TÄNA AJALOOS, 11. juuli ⟩ Avati Tallinna teletorn

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.^[1] The special case $\lambda =1$ coincides with the continuous Bernoulli distribution^[2].

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.