Conway–Maxwell–binomial distribution explained

Conway–Maxwell–binomial

Type:

mass

Parameters:

n\in\{1,2,\ldots\},

0\leqp\leq1,

-infty<\nu<infty

Support:

x\in\{0,1,2,...,n\}

Pdf:

	1
	C_n,p,\nu

\binom{n}{x}^\nup^j(1-p)^n-x

Cdf:

	x
\sum
	i=0

\Pr(X=i)

Mean:

Not listed

Median:

No closed form

Mode:

See text

Variance:

Not listed

Skewness:

Not listed

Kurtosis:

Not listed

Entropy:

Not listed

Mgf:

See text

Char:

See text

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,.

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

\Pr(Y=j)=	1
	C_n,p,\nu

\binom{n}{j}^\nup^j(1-p)^n-j, j\in\{0,1,\ldots,n\},

where

n\inN=\{1,2,\ldots\}

0\leqp\leq1

and

-infty<\nu<infty

. The normalizing constant

C_n,p,\nu

is defined by

C_n,p,\nu

	n\binom{n}{i}
=\sum
	i=0

^\nup^i(1-p)^n-i.

has the above mass function, then we write

Y\sim\operatorname{CMB}(n,p,\nu)

The case

\nu=1

is the usual binomial distribution

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

X_1\sim\operatorname{CMP}(λ_1,\nu)

and

X_2\sim\operatorname{CMP}(λ_2,\nu)

are independent, then

X_1|X_1+X_{2=n\sim\operatorname{CMB}(n,λ}_1/(λ_1+λ_2),\nu)

Sum of possibly associated Bernoulli random variables

The random variable

Y\sim\operatorname{CMB}(n,p,\nu)

may be written ^[1] as a sum of exchangeable Bernoulli random variables

Z_1,\ldots,Z_n

satisfying

\Pr(Z_1=z_1,\ldots,Z_n=z

n)=	1
	C_n,p,\nu

\binom{n}{k}^\nu-1p^k(1-p)^n-k,

where

k=z_{1+ … +z}_n

. Note that

\operatorname{E}Z_1\not=p

in general, unless

\nu=1

Generating functions

Let

	n
T(x,\nu)=\sum
	k=0

x^{k\binom{n}{k}}^\nu.

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

G(t)=	T(tp/(1-p),\nu)
	T(p(1-p),\nu)

M(t)=	T(e^{tp/(1-p),\nu)}
	T(p(1-p),\nu)

\varphi(t)=	T(e^itp/(1-p),\nu)
	T(p(1-p),\nu)

Moments

For general

\nu

, there do not exist closed form expressions for the moments of the CMB distribution. The following neat formula is available, however.^[3] Let

(j)_{r=j(j-1) … (j-r+1)}

denote the falling factorial. Let

Y\sim\operatorname{CMB}(n,p,\nu)

, where

\nu>0

. Then

\nu]=	C_n-r,p,\nu
	C_n,p,\nu

\operatorname{E}[((Y)

	\nu
((n)
	r)

p^r,

for

r=1,\ldots,n-1

Mode

Let

Y\sim\operatorname{CMB}(n,p,\nu)

and define

n+1

1+\left(	1-p	\right)^1/\nu
	p

Then the mode of

\lfloora\rfloor

is not an integer. Otherwise, the modes of

are

and

a-1

.^[3]

Stein characterisation

Let

Y\sim\operatorname{CMB}(n,p,\nu)

, and suppose that

f:Z^+\mapstoR

is such that

\operatorname{E}|f(Y+1)|<infty

and

\operatorname{E}|Y^\nuf(Y)|<infty

. Then ^[3]

\operatorname{E}[p(n-Y)^\nuf(Y+1)-(1-p)Y^\nuf(Y)]=0.

Approximation by the Conway–Maxwell–Poisson distribution

Fix

λ>0

and

\nu>0

and let

	\nu,\nu)
Y
	n\simCMB(n,λ/n

Then

Y_n

converges in distribution to the

CMP(λ,\nu)

distribution as

n → infty

.^[3] This result generalises the classical Poisson approximation of the binomial distribution.

Conway–Maxwell–Poisson binomial distribution

Let

X_1,\ldots,X_n

be Bernoulli random variables with joint distribution given by

\Pr(X_1=x_1,\ldots,X_n=x

n)=	1
	C_n'

\binom{n}{k}^\nu-1

	x_j
\prod
	j

	1-x_j
(1-p
	j)

where

k=x_{1+ … +x}_n

and the normalizing constant

	\prime
C
	n

is given by

C_n'=\sum

	n

	k=0

\binom{n}{k}^\nu-1

\sum
	A\inF_k

\prod_i\inp_i

\prod
	j\inA^c

(1-p_j),

where

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

Let

W=X_{1+ … +X}_n

. Then

has mass function

\Pr(W=k)=	1
	C_n'

\binom{n}{k}^\nu-1

\sum
	A\inF_k

\prod_i\inp_i\prod


	j\inA^c

(1-p_j),

for

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

\nu=1

is the usual Poisson binomial distribution and the case

p_{1= … =p}_n=p

is the

\operatorname{CMB}(n,p,\nu)

distribution.

References

Shmueli G., Minka T., Kadane J.B., Borle S., and Boatwright, P.B. "A useful distribution for fitting discrete data: revival of the Conway–Maxwell–Poisson distribution." Journal of the Royal Statistical Society: Series C (Applied Statistics) 54.1 (2005): 127–142.https://dx.doi.org/10.1111/j.1467-9876.2005.00474.x
Kadane, J.B. " Sums of Possibly Associated Bernoulli Variables: The Conway–Maxwell–Binomial Distribution." Bayesian Analysis 11 (2016): 403–420.
Daly, F. and Gaunt, R.E. " The Conway–Maxwell–Poisson distribution: distributional theory and approximation." ALEA Latin American Journal of Probabability and Mathematical Statistics 13 (2016): 635–658.