Draft:Continuous binomial distribution

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

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

where the log-partition function is

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

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

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

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

When ${\displaystyle \lambda }$ is fixed, the cobin distribution belongs to a one-parameter natural exponential family in ${\displaystyle \theta }$.^[1]

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

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

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

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

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

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.^[1]