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Revision as of 06:17, 21 April 2013 by Illia Connell(talk | contribs)(→References: standardize journal name, replaced: Journal of the Royal Statistical Society: Series A → Journal of the Royal Statistical Society, Series A, Journal of the Royal Statistical Society: Series D → Journal o using AWB)
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.)
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 from two Poisson-distributed variables with means and is given by:
for (and zero otherwise). The Skellam probability mass function for the difference of two counts is the cross-correlation of two Poisson distributions: (Skellam, 1946)
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
so that:
where Ik(z) is the modified Bessel function of the first kind. The special case for is given by Irwin (1937):
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 .
Properties
As it is a discrete probability function, the Skellam probability mass function is normalized:
It follows that the pgf, , for a Skellam probability function will be:
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.
(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:
These special results can easily be extended to the more general case of
different means.
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]