Skellam distribution

Skellam

Probability mass function Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index k. (Note that the function is only defined at integer values of k. The connecting lines do not indicate continuity.)
Parameters	$\mu_1\ge 0,~~\mu_2\ge 0$
Support	$\{\ldots, -2,-1,0,1,2,\ldots\}$
PMF	$e^{-(\mu_1\!+\!\mu_2)} \left(\frac{\mu_1}{\mu_2}\right)^{k/2}\!\!I_{|k|}(2\sqrt{\mu_1\mu_2})$
Mean	$\mu_1-\mu_2\,$
Median	N/A
Variance	$\mu_1+\mu_2\,$
Skewness	$\frac{\mu_1-\mu_2}{(\mu_1+\mu_2)^{3/2}}$
Ex. kurtosis	$1/(\mu_1+\mu_2)\,$
MGF	$e^{-(\mu_1+\mu_2)+\mu_1e^t+\mu_2e^{-t}}$
CF	$e^{-(\mu_1+\mu_2)+\mu_1e^{it}+\mu_2e^{-it}}$

The Skellam distribution is the discrete probability distribution of the difference n₁ − n₂ of two statistically independent random variables N₁ and N₂ each having Poisson distributions with different expected values μ₁ and μ₂. It is useful in describing the statistics of the difference of two images with simple photon noise, as well as describing the point spread distribution in certain sports where all scored points are equal, such as baseball, hockey and soccer.

The distribution is also applicable to a special case of the difference of dependent Poisson random variables, but just the obvious case where the two variables have a common additive random contribution which is cancelled by the differencing: see Karlis & Ntzoufras (2003) for details and an application.

The probability mass function for the Skellam distribution for a count difference k = n₁ − n₂ from two Poisson-distributed variables with means μ₁ and μ₂ is given by:

$f(k;\mu_1,\mu_2)= e^{-(\mu_1+\mu_2)} \left({\mu_1\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{\mu_1\mu_2})$

where I_k(z) is the modified Bessel function of the first kind.

Derivation

Note that the probability mass function of a Poisson distribution for a count n with mean μ is given by

$f(n;\mu)={\mu^n\over n!}e^{-\mu}.\,$

for $n \ge 0$ (and zero otherwise). The Skellam probability mass function for the difference of two counts k = n₁ − n₂ is the cross-correlation of two Poisson distributions: (Skellam, 1946)

$f(k;\mu_1,\mu_2) =\sum_{n=-\infty}^\infty \!f(k\!+\!n;\mu_1)f(n;\mu_2)$

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

Since the Poisson distribution is zero for negative values of the count, all terms with negative factorials in the above sum are set to zero. It can be shown that the above sum implies that

$\frac{f(k;\mu_1,\mu_2)}{f(-k;\mu_1,\mu_2)}=\left(\frac{\mu_1}{\mu_2}\right)^k$

so that:

$f(k;\mu_1,\mu_2)= e^{-(\mu_1+\mu_2)} \left({\mu_1\over\mu_2}\right)^{k/2}I_{|k|}(2\sqrt{\mu_1\mu_2})$

where I _k(z) is the modified Bessel function of the first kind. The special case for μ₁ = μ₂( = μ) is given by Irwin (1937):

$f\left(k;\mu,\mu\right) = e^{-2\mu}I_{|k|}(2\mu).$

Note also that, using the limiting values of the modified Bessel function for small arguments, we can recover the Poisson distribution as a special case of the Skellam distribution for μ₂ = 0.

Properties

As it is a discrete probability function, the Skellam probability mass function is normalized:

$\sum_{k=-\infty}^\infty f(k;\mu_1,\mu_2)=1.$

We know that the probability generating function (pgf) for a Poisson distribution is:

$G\left(t;\mu\right)= e^{\mu(t-1)}.$

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

$G(t;\mu_1,\mu_2) = \sum_{k=0}^\infty f(k;\mu_1,\mu_2)t^k$

$= G\left(t;\mu_1\right)G\left(1/t;\mu_2\right)\,$

$= e^{-(\mu_1+\mu_2)+\mu_1 t+\mu_2/t}.$

Notice that the form of the probability generating function implies that the distribution of the sums or the differences of any number of independent Skellam-distributed variables are again Skellam-distributed. It is sometimes claimed that any linear combination of two Skellam-distributed variables are again Skellam-distributed, but this is clearly not true since any multiplier other than +/-1 would change the support of the distribution.

The moment-generating function is given by:

$M\left(t;\mu_1,\mu_2\right) = G(e^t;\mu_1,\mu_2)$

$= \sum_{k=0}^\infty { t^k \over k!}\,m_k$

which yields the raw moments m_k . Define:

$\Delta\ \stackrel{\mathrm{def}}{=}\ \mu_1-\mu_2\,$

$\mu\ \stackrel{\mathrm{def}}{=}\ (\mu_1+\mu_2)/2.\,$

Then the raw moments m_k are

$m_1=\left.\Delta\right.\,$

$m_2=\left.2\mu+\Delta^2\right.\,$

$m_3=\left.\Delta(1+6\mu+\Delta^2)\right.\,$

The central moments M _k are

$M_2=\left.2\mu\right.,\,$

$M_3=\left.\Delta\right.,\,$

$M_4=\left.2\mu+12\mu^2\right..\,$

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

$\left.\right.E(n)=\Delta\,$

$\sigma^2=\left.2\mu\right.\,$

$\gamma_1=\left.\Delta/(2\mu)^{3/2}\right.\,$

$\gamma_2=\left.1/2\mu\right..\,$

The cumulant-generating function is given by:

$K(t;\mu_1,\mu_2)\ \stackrel{\mathrm{def}}{=}\ \ln(M(t;\mu_1,\mu_2)) = \sum_{k=0}^\infty { t^k \over k!}\,\kappa_k$

which yields the cumulants:

$\kappa_{2k}=\left.2\mu\right.$

$\kappa_{2k+1}=\left.\Delta\right. .$

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

$f(k;\mu,\mu)\sim {1\over\sqrt{4\pi\mu}}\left[1+\sum_{n=1}^\infty (-1)^n{\{4k^2-1^2\}\{4k^2-3^2\}\cdots\{4k^2-(2n-1)^2\} \over n!\,2^{3n}\,(2\mu)^n}\right]$

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

$f(k;\mu,\mu)\sim {e^{-k^2/4\mu}\over\sqrt{4\pi\mu}}.$

These special results can easily be extended to the more general case of different means.

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. [1]
• 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
• Karlis D. and Ntzoufras I. (2006). Bayesian analysis of the differences of count data. Statistics in Medicine, 25, 1885–1905. [2]
• 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. [3]