In probability theory and statistics, the **negative multinomial distribution** is a generalization of the negative binomial distribution (NB(*r*, *p*)) to more than two outcomes.^{[1]}

Suppose we have an experiment that generates *m*+1≥2 possible outcomes, {*X*_{0},...,*X*_{m}}, each occurring with non-negative probabilities {*p*_{0},...,*p*_{m}} respectively. If sampling proceeded until *n* observations were made, then {*X*_{0},...,*X*_{m}} would have been multinomially distributed. However, if the experiment is stopped once *X*_{0} reaches the predetermined value *x*_{0}, then the distribution of the *m*-tuple {*X*_{1},...,*X*_{m}} is *negative multinomial*. These variables are not multinomially distributed because their sum *X*_{1}+...+*X*_{m} is not fixed, being a draw from a negative binomial distribution.

## Properties

### Marginal distributions

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 *p'*s properly scaled so as to add to one.

The univariate marginal is the negative binomial distribution.

### Independent sums

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.

### Aggregation

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.

### Correlation matrix

The entries of the correlation matrix are

## Parameter estimation

### Method of Moments

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

## Related distributions

- Negative binomial distribution
- Multinomial distribution
- Inverted Dirichlet distribution, a conjugate prior for the negative multinomial
- Dirichlet negative multinomial distribution

## References

**^**Le Gall, F. The modes of a negative multinomial distribution, Statistics & Probability Letters, Volume 76, Issue 6, 15 March 2006, Pages 619-624, ISSN 0167-7152, 10.1016/j.spl.2005.09.009.

Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi- nomial distribution. Biometrics 53: 971-82.

## Further reading

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1997). "Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions". *Discrete Multivariate Distributions*. Wiley. ISBN 978-0-471-12844-1.