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Why is the negbin distribution required when the analyzed count data is bounded? I don't really understand the following:

"The Poisson distribution can form the basis for some analyses of count data and in this case Poisson regression may be used. This is a special case of the class of generalized linear models which also contains specific forms of model capable of using the binomial distribution (binomial regression, logistic regression) or the negative binomial distribution where the assumptions of the Poisson model are violated, in particular when the range of count values is limited or when overdispersion is present." --- Wikipedia

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    $\begingroup$ Neither "limited" nor "overdispersion" imply "bounded." The Wikipedia article on overdispersion points out that the negative binomial is a gamma mixture of Poissons, allowing it to handle overdispersion. The article you cite is in a nascent phase; it uses "limited" only in a vague colloquial way that one hopes will be improved as the article evolves. $\endgroup$
    – whuber ♦
    Jun 1, 2011 at 19:27

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In a Poisson distribution, the variance is equal to the mean. The negative binomial distribution has a variance that is greater than the mean by some factor -- hence it's "overdispersed" relative to the Poisson.

In marketing theory (see Ehrenberg's Repeat Buying), purchases by a given individual have a Poisson distribution, with individual lambdas. But since your lambda and my lambda are different values, the overall variance is higher. In a negative binomial, the lambdas are assumed to follow a gamma distribution.

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  • $\begingroup$ Hey! Could you give me the page where it's been said, that the poisson-gamma mixture model include that there are different means, or lambdas, for the individuals? Thank you for your answer, it sounds interesting! $\endgroup$
    – MarkDollar
    Jul 3, 2011 at 11:43
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    $\begingroup$ Schmittlein et al. is one essec.edu/faculty/… (see first paragraph). This will lead back eventually to Ehrenberg, Repeat Buying, section 7.2 (compound Poisson model). marketplanet.ru/filestore/0041/0037/939/repeat_buying.pdf $\endgroup$
    – zbicyclist
    Jul 5, 2011 at 20:21

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