Conway–Maxwell–binomial distribution

In probability theory and statistics, the Conway–Maxwell–binomial (CMB) distribution is a three parameter discrete probability distribution that generalises the binomial distribution in an analogous manner to the way that the Conway–Maxwell–Poisson distribution generalises the Poisson distribution. The CMB distribution can be used to model both positive and negative association among the Bernoulli summands,.^[1][2]

Conway–Maxwell–binomial

Parameters	${\displaystyle n\in \{1,2,\ldots \},}$ ${\displaystyle 0\leq p\leq 1,}$ ${\displaystyle -\infty <\nu <\infty }$
Support	${\displaystyle x\in \{0,1,2,\dots ,n\}}$
PMF	${\displaystyle {\frac {1}{C_{n,p,\nu }}}{\binom {n}{x}}^{\nu }p^{j}(1-p)^{n-x}}$
CDF	${\displaystyle \sum _{i=0}^{x}\Pr(X=i)}$
Mean	Not listed
Median	No closed form
Mode	See text
Variance	Not listed
Skewness	Not listed
Excess kurtosis	Not listed
Entropy	Not listed
MGF	See text
CF	See text

The distribution was introduced by Shumeli et al. (2005),^[1] and the name Conway–Maxwell–binomial distribution was introduced independently by Kadane (2016) ^[2] and Daly and Gaunt (2016).^[3]

Probability mass function

The Conway–Maxwell–binomial (CMB) distribution has probability mass function

${\displaystyle \Pr(Y=j)={\frac {1}{C_{n,p,\nu }}}{\binom {n}{j}}^{\nu }p^{j}(1-p)^{n-j}\,,\qquad j\in \{0,1,\ldots ,n\},}$

where ${\displaystyle n\in \mathbb {N} =\{1,2,\ldots \}}$, ${\displaystyle 0\leq p\leq 1}$ and ${\displaystyle -\infty <\nu <\infty }$. The normalizing constant ${\displaystyle C_{n,p,\nu }}$ is defined by

${\displaystyle C_{n,p,\nu }=\sum _{i=0}^{n}{\binom {n}{i}}^{\nu }p^{i}(1-p)^{n-i}.}$

If a random variable ${\displaystyle Y}$ has the above mass function, then we write ${\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )}$.

The case ${\displaystyle \nu =1}$ is the usual binomial distribution ${\displaystyle Y\sim \operatorname {Bin} (n,p)}$.

Relation to Conway–Maxwell–Poisson distribution

The following relationship between Conway–Maxwell–Poisson (CMP) and CMB random variables ^[1] generalises a well-known result concerning Poisson and binomial random variables. If ${\displaystyle X_{1}\sim \operatorname {CMP} (\lambda _{1},\nu )}$ and ${\displaystyle X_{2}\sim \operatorname {CMP} (\lambda _{2},\nu )}$ are independent, then ${\displaystyle X_{1}\,|\,X_{1}+X_{2}=n\sim \operatorname {CMB} (n,\lambda _{1}/(\lambda _{1}+\lambda _{2}),\nu )}$.

Sum of possibly associated Bernoulli random variables

The random variable ${\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )}$ may be written ^[1] as a sum of exchangeable Bernoulli random variables ${\displaystyle Z_{1},\ldots ,Z_{n}}$ satisfying

${\displaystyle \Pr(Z_{1}=z_{1},\ldots ,Z_{n}=z_{n})={\frac {1}{C_{n,p,\nu }}}{\binom {n}{k}}^{\nu -1}p^{k}(1-p)^{n-k},}$

where ${\displaystyle k=z_{1}+\cdots +z_{n}}$. Note that ${\displaystyle \operatorname {E} Z_{1}\not =p}$ in general, unless ${\displaystyle \nu =1}$.

Generating functions

Let

${\displaystyle T(x,\nu )=\sum _{k=0}^{n}x^{k}{\binom {n}{k}}^{\nu }.}$

Then, the probability generating function, moment generating function and characteristic function are given, respectively, by:^[2]

${\displaystyle G(t)={\frac {T(tp/(1-p),\nu )}{T(p(1-p),\nu )}},}$

${\displaystyle M(t)={\frac {T(\mathrm {e} ^{t}p/(1-p),\nu )}{T(p(1-p),\nu )}},}$

${\displaystyle \varphi (t)={\frac {T(\mathrm {e} ^{\mathrm {i} t}p/(1-p),\nu )}{T(p(1-p),\nu )}}.}$

Moments

For general ${\displaystyle \nu }$, there do not exist closed form expressions for the moments of the CMB distribution. Having said that, the following mathematical relationship holds:^[3]

Let ${\displaystyle (j)_{r}=j(j-1)\cdots (j-r+1)}$ denote the falling factorial. If ${\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )}$, where ${\displaystyle \nu >0}$, then

${\displaystyle \operatorname {E} [((Y)_{r})^{\nu }]={\frac {C_{n-r,p,\nu }}{C_{n,p,\nu }}}((n)_{r})^{\nu }p^{r}\,,}$

for ${\displaystyle r=1,\ldots ,n-1}$.

Mode

Let ${\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )}$ and define

${\displaystyle a={\frac {n+1}{1+\left({\frac {1-p}{p}}\right)^{1/\nu }}}.}$

Then the mode of ${\displaystyle Y}$ is ${\displaystyle \lfloor a\rfloor }$ if ${\displaystyle a}$ is not an integer. Otherwise, the modes of ${\displaystyle Y}$ are ${\displaystyle a}$ and ${\displaystyle a-1}$.^[3]

Stein characterisation

Let ${\displaystyle Y\sim \operatorname {CMB} (n,p,\nu )}$, and suppose that ${\displaystyle f:\mathbb {Z} ^{+}\mapsto \mathbb {R} }$ is such that ${\displaystyle \operatorname {E} |f(Y+1)|<\infty }$ and ${\displaystyle \operatorname {E} |Y^{\nu }f(Y)|<\infty }$. Then ^[3]

${\displaystyle \operatorname {E} [p(n-Y)^{\nu }f(Y+1)-(1-p)Y^{\nu }f(Y)]=0.}$

Approximation by the Conway–Maxwell–Poisson distribution

Fix ${\displaystyle \lambda >0}$ and ${\displaystyle \nu >0}$ and let ${\displaystyle Y_{n}\sim \mathrm {CMB} (n,\lambda /n^{\nu },\nu )}$ Then ${\displaystyle Y_{n}}$ converges in distribution to the ${\displaystyle \mathrm {CMP} (\lambda ,\nu )}$ distribution as ${\displaystyle n\rightarrow \infty }$.^[3] This result generalises the classical Poisson approximation of the binomial distribution.

Conway–Maxwell–Poisson binomial distribution

Let ${\displaystyle X_{1},\ldots ,X_{n}}$ be Bernoulli random variables with joint distribution given by

${\displaystyle \Pr(X_{1}=x_{1},\ldots ,X_{n}=x_{n})={\frac {1}{C_{n}'}}{\binom {n}{k}}^{\nu -1}\prod _{j=1}^{n}p_{j}^{x_{j}}(1-p_{j})^{1-x_{j}},}$

where ${\displaystyle k=x_{1}+\cdots +x_{n}}$ and the normalizing constant ${\displaystyle C_{n}^{\prime }}$ is given by

${\displaystyle C_{n}'=\sum _{k=0}^{n}{\binom {n}{k}}^{\nu -1}\sum _{A\in F_{k}}\prod _{i\in A}p_{i}\prod _{j\in A^{c}}(1-p_{j}),}$

where

${\displaystyle F_{k}=\left\{A\subseteq \{1,\ldots ,n\}:|A|=k\right\}.}$

Let ${\displaystyle W=X_{1}+\cdots +X_{n}}$. Then ${\displaystyle W}$ has mass function

${\displaystyle \Pr(W=k)={\frac {1}{C_{n}'}}{\binom {n}{k}}^{\nu -1}\sum _{A\in F_{k}}\prod _{i\in A}p_{i}\prod _{j\in A^{c}}(1-p_{j}),}$

for ${\displaystyle k=0,1,\ldots ,n}$. This distribution generalises the Poisson binomial distribution in a way analogous to the CMP and CMB generalisations of the Poisson and binomial distributions. Such a random variable is therefore said ^[3] to follow the Conway–Maxwell–Poisson binomial (CMPB) distribution. This should not be confused with the rather unfortunate terminology Conway–Maxwell–Poisson–binomial that was used by ^[1] for the CMB distribution.

The case ${\displaystyle \nu =1}$ is the usual Poisson binomial distribution and the case ${\displaystyle p_{1}=\cdots =p_{n}=p}$ is the ${\displaystyle \operatorname {CMB} (n,p,\nu )}$ distribution.