Dirichlet_negative_multinomial_distribution
Dirichlet negative multinomial distribution
Probability multivariate distribution
In probability theory and statistics, the Dirichlet negative multinomial distribution is a multivariate distribution on the non-negative integers. It is a multivariate extension of the beta negative binomial distribution. It is also a generalization of the negative multinomial distribution (NM(k, p)) allowing for heterogeneity or overdispersion to the probability vector. It is used in quantitative marketing research to flexibly model the number of household transactions across multiple brands.
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If parameters of the Dirichlet distribution are , and if
where
then the marginal distribution of X is a Dirichlet negative multinomial distribution:
In the above, is the negative multinomial distribution and is the Dirichlet distribution.