Notation
Parameters
Support
PMF
where , and Γ(x) is the Gamma function and B is the beta function.
Mean for
Variance for
MGF does not exist
CF
where is the Lauricella function

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 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[1] 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 (the variables that one wants to marginalize out) from the vector. The joint distribution of the remaining random variates is where is the vector with the removed 's. The univariate marginals are said to be beta negative binomially distributed.

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 expression does not depend on or .

Aggregation

If

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


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 . Therefore the moment generating function does not exist.

Applications

Dirichlet negative multinomial as a Pólya urn model

In the case when the parameters and are positive integers the Dirichlet negative multinomial can also be motivated by an urn model - or more specifically a basic Pólya urn model. Consider an urn initially containing balls of various colors including red balls (the stopping color). The vector gives the respective counts of the other balls of various non-red colors. 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. The process is repeated over and over, until red colored balls are drawn. The random vector of observed draws of the other 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 of the other colors.

See also

References

  1. Farewell, Daniel & Farewell, Vernon. (2012). Dirichlet negative multinomial regression for overdispersed correlated count data. Biostatistics (Oxford, England). 14. 10.1093/biostatistics/kxs050.
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