In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x<sub>0</sub>, p)) to more than two outcomes.
As with the univariate negative binomial distribution, if the parameter is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates m+1âÂÂ¥2 possible outcomes, {X<sub>0</sub>,...,X<sub>m</sub>}, each occurring with non-negative probabilities {p<sub>0</sub>,...,p<sub>m</sub>} respectively. If sampling proceeded until n observations were made, then {X<sub>0</sub>,...,X<sub>m</sub>} would have been multinomially distributed. However, if the experiment is stopped once X<sub>0</sub> reaches the predetermined value x<sub>0</sub> (assuming x<sub>0</sub> is a positive integer), then the distribution of the m-tuple {X<sub>1</sub>,...,X<sub>m</sub>} is negative multinomial. These variables are not multinomially distributed because their sum X<sub>1</sub>+...+X<sub>m</sub> is not fixed, being a draw from a negative binomial distribution.
If m-dimensional x is partitioned as follows
and accordingly
and let
The marginal distribution of is . That is the marginal distribution is also negative multinomial with the removed and the remaining ps properly scaled so as to add to one.
The univariate marginal is said to have a negative binomial distribution.
The conditional distribution of given is . That is,
If and If are independent, then . Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.
If
then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum,
This aggregation property may be used to derive the marginal distribution of mentioned above.
The entries of the correlation matrix are
If we let the mean vector of the negative multinomial be
then it is easy to show through properties of determinants that . From this, it can be shown that
and
Substituting sample moments yields the method of moments estimates
and
Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi- nomial distribution. Biometrics 53: 971âÂÂ82.