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.

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

This fact leads to an analytically tractable compound distribution.

For a random vector of category counts

dimensional vectors created by appending the scalars

is the multivariate version of the beta function.

We can write this equation explicitly as Alternative formulations exist.

This can also be written To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the irrelevant

The joint distribution of the remaining random variates is

The univariate marginals are said to be beta negative binomially distributed.

where and That is, The conditional distribution of a Dirichlet negative multinomial distribution on

is Dirichlet-multinomial distribution with parameters

That is Notice that the expression does not depend on

If then, if the random variables with positive subscripts i and j are dropped from the vector and replaced by their sum,

the entries of the correlation matrix are The Dirichlet negative multinomial is a heavy tailed distribution.

and it has infinite covariance matrix for

Therefore the moment generating function does not exist.

are positive integers the Dirichlet negative multinomial can also be motivated by an urn model - or more specifically a basic Pólya urn model.

red balls (the stopping color).

gives the respective counts of the other balls of various

At each step of the model, a ball is drawn at random from the urn and replaced, along with one additional ball of the same color.

red colored balls are drawn.

non-red colors are distributed according to a

Note, at the end of the experiment, the urn always contains the fixed number

of red balls while containing the random number