=0}^\infty P_k(\mu_1,\mu_2)t^k<math>

<math>= G\left(t,\mu_1\right)G\left(1/t,\mu_2\right)\,<math>

<math>= e^{-(\mu_1+\mu_2)+\mu_1 t+\mu_2/t}<math>

Notice that the form of the generating function implies that the distribution of the sums or the differences or, in fact, any linear combination of two Skellam-distributed variables are again Skellam-distributed. The moment-generating function is given by:

<math>M\left(t,\mu_1,\mu_2\right) = G(e^t,\mu_1,\mu_2)<math>

<math> = \sum_{k=0}^\infty { t^k \over k!}\,m_k<math>

which yields the raw moments m _k. Define:

<math>\Delta\equiv\mu_1-\mu_2<math>

<math>\mu\equiv (\mu_1+\mu_2)/2<math>

Then the raw moments m_k are

<math>m_1=\left.\Delta\right.<math>

<math>m_2=\left.2\mu+\Delta^2\right.<math>

<math>m_3=\left.\Delta(1+6\mu+\Delta^2)\right.<math>

The central moments M _k are

<math>M_2=\left.2\mu\right.<math>

<math>M_3=\left.\Delta\right.<math>

<math>M_4=\left.2\mu+12\mu^2\right.<math>

The mean, variance, skewness, and kurtosis excess are respectively:

<math>\left.\right.E(n)=\Delta<math>

<math>\sigma^2=\left.2\mu\right.<math>

<math>\gamma_1=\left.\Delta/(2\mu)^{3/2}\right.<math>

<math>\gamma_2=\left.1/2\mu\right.<math>

The cumulant-generating function is given by:

<math>

 K(t,\mu_1,\mu_2)\equiv \ln(M(t,\mu_1,\mu_2))
 = \sum_{k=0}^\infty { t^k \over k!}\,\kappa_k
 <math>

which yields the cumulants:

<math>\kappa_{2k}=\left.2\mu\right.<math>

<math>\kappa_{2k+1}=\left.\Delta\right. .<math>

For the special case when μ₁=μ₂, an asymptotic expansion of the modified Bessel function of the first kind yields for large μ:

<math>

 P_n(\mu,\mu)\sim
 {1\over\sqrt{4\pi\mu}}\left[1+\sum_{k=1}^\infty
 (-1)^k{\{4n^2-1^2\}\{4n^2-3^2\}\ldots\{4n^2-(2k-1)^2\}
 \over k!\,2^{3k}\,(2\mu)^k}\right]
 <math>

(Abramowitz & Stegun 1972, p. 377). Also, for this special case, when n is also large, and of order of the square root of 2μ, the distribution tends to a normal distribution:

<math>

 P_n(\mu,\mu)\sim
 {e^{-n^2/4\mu}\over\sqrt{4\pi\mu}}.
 <math>

These special results can easily be extended to the more general case of different means. It is also notable that if the two variates which are being differenced are correlated, the distribution of their differences is also a Skellam distribution, but the μ₁ and μ₂ parameters are somewhat differently interpreted (Karlis & Ntzoufras, 2003).

References

• Abramowitz, M. and Stegun, I. A. (Eds.). 1972. Modified Bessel functions I and K. Sections 9.6–9.7 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing, pp. 374–378. New York: Dover.

• Irwin, J. O. 1937. The frequency distribution of the difference between two independent variates following the same Poisson distribution. Journal of the Royal Statistical Society: Series A 100 (3): 415–416.

• Karlis, D. and Ntzoufras, I. 2003. Analysis of sports data using bivariate Poisson models. Journal of the Royal Statistical Society: Series D (The Statistician) 52 (3): 381–393. doi:10.1111/1467-9884.00366 (http://dx.doi.org/10.1111/1467-9884.00366)

• Karlis, D. and Ntzoufras, J. Bayesian analysis of paired count data. Unpublished manuscript. http://www.ba.aegean.gr/ntzoufras/papers/11_poisdif.pdf

• Skellam, J. G. 1946. The frequency distribution of the difference between two Poisson variates belonging to different populations. Journal of the Royal Statistical Society: Series A 109 (3): 296.