Editing Skellam distribution

{{Short description|Discrete probability distribution}}
{{Probability distribution|
  name       =Skellam|
  type       =mass|
  pdf_image  =[[Image:Skellam distribution.svg|325px|Examples of the probability mass function for the Skellam distribution.]]<br /><small>Examples of the probability mass function for the Skellam distribution. The horizontal axis is the index ''k''. (The function is only defined at integer values of ''k''. The connecting lines do not indicate continuity.)</small>|
  cdf_image  =|
  parameters =<math>\mu_1\ge 0,~~\mu_2\ge 0</math>|
  support    =<math>k \in \{\ldots, -2,-1,0,1,2,\ldots\}</math>|
  pdf        =<math>e^{-(\mu_1\!+\!\mu_2)}
\left(\frac{\mu_1}{\mu_2}\right)^{k/2}\!\!I_{k}(2\sqrt{\mu_1\mu_2})</math>|
  cdf        =|
  mean       =<math>\mu_1-\mu_2\,</math>|
  median     =N/A|
  mode       =|
  variance   =<math>\mu_1+\mu_2\,</math>|
  skewness   =<math>\frac{\mu_1-\mu_2}{(\mu_1+\mu_2)^{3/2}}</math>|
  kurtosis   =<math>\frac{1}{\mu_1+\mu_2}</math>|
  entropy    =|
  mgf        =<math>e^{-(\mu_1+\mu_2)+\mu_1e^t+\mu_2e^{-t}}</math>|
  char       =<math>e^{-(\mu_1+\mu_2)+\mu_1e^{it}+\mu_2e^{-it}}</math>
}}

The '''Skellam distribution''' is the [[discrete probability distribution]] of the difference <math>N_1-N_2</math> of two [[statistically independent]] [[random variable]]s <math>N_1</math> and <math>N_2,</math> each [[Poisson distribution|Poisson-distributed]] with respective [[expected value]]s <math>\mu_1</math> and <math>\mu_2</math>. It is useful in describing the statistics of the difference of two images with simple [[photon noise]], as well as describing the [[Spread betting|point spread]] distribution in sports where all scored points are equal, such as [[baseball]], [[ice hockey|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 difference <math>K=N_1-N_2</math> between two independent Poisson-distributed random variables with means <math>\mu_1</math> and <math>\mu_2</math> is given by:

:<math>
  p(k;\mu_1,\mu_2) = \Pr\{K=k\} = e^{-(\mu_1+\mu_2)}
  \left({\mu_1\over\mu_2}\right)^{k/2}I_{k}(2\sqrt{\mu_1\mu_2})
</math>

where ''I<sub>k</sub>''(''z'') is the [[Bessel function#Modified Bessel functions : I.CE.B1.2C K.CE.B1|modified Bessel function]] of the first kind. Since ''k'' is an integer we have that ''I<sub>k</sub>''(''z'')=''I<sub>|k|</sub>''(''z'').

== Derivation ==

The [[probability mass function]] of a [[Poisson distribution|Poisson-distributed]] random variable with mean μ is given by

:<math>
 p(k;\mu)={\mu^k\over k!}e^{-\mu}.\,
 </math>

for <math>k \ge 0</math> (and zero otherwise). The Skellam probability mass function for the difference of two independent counts <math>K=N_1-N_2</math> is the [[convolution]] of two Poisson distributions: ([[John Gordon Skellam|Skellam]], 1946)

:<math>
\begin{align}
  p(k;\mu_1,\mu_2)
  & =\sum_{n=-\infty}^\infty
  p(k+n;\mu_1)p(n;\mu_2) \\
  & =e^{-(\mu_1+\mu_2)}\sum_{n=\max(0,-k)}^\infty
  {{\mu_1^{k+n}\mu_2^n}\over{n!(k+n)!}}
\end{align}
  </math>
Since the Poisson distribution is zero for negative values of the count <math>(p(N<0;\mu)=0)</math>, the second sum is only taken for those terms where <math>n\ge0</math> and <math>n+k\ge0</math>. It can be shown that the above sum implies that

:<math>\frac{p(k;\mu_1,\mu_2)}{p(-k;\mu_1,\mu_2)}=\left(\frac{\mu_1}{\mu_2}\right)^k</math>

so that:

:<math>
  p(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})
  </math>

where ''I''&nbsp;<sub>k</sub>(z) is the [[Bessel function#Modified Bessel functions : I.CE.B1.2C K.CE.B1|modified Bessel function]] of the first kind. The special case for <math>\mu_1=\mu_2(=\mu)</math> is given by Irwin (1937):

:<math>
  p\left(k;\mu,\mu\right) = e^{-2\mu}I_{|k|}(2\mu).
 </math>

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 <math>\mu_2=0</math>.

== Properties ==

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

:<math>
  \sum_{k=-\infty}^\infty p(k;\mu_1,\mu_2)=1.
  </math>

We know that the [[Probability-generating function|probability generating function]] (pgf) for a [[Poisson distribution]] is:

:<math>
  G\left(t;\mu\right)= e^{\mu(t-1)}.
  </math>

It follows that the pgf, <math>G(t;\mu_1,\mu_2)</math>, for a Skellam probability mass function will be:

:<math>
\begin{align}
G(t;\mu_1,\mu_2) & = \sum_{k=-\infty}^\infty p(k;\mu_1,\mu_2)t^k \\[4pt]
& = G\left(t;\mu_1\right)G\left(1/t;\mu_2\right) \\[4pt]
& = e^{-(\mu_1+\mu_2)+\mu_1 t+\mu_2/t}.
\end{align}
</math>

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 <math>\pm 1</math> would change the [[support (mathematics)|support]] of the distribution and alter the pattern of [[Moment (mathematics)|moments]] in a way that no Skellam distribution can satisfy.

The [[moment-generating function]] is given by:

:<math>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</math>

which yields the raw moments ''m''<sub>''k''</sub>&nbsp;. Define:

:<math>\Delta\ \stackrel{\mathrm{def}}{=}\  \mu_1-\mu_2\,</math>

:<math>\mu\ \stackrel{\mathrm{def}}{=}\   (\mu_1+\mu_2)/2.\,</math>

Then the raw moments ''m''<sub>''k''</sub> 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 [[moment about the mean|central moments]] ''M''<sub> ''k''</sub> 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 [[expected value|mean]], [[variance]], [[skewness]], and [[kurtosis|kurtosis excess]] are respectively:

:<math>
\begin{align}
\operatorname E(n) & = \Delta, \\[4pt]
\sigma^2 & =2\mu, \\[4pt]
\gamma_1 & =\Delta/(2\mu)^{3/2}, \\[4pt]
\gamma_2 & = 1/2.
\end{align}
</math>

The [[cumulant-generating function]] is given by:

:<math>
  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
  </math>

which yields the [[cumulant]]s:

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

For the special case when μ<sub>1</sub> = μ<sub>2</sub>, an
[[asymptotic expansion]] of the [[Bessel function|modified Bessel function of the first kind]] yields for large μ:

:<math>
  p(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].
  </math>

(Abramowitz & Stegun 1972, p.&nbsp;377).  Also, for this special case, when ''k'' is also large, and of [[Big O notation|order]] of the square root of 2μ, the distribution tends to a [[normal distribution]]:

:<math>
  p(k;\mu,\mu)\sim
  {e^{-k^2/4\mu}\over\sqrt{4\pi\mu}}.
  </math>

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

===Bounds on weight above zero===
If <math>X \sim \operatorname{Skellam} (\mu_1, \mu_2) </math>, with <math>\mu_1 < \mu_2</math>, then

::<math>
\frac{\exp(-(\sqrt{\mu_1} -\sqrt{\mu_2})^2  )}{(\mu_1 + \mu_2)^2} - \frac{e^{-(\mu_1 + \mu_2)}}{2\sqrt{\mu_1 \mu_2}} - \frac{e^{-(\mu_1 + \mu_2)}}{4\mu_1 \mu_2} \leq \Pr\{X  \geq 0\} \leq \exp (- (\sqrt{\mu_1} -\sqrt{\mu_2})^2)
</math>

Details can be found in [[Poisson distribution#Poisson races]]

==References==
*{{cite book|editor1-last=Abramowitz|editor1-first=Milton|editor2-last=Stegun|editor2-first=Irene A.|title=Handbook of mathematical functions with formulas, graphs, and mathematical tables|date=June 1965|publisher=Dover Publications|isbn=0486612724|url=http://store.doverpublications.com/0486612724.html|edition=Unabridged and unaltered republ. [der Ausg.] 1964, 5. Dover printing|access-date=27 September 2012|pages=374–378}}
*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&ndash;416. {{JSTOR|2980526}}
*Karlis, D. and Ntzoufras, I. (2003) "Analysis of sports data using bivariate Poisson models". ''Journal of the Royal Statistical Society, Series D'', 52 (3), 381&ndash;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&ndash;1905. [http://stat-athens.aueb.gr/~jbn/papers/paper11.htm]
*[[John Gordon Skellam|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. {{JSTOR|2981372}}

== See also ==
* [[Ratio_distribution#Poisson_and_truncated_Poisson_distributions|Ratio distribution for (truncated) Poisson distributions]]


{{ProbDistributions|Skellam distribution}}

[[Category:Discrete distributions]]
[[Category:Poisson distribution]]
[[Category:Infinitely divisible probability distributions]]