Continuous binomial distribution: Difference between revisions
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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 model|exponential dispersion family]]. | 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 model|exponential dispersion family]]. The name ''continuous binomial'' (cobin) was adopted in 2026 in developing [[generalized linear model]]s for continuous proportional data, as an alternative to [[beta regression]] <ref name ="lee2026">{{Cite journal |last=Lee |first=Changwoo J. |last2=Dahl |first2=Benjamin K. |last3=Ovaskainen |first3=Otso |last4=Dunson |first4=David B. |date=2026-05-18 |title=Scalable and robust regression models for continuous proportional data |url=https://pmc.ncbi.nlm.nih.gov/articles/PMC13188389/ |journal=Journal of the American Statistical Association |doi=10.1080/01621459.2026.2626081 |issn=0162-1459 |pmc=13188389 |pmid=42169758}}</ref>. | ||
The special case <math>\lambda=1</math> coincides with the [[continuous Bernoulli distribution]]<ref>{{Cite journal |last=Loaiza-Ganem |first=Gabriel |last2=Cunningham |first2=John |date=2019 |title=The continuous Bernoulli: fixing a pervasive error in variational autoencoders |url=https://proceedings.neurips.cc/paper/2019/hash/f82798ec8909d23e55679ee26bb26437-Abstract.html |journal=Advances in Neural Information Processing Systems |publisher=Curran Associates, Inc. |volume=32}}</ref>. | |||
==Definition== | ==Definition== | ||
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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>. | 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>. | When <math>\lambda</math> is fixed, the cobin distribution belongs to a one-parameter [[natural exponential family]] in <math>\theta</math>. | ||
==Earlier studies== | |||
The corresponding two-parameter distributional family appeared in earlier statistical literature<ref name="lee2026" /> without the name ''continuous binomial distribution''. For example, Bates (1955)<ref>{{cite journal |last=Bates |first=Grace E. |year=1955 |title=Joint Distributions of Time Intervals for the Occurrence of Successive Accidents in a Generalized Polya Scheme |journal=The Annals of Mathematical Statistics |volume=26 |issue=4 |pages=705–720 |doi=10.1214/aoms/1177728429}}</ref> studied the same distribution in the context of modeling time intervals between successive accidents and hypothesis testing against uniformity. | |||
==Related distributions== | ==Related distributions== | ||
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The cobin distribution has been proposed as a response distribution for [[generalized linear model]]s of continuous proportional data, as an alternative to beta regression, including extensions with random effects. | The cobin distribution has been proposed as a response distribution for [[generalized linear model]]s of continuous proportional data, as an alternative to beta regression, including extensions with random effects. | ||
==References== | |||
{{reflist}} | {{reflist}} | ||
Latest revision as of 23:13, 7 November 2026
Template:Infobox probability 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. The name continuous binomial (cobin) was adopted in 2026 in developing generalized linear models for continuous proportional data, as an alternative to beta regression [1]. The special case <math>\lambda=1</math> coincides with the continuous Bernoulli distribution[2].
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>.
Earlier studies
The corresponding two-parameter distributional family appeared in earlier statistical literature[1] without the name continuous binomial distribution. For example, Bates (1955)[3] studied the same distribution in the context of modeling time intervals between successive accidents and hypothesis testing against uniformity.
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
- <math>
\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.
References
- ↑ 1.0 1.1 Lee, Changwoo J.; Dahl, Benjamin K.; Ovaskainen, Otso; Dunson, David B. (2026-05-18). "Scalable and robust regression models for continuous proportional data". Journal of the American Statistical Association. doi:. ISSN 0162-1459. PMID 42169758. PMC: 13188389. https://pmc.ncbi.nlm.nih.gov/articles/PMC13188389/.
- ↑ Loaiza-Ganem, Gabriel; Cunningham, John (2019). "The continuous Bernoulli: fixing a pervasive error in variational autoencoders". Advances in Neural Information Processing Systems (Curran Associates, Inc.) 32. https://proceedings.neurips.cc/paper/2019/hash/f82798ec8909d23e55679ee26bb26437-Abstract.html.
- ↑ Bates, Grace E. (1955). "Joint Distributions of Time Intervals for the Occurrence of Successive Accidents in a Generalized Polya Scheme". The Annals of Mathematical Statistics 26 (4): 705–720. doi:.