User:BeyondNormality/Additive binomial distribution

Additive binomial distribution

Notation	$\mathrm {AdditiveBinomial} (a,n,p)$
Parameters	n ∈ $\mathbb {N} _{0}$ 0 < p < 1 a (see article) q=1-p
Support	x ∈ { 0, 1, 2, ... , n}
PMF	${\displaystyle }$
CDF	${\displaystyle }$
Mean	${\displaystyle }$
Median	${\displaystyle }$
Mode	${\displaystyle }$
Variance	${\displaystyle }$
Skewness	${\displaystyle }$
Excess kurtosis	${\displaystyle }$
Entropy	${\displaystyle }$
MGF	${\displaystyle }$
CF	${\displaystyle }$
PGF	${\displaystyle }$

Also known as the additive generalization of the binomial distribution and correlated binomial distribution, the probability mass function of the additive binomial distribution is given by

$\mathrm {AdditiveBinomial} (x;a,n,p)\equiv \Pr(X=x)={\binom {n}{x}}p^{x}q^{n-x}\left(1+{\frac {a}{2}}\left({\frac {(x-1)x}{p}}+{\frac {(n-x)(n-x-1)}{q}}-n(n-1)\right)\right)$

• $x=0,1,2,\dots ,n$
• $n\in \mathbb {N} _{0}$
• $0<p<1$
• $q=1-p$
• ${\begin{cases}-\min \left({\frac {p}{q}},{\frac {q}{p}}\right)\leq a\leq 1&n=2,\\{\frac {-2}{n(n-1)}}\min \left({\frac {p}{q}},{\frac {q}{p}}\right)\leq a\leq 2\left(n+{\frac {(q-p)^{2}}{4pq}}\right)^{-1}&n\in \mathbb {N} -\{1,2\}\end{cases}}$

Expected Value

$\mathbb {E} (X)=np$

Variance

$\operatorname {Var} (X)={\frac {an(1-p)^{n}\,_{4}F_{3}\left(2,2,2,1-n;1,1,1;{\frac {p}{p-1}}\right)+n(p-1)\left(a((n-1)p(p(n((n-5)p+6)+4(p-2))+5)+1)-2(p-1)^{2}p\right)}{2(p-1)^{2}}}$

Recurrence Relation

$\left\{p(x-n)\left(an^{2}p^{2}-anp^{2}-2anpx-2anp+2apx+2ap+ax^{2}+ax-2p^{2}+2p\right)\Pr(x)-(p-1)(x+1)\left(an^{2}p^{2}-anp^{2}-2anpx+2apx+ax^{2}-ax-2p^{2}+2p\right)\Pr(x+1)=0\right\}$

Moment Generating Function

$M_{X}(t)=\left(pe^{t}+q\right)^{n-2}\left({\frac {1}{2}}a(n-1)npq\left(e^{t}-1\right)^{2}+\left(pe^{t}+q\right)^{2}\right)$

Characteristic Function

$\varphi _{X}(t)=\left(pe^{it}+q\right)^{n-2}\left({\frac {1}{2}}a(n-1)npq\left(e^{it}-1\right)^{2}+\left(pe^{it}+q\right)^{2}\right)$

Probability Generating Function

$G(t)=(q+pt)^{n-2}\left((q+pt)^{2}+{\frac {1}{2}}pq(1-t)^{2}(an(n-1))\right)$

Interrelations

Symbol	Meaning
$\sim$	$X\sim Y$: the random variable X is distributed as the random variable Y
$\equiv$	the distribution in the title is identical with this distribution
$\Leftarrow$	the distribution in title is a special case of this distribution
$\Rightarrow$	this distribution is a special case of the distribution in the title
$\leftarrow$	this distribution converges to the distribution in the title
$\rightarrow$	the distribution in the title converges to this distribution

Relationship	Distribution	When
$\leftarrow$	Paul-negative hypergeometric $\left(a,n,\alpha ,\beta \right)$	$a\rightarrow \infty \qquad \beta \rightarrow \infty \qquad {\frac {\alpha }{\alpha +\beta }}\rightarrow p$
$\Rightarrow$	binomial $\left(n,p\right)$	$a=0$
$\Rightarrow$	deterministic $\left(0\right)$	$n=0$
$\Rightarrow$	zero-one $\left(p\right)$	$n=1$
$\rightarrow$	Poisson $\left(a'\right)$	$n\rightarrow \infty \qquad p\rightarrow 0\qquad np\rightarrow a'$

References

• Altham, P. (1978). Two generalizations of the binomial distribution. Applied Statistics 27, 162-167
• Altmann-Fitter (1994). Iterative Anpassung diskreter Wahrscheinlichkeitsverteilungen. Lüdenscheid:RAM-Verlag.
• Bahadur, R.R. (1961). A representation of the joint distribution of responses to n dichotomous items. In: Solomon. H (ed.), Studies in Item Analysis and Prediction: 158-168. Stanford: Stanford University
• Engel, B., Brake, J. de (1993). Analysis of embryonic development with a model for under-or overdispersion relative to binomial variation. Biometrics 49, 269-279.
• Haseman, J.K. Kupper, L.L. (1979). Analysis of dichotomous response data from certain toxicological experiments. Biometrics 35, 281-293
• Johnson, N.L., Kotz, S., Kemp, A.W. (1992). Univariate Discrete Distributions. New York: Wiley. pg 148ff
• Kupper, L.L., Haseman, J.K. (1978). The use of a correlated binomial model for the analysis of certain toxicological experiments. Biometrica 34, 69-76.
• Lord, F.M. (1965). A strong true-score theory, with applications. Psychometrika 30, 239-270.
• Paul, S.R. (1982). Analysis of proportions of affected foetuses in teratological experiments. Biometrics 38, 361-370
• Paul, S.R. (1985). A three-parameter generalization of the binomial distribution. Communications in Statistics - Theory and Methods 14, 1497-1506
• Paul, S.R. (1987). On the beta-correlated binomial (BCB) distribution - a three parameter generalization of the binomial distribution. Communications in Statistics - Theory and Methods 16, 1473-1478
• Rudolpher, S.M. (1990). A Markov chain model of extrabinomial variations. Biometrika 77, 255-264
• Tarone, R.F. (1979). Testing the goodness of fit of the binomial distribution. Biometrika 66, 585-590
• Wilcox, R.R. (1981). A review of the beta-binomial model and its extensions. J. of Educational Statistics 6, 3-32
• Wimmer, G., Altmann. (1999). Thesaurus of univariate discrete probability distributions. Stamm; 1. ed (1999), pg 2