Continuous binomial distribution

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 <math>\lambda=1</math> coincides with the continuous Bernoulli distribution.

Definition

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

<math>

f(x;\theta,\lambda)=h(x;\lambda)\exp\!\big(\lambda\theta x-\lambda B(\theta)\big),\qquad 0\le x\le 1, </math> where the log-partition function is

<math>

B(\theta)= \begin{cases} \log\!\left(\frac{e^\theta-1}{\theta}\right), & \theta\ne 0,\\ 0, & \theta=0, \end{cases} </math> and the base measure <math>h(x;\lambda)</math> is

<math>

h(x;\lambda)=\frac{\lambda}{(\lambda-1)!}\sum_{k=0}^{\lambda}(-1)^k{\lambda\choose k}\,\max(0,\lambda x-k)^{\lambda-1}, </math> with <math>h(x;1)=1</math>. The function <math>h(x;\lambda)/\lambda</math> coincides with the probability density function of the Irwin–Hall distribution with parameter <math>n=\lambda</math>, evaluated at <math>\lambda x</math>.

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

Related distributions

• Continuous Bernoulli distribution: when <math>\lambda=1</math>, the cobin distribution reduces to the continuous Bernoulli distribution.

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

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

• If <math>X_1,\dots,X_\lambda</math> are independent and identically distributed continuous Bernoulli random variables with common natural parameter <math>\theta</math>, then

<math>

\bar{X}=\frac{1}{\lambda}\sum_{i=1}^{\lambda} X_i \sim \mathrm{cobin}(\theta,\lambda^{-1}). </math>

Properties

Mean and variance

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

• <math>\operatorname{E}(X)=B'(\theta)=\frac{e^\theta}{e^\theta-1}-\frac{1}{\theta}</math>, for <math>\theta\ne 0</math>.
• <math>\operatorname{Var}(X)=\frac{1}{\lambda}B(\theta)=\frac{1}{\lambda}\left(\frac{1}{\theta^2}-\frac{e^\theta}{(e^\theta-1)^2}\right)</math>, for <math>\theta\ne 0</math>.

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

Sufficient statistic for the mean

If <math>X_1,\ldots,X_n</math> are independent and identically distributed continuous binomial random variables with common natural parameter <math>\theta</math> and fixed inverse dispersion parameter <math>\lambda</math>, then the sample mean

<math>

\bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_i </math> is a sufficient statistic for <math>\theta</math>.

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

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\sum_{i=1}^{n}\log\!\left(\frac{X_i}{1-X_i}\right), </math> not the sample mean <math> \bar{X} </math>.

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.