Dirichlet negative multinomial 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 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 several 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 is the Dirichlet distribution.

Motivation

Dirichlet negative multinomial as a compound distribution

The Dirichlet distribution is a conjugate distribution to the negative multinomial distribution. This fact leads to an analytically tractable compound distribution.
For a random vector of category counts, distributed according to a negative multinomial distribution, the compound distribution is obtained by integrating on the distribution for p which can be thought of as a random vector following a Dirichlet distribution:
which results in the following formula:
where and are the dimensional vectors created by appending the scalars and to the dimensional vectors and respectively and is the multivariate version of the beta function. We can write this equation explicitly as
Alternative formulations exist. One convenient representation is
where and.
This can also be written

Properties

Marginal distributions

To obtain the marginal distribution over a subset of Dirichlet negative multinomial random variables, one only needs to drop the irrelevant 's from the vector. The joint distribution of the remaining random variates is where is the vector with the removed 's.

Conditional distributions

If m-dimensional x is partitioned as follows
and accordingly
then the conditional distribution of on is where
and
That is,

Conditional on the sum

The conditional distribution of a Dirichlet negative multinomial distribution on is Dirichlet-multinomial distribution with parameters and. That is
Notice that the equation does not depend on or.

Correlation matrix

For the entries of the correlation matrix are

Heavy tailed

The Dirichlet negative multinomial is a heavy tailed distribution. It does not have a finite mean for and it has infinite covariance matrix for. It therefore has undefined moment generating function.

Aggregation

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

Applications

Dirichlet negative multinomial as a urn model

The Dirichlet negative multinomial can also be motivated by an urn model in the case when is a positive integer. Consider a sequence of independent and identically distributed multinomial trials, each of which has outcomes. Call one of the outcomes a “success”, and suppose it has probability. The other outcomes – called "failures" - have probabilities. If the vector counts the m types of failures before the success is observed, then the have negative mulitnomial distribution with parameters.
If the parameters are themselves sampled from a Dirichlet distribution with parameters, then the resulting distribution of is Dirichlet negative multinomial. The resultant distribution has parameters.