Skellam distribution

Skellam
Probability mass function 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.)
Parameters	$\mu _{1}\geq 0,~~\mu _{2}\geq 0$
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	$e^{{-(\mu _{1}+\mu _{2})+\mu _{1}e^{t}+\mu _{2}e^{{-t}}}}$
CF	$e^{{-(\mu _{1}+\mu _{2})+\mu _{1}e^{{it}}+\mu _{2}e^{{-it}}}}$

The Skellam distribution is the discrete probability distribution of the difference $N_{1}-N_{2}$ of two statistically independent random variables $N_{1}$ and $N_{2},$ each Poisson-distributed with respective expected values $\mu _{1}$ and $\mu _{2}$ 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 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 difference $K=N_{1}-N_{2}$ between two independent Poisson-distributed random variables with means $\mu _{1}$ and $\mu _{2}$ is given by:

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}}})

where I_k(z) is the modified Bessel function of the first kind. Since k is an integer we have that I_k(z)=I_|k|(z).

Derivation

Note that the probability mass function of a Poisson-distributed random variable with mean μ is given by

p(k;\mu )={\mu ^{k} \over k!}e^{-\mu }.\,

for $k\geq 0$ (and zero otherwise). The Skellam probability mass function for the difference of two independent counts $K=N_{1}-N_{2}$ is the convolution of two Poisson distributions: (Skellam, 1946)

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)!}}

Since the Poisson distribution is zero for negative values of the count $(p(N<0;\mu )=0)$ , the second sum is only taken for those terms where $n>=0$ and $n+k>=0$ . It can be shown that the above sum implies that

{\frac {p(k;\mu _{1},\mu _{2})}{p(-k;\mu _{1},\mu _{2})}}=\left({\frac {\mu _{1}}{\mu _{2}}}\right)^{k}

so that:

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}}})

where I _k(z) is the modified Bessel function of the first kind. The special case for $\mu _{1}=\mu _{2}(=\mu )$ is given by Irwin (1937):

p\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 $\mu _{2}=0$ .

Properties

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

\sum _{k=-\infty }^{\infty }p(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, $G(t;\mu _{1},\mu _{2})$ , for a Skellam probability mass function will be:

G(t;\mu _{1},\mu _{2})=\sum _{k=0}^{\infty }p(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 $\pm 1$ would change the support of the distribution and alter the pattern of moments in a way that no Skellam distribution can satisfy.

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:

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].

(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:

p(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.

The following recurrence relation holds. Let $P(k)=p(k;\mu _{1},\mu _{2})$ be the probability mass function for a Skellam-distributed random variable with parameters $\mu _{1}$ and $\mu _{2}$ . Then

\left\{\begin{array}{l} -\mu _1 P(k)+\mu _2 P(k+2)+(k+1) P(k+1)=0, \\[10pt] P(0)=e^{-\mu _1-\mu _2} \, _0\tilde{F}_1\left(;1;\mu _1 \mu _2\right), \\[10pt] P(1)=e^{-\mu _1-\mu _2} \mu _1 \, _0\tilde{F}_1\left(;2;\mu _1 \mu _2\right) \end{array}\right\}

where $_{0}{\tilde {F}}_{1}$ denotes the regularized hypergeometric function.

Bounds on weight above zero

If $X\sim Skellam(\mu _{1},\mu _{2})$ , with $\mu _{1}<\mu _{2}$ , then

{\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})

Details can be found in Poisson distribution#Poisson Races

References

Abramowitz, Milton; Stegun, Irene A., eds. (June 1965). Handbook of mathematical functions with formulas, graphs, and mathematical tables (Unabridged and unaltered republ. [der Ausg.] 1964, 5. Dover printing ed.). Dover Publications. pp. 374–378. ISBN 0486612724. Retrieved 27 September 2012.
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, 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.
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.

Probability distributions

List

Discrete univariate with finite support	Benford Bernoulli beta-binomial binomial categorical hypergeometric Poisson binomial Rademacher discrete uniform Zipf Zipf–Mandelbrot

Discrete univariate with infinite support	beta negative binomial Borel Conway–Maxwell–Poisson discrete phase-type Delaporte extended negative binomial Gauss–Kuzmin geometric logarithmic negative binomial parabolic fractal Poisson Skellam Yule–Simon zeta

Continuous univariate supported on a bounded interval	arcsine ARGUS Balding–Nichols Bates beta beta rectangular Irwin–Hall Kumaraswamy logit-normal noncentral beta raised cosine reciprocal triangular U-quadratic uniform Wigner semicircle

Continuous univariate supported on a semi-infinite interval	Benini Benktander 1st kind Benktander 2nd kind beta prime Burr chi-squared chi Dagum Davis exponential-logarithmic Erlang exponential F folded normal Flory–Schulz Fréchet gamma gamma/Gompertz generalized inverse Gaussian Gompertz half-logistic half-normal Hotelling's T-squared hyper-Erlang hyperexponential hypoexponential inverse chi-squared scaled inverse chi-squared inverse Gaussian inverse gamma Kolmogorov Lévy log-Cauchy log-Laplace log-logistic log-normal Lomax matrix-exponential Maxwell–Boltzmann Maxwell–Jüttner Mittag-Leffler Nakagami noncentral chi-squared Pareto phase-type poly-Weibull Rayleigh relativistic Breit–Wigner Rice shifted Gompertz truncated normal type-2 Gumbel Weibull Discrete Weibull Wilks's lambda

Continuous univariate supported on the whole real line	Cauchy exponential power Fisher's z Gaussian q generalized normal generalized hyperbolic geometric stable Gumbel Holtsmark hyperbolic secant Johnson's S_U Landau Laplace asymmetric Laplace logistic noncentral t normal (Gaussian) normal-inverse Gaussian skew normal slash stable Student's t type-1 Gumbel Tracy–Widom variance-gamma Voigt

Continuous univariate with support whose type varies	generalized extreme value generalized Pareto Tukey lambda q-Gaussian q-exponential q-Weibull shifted log-logistic

Mixed continuous-discrete univariate	rectified Gaussian

Multivariate (joint)	Discrete Ewens multinomial Dirichlet-multinomial negative multinomial Continuous Dirichlet generalized Dirichlet multivariate normal multivariate stable multivariate t normal-inverse-gamma normal-gamma Matrix-valued inverse matrix gamma inverse-Wishart matrix normal matrix t matrix gamma normal-inverse-Wishart normal-Wishart Wishart

Directional	Univariate (circular) directional Circular uniform univariate von Mises wrapped normal wrapped Cauchy wrapped exponential wrapped asymmetric Laplace wrapped Lévy Bivariate (spherical) Kent Bivariate (toroidal) bivariate von Mises Multivariate von Mises–Fisher Bingham

Degenerate and singular	Degenerate Dirac delta function Singular Cantor

Families	Circular compound Poisson elliptical exponential natural exponential location-scale maximum entropy mixture Pearson Tweedie wrapped

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