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>{{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 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. Peritz and Shlom (1969)<ref>{{cite journal |last=Peritz |first=E. |last2=Shlom |first2=E. |year=1969 |title=Testing for Linear Contagion—Inverse Sampling |journal=Journal of the Royal Statistical Society, Series B (Methodological) |volume=31 |issue=2 |pages=388–395 |doi=10.1111/j.2517-6161.1969.tb00799.x}}</ref> derived an equivalent distribution in developing statistical tests based on accident occurrence times. | |||
==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}} | ||