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 coincides with the continuous Bernoulli distribution.
Definition A random variable is said to follow a continuous binomial (cobin) distribution with natural parameter and inverse dispersion , written , if it has density on given by
where the log-partition function is
and the base measure is
with . The function coincides with the probability density function of the Irwin–Hall distribution with parameter , evaluated at .
When is fixed, the cobin distribution belongs to a one-parameter natural exponential family in .
Related distributions
Continuous Bernoulli distribution: when , the cobin distribution reduces to the continuous Bernoulli distribution.
Bates distribution: when , the density reduces to , corresponding to the distribution of the mean of independent random variables (equivalently, a scaled Irwin–Hall distribution or Bates distribution).
Uniform distribution: when and , the distribution reduces to the continuous uniform distribution on .
If are independent and identically distributed continuous Bernoulli random variables with common natural parameter , then
Properties
Mean and variance The mean and variance of can be expressed in terms of derivatives of :
, for . , for .
If , then and .
Sufficient statistic for the mean If are independent and identically distributed continuous binomial random variables with common natural parameter and fixed inverse dispersion parameter , then the sample mean
is a sufficient statistic for .
This is in contrast with the beta distribution: under a mean–precision parameterisation with fixed , a sufficient statistic for the mean is
not the sample mean .
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.