Continuous binomial distribution

Continuous binomial (cobin)
Parameters	$\theta \in \mathbb {R}$ (natural parameter) $\lambda \in \{1,2,3,\dots \}$ (inverse dispersion)
Support	$x\in [0,1]$ if $\lambda =1$, $x\in (0,1)$ if $\lambda \geq 2$
PDF	$f(x;\theta ,\lambda )=h(x;\lambda )\exp \!{\big (}\lambda \theta x-\lambda B(\theta ){\big )},\quad 0\leq x\leq 1,$ with $B(\theta )={\begin{cases}\log \!\left({\frac {e^{\theta }-1}{\theta }}\right),&\theta \neq 0,\\0,&\theta =0,\end{cases}}$ and $h(x;\lambda )={\frac {\lambda }{(\lambda -1)!}}\sum _{k=0}^{\lambda }(-1)^{k}{\lambda \choose k}\,\max(0,\lambda x-k)^{\lambda -1}.$
Mean	$B'(\theta )={\begin{cases}{\frac {e^{\theta }}{e^{\theta }-1}}-{\frac {1}{\theta }},&\theta \neq 0,\\{\frac {1}{2}},&\theta =0,\end{cases}}$
Variance	${\frac {1}{\lambda }}B''(\theta )={\begin{cases}{\frac {1}{\lambda }}\left({\frac {1}{\theta ^{2}}}-{\frac {e^{\theta }}{(e^{\theta }-1)^{2}}}\right),&\theta \neq 0,\\{\frac {1}{12\lambda }},&\theta =0.\end{cases}}$

In probability theory and statistics, the continuous binomial distribution (also called the cobin distribution) is a family of continuous probability distributions on the unit interval that belongs to an exponential dispersion family. It was introduced as a response distribution for generalized linear models for continuous proportional data, proposed as an alternative to beta regression. The special case $\lambda =1$ coincides with the continuous Bernoulli distribution.

Definition

A random variable $X$ is said to follow a continuous binomial (cobin) distribution with natural parameter $\theta$ and inverse dispersion $\lambda \in \{1,2,\dots \}$, written $X\sim \mathrm {cobin} (\theta ,\lambda ^{-1})$, if it has density on $[0,1]$ given by

$f(x;\theta ,\lambda )=h(x;\lambda )\exp \!{\big (}\lambda \theta x-\lambda B(\theta ){\big )},\qquad 0\leq x\leq 1,$

where the log-partition function is

$B(\theta )={\begin{cases}\log \!\left({\frac {e^{\theta }-1}{\theta }}\right),&\theta \neq 0,\\0,&\theta =0,\end{cases}}$

and the base measure $h(x;\lambda )$ is

$h(x;\lambda )={\frac {\lambda }{(\lambda -1)!}}\sum _{k=0}^{\lambda }(-1)^{k}{\lambda \choose k}\,\max(0,\lambda x-k)^{\lambda -1},$

with $h(x;1)=1$. The function $h(x;\lambda )/\lambda$ coincides with the probability density function of the Irwin–Hall distribution with parameter $n=\lambda$, evaluated at $\lambda x$.

When $\lambda$ is fixed, the cobin distribution belongs to a one-parameter natural exponential family in $\theta$.

Related distributions

• Continuous Bernoulli distribution: when $\lambda =1$, the cobin distribution reduces to the continuous Bernoulli distribution.

• Bates distribution: when $\theta =0$, the density reduces to $h(x;\lambda )$, corresponding to the distribution of the mean of $\lambda$ independent $\mathrm {Uniform} (0,1)$ random variables (equivalently, a scaled Irwin–Hall distribution or Bates distribution).

• Uniform distribution: when $\lambda =1$ and $\theta =0$, the distribution reduces to the continuous uniform distribution on $[0,1]$.

• If $X_{1},\dots ,X_{\lambda }$ are independent and identically distributed continuous Bernoulli random variables with common natural parameter $\theta$, then

${\bar {X}}={\frac {1}{\lambda }}\sum _{i=1}^{\lambda }X_{i}\sim \mathrm {cobin} (\theta ,\lambda ^{-1}).$

Properties

Mean and variance

The mean and variance of $X\sim \mathrm {cobin} (\theta ,\lambda ^{-1})$ can be expressed in terms of derivatives of $B(\theta )$:

• $\operatorname {E} (X)=B'(\theta )={\frac {e^{\theta }}{e^{\theta }-1}}-{\frac {1}{\theta }}$, for $\theta \neq 0$.
• $\operatorname {Var} (X)={\frac {1}{\lambda }}B''(\theta )={\frac {1}{\lambda }}\left({\frac {1}{\theta ^{2}}}-{\frac {e^{\theta }}{(e^{\theta }-1)^{2}}}\right)$, for $\theta \neq 0$.

If $\theta =0$, then $\operatorname {E} (X)={\frac {1}{2}}$ and $\operatorname {Var} (X)={\frac {1}{12\lambda }}$.

Sufficient statistic for the mean

If $X_{1},\ldots ,X_{n}$ are independent and identically distributed continuous binomial random variables with common natural parameter $\theta$ and fixed inverse dispersion parameter $\lambda$, then the sample mean

${\bar {X}}={\frac {1}{n}}\sum _{i=1}^{n}X_{i}$

is a sufficient statistic for $\theta$.

This is in contrast with the beta distribution: under a mean–precision parameterisation $X_{i}\sim \mathrm {Beta} (\mu \phi ,(1-\mu )\phi )$ with fixed $\phi$, a sufficient statistic for the mean $\mu$ is

$\sum _{i=1}^{n}\log \!\left({\frac {X_{i}}{1-X_{i}}}\right),$

not the sample mean ${\bar {X}}$.

Applications

The cobin distribution has been proposed as a response distribution for generalized linear models of continuous proportional data, as an alternative to beta regression, including extensions with random effects.