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]]. It was introduced as a response distribution for [[generalized linear model]]s for continuous proportional data, proposed as an alternative to [[beta regression]].{{ | 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]]. It was introduced as a response distribution for [[generalized linear model]]s for continuous proportional data, proposed as an alternative to [[beta regression]].<ref>{{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== | ||
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 | 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> | :<math> | ||
f(x;\theta,\lambda)=h(x;\lambda)\exp\!\big(\lambda\theta x-\lambda B(\theta)\big),\qquad 0\le x\le 1, | f(x;\theta,\lambda)=h(x;\lambda)\exp\!\big(\lambda\theta x-\lambda B(\theta)\big),\qquad 0\le x\le 1, | ||
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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>. | ||
==Related distributions== | ==Related distributions== | ||
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\bar{X}=\frac{1}{\lambda}\sum_{i=1}^{\lambda} X_i | \bar{X}=\frac{1}{\lambda}\sum_{i=1}^{\lambda} X_i | ||
\sim \mathrm{cobin}(\theta,\lambda^{-1}). | \sim \mathrm{cobin}(\theta,\lambda^{-1}). | ||
</math> | </math> | ||
==Properties== | ==Properties== | ||
===Mean and variance=== | ===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>: | 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{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>. | * <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>. | 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=== | ===Sufficient statistic for the mean=== | ||
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==Applications== | ==Applications== | ||
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== | ==References== | ||