View source for User:BeyondNormality/Additive binomial distribution

{{Probability distribution
  | name       = Additive binomial distribution
  | type       = mass
  | pdf_image  = 
  | cdf_image  = 
  | notation   = <math>\mathrm{AdditiveBinomial}(a,n,p)</math>
  | parameters = n ∈ <math> \mathbb{N}_0 </math> <br> 0 < p < 1 <br> a (see article) <br> q=1-p
  | support    = ''x'' ∈ { 0, 1, 2,  ... , n}
  | pdf        = <math></math>
  | cdf        = <math></math>
  | mean       = <math></math>
  | median     = <math></math>
  | mode       = <math></math>
  | variance   = <math></math>
  | skewness   = <math></math>
  | kurtosis   = <math></math>
  | entropy    = <math></math>
  | pgf        = <math></math>
  | mgf        = <math></math>
  | char       = <math></math>
}}

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


: <math>
    \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) 
  </math>

::* <math> x = 0, 1, 2, \dots ,n  </math>
::* <math> n  \in \mathbb{N}_0  </math>
::* <math> 0 < p < 1 </math>
::* <math> q = 1 - p </math>
::* <math> \begin{cases}
-\min \left( \frac{p}{q} , \frac{q}{p} \right) \le a \le 1 & n = 2, \\
\frac{-2}{n(n-1)} \min \left( \frac{p}{q} , \frac{q}{p} \right) \le a \le 2 \left(n+\frac{(q-p)^2}{4 p q}\right)^{-1} & n  \in \mathbb{N}-\{1,2\}
\end{cases} </math>


'''Expected Value'''

<math> \mathbb E(X) = n p </math>


'''Variance'''

<math>
\operatorname{Var}(X) = \frac{a n (1-p)^n \, _4F_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}
</math>


'''Recurrence Relation'''

<math>
\left\{p (x-n) \left(a n^2 p^2-a n p^2-2 a n p x-2 a n p+2 a p x+2 a p+a x^2+a x-2 p^2+2 p\right) \Pr (x) -(p-1) (x+1) \left(a n^2 p^2-a n p^2-2 a n p x+2 a p x+a
   x^2-a x-2 p^2+2 p\right) \Pr (x+1)  =0\right\}
</math>


'''Moment Generating Function'''

: <math> M_X(t)=\left(p e^t+q\right)^{n-2} \left(\frac{1}{2} a (n-1) n p q \left(e^t-1\right)^2+\left(p e^t+q\right)^2\right) </math>


'''Characteristic Function'''

: <math> \varphi_X(t)=\left(p e^{i t}+q\right)^{n-2} \left(\frac{1}{2} a (n-1) n p q \left(e^{i t}-1\right)^2+\left(p e^{i t}+q\right)^2\right) </math>


'''Probability Generating Function'''

: <math> G(t) = (q+p t)^{n-2} \left((q+p t)^2+\frac{1}{2} p q (1-t)^2 (a n (n-1))\right) </math>


==Interrelations==


{| class="wikitable"
|-
! Symbol !! Meaning
|-
| align="center" | <math> \sim </math>  || <math> X \sim Y </math>: the random variable X is distributed as the random variable Y
|-
| align="center" | <math> \equiv </math> || the distribution in the title is identical with this distribution
|-
| align="center" | <math> \Leftarrow </math> || the distribution in title is a special case of this distribution 
|-
| align="center" | <math> \Rightarrow </math> || this distribution is a special case of the distribution in the title 
|-
| align="center" | <math> \leftarrow </math> || this distribution converges to the distribution in the title 
|-
| align="center" | <math> \rightarrow </math> || the distribution in the title converges to this distribution 
|}

{| class="wikitable"
|-
! Relationship !! Distribution !! When
|-
|  align="center" | <math> \leftarrow </math>  
| Paul-negative hypergeometric <math>\left( a,n,\alpha,\beta \right)</math>
| <math> a \rightarrow \infty \qquad \beta \rightarrow \infty \qquad \frac{\alpha}{\alpha+\beta} \rightarrow p </math>
|-
|  align="center" | <math> \Rightarrow </math> 
| binomial <math> \left( n, p \right) </math>
| <math> a = 0 </math>
|-
|  align="center" | <math> \Rightarrow </math>
| deterministic <math>\left( 0 \right)</math>
| <math> n=0 </math>
|-
|  align="center" | <math> \Rightarrow </math>
| zero-one <math>\left( p \right)</math>
| <math> n=1 </math>
|-
|  align="center" | <math> \rightarrow </math>
| Poisson <math>\left( a' \right)</math>
| <math> n \rightarrow \infty \qquad p \rightarrow 0 \qquad n p \rightarrow a' </math>
|}

==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