Negative Multinomial Distribution

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(''x''₀, ''p'')) to more than two outcomes.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

As with the univariate negative binomial distribution, if the parameter $x_0$ is a positive integer, the negative multinomial distribution has an urn model interpretation. Suppose we have an experiment that generates ''m''+1≥2 possible outcomes, , each occurring with non-negative probabilities respectively. If sampling proceeded until ''n'' observations were made, then would have been multinomially distributed. However, if the experiment is stopped once ''X''₀ reaches the predetermined value ''x''₀ (assuming ''x''₀ is a positive integer), then the distribution of the ''m''-tuple 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
:$\mathbf = \begin \mathbf^ \\ \mathbf^ \end \text\begin n \times 1 \\ (m-n) \times 1 \end$
and accordingly $\boldsymbol$
:$\boldsymbol p = \begin \boldsymbol p^ \\ \boldsymbol p^ \end \text\begin n \times 1 \\ (m-n) \times 1 \end$

and let

:$q = 1-\sum_i p_i^=p_0+\sum_i p_i^$

The marginal distribution of $\boldsymbol X^$ is $\mathrm(x_0,p_0/q, \boldsymbol p^/q )$. That is the marginal distribution is also negative multinomial with the $\boldsymbol p^$ removed and the remaining ''ps properly scaled so as to add to one.

The univariate marginal $m=1$ is said to have a negative binomial distribution.

Conditional distributions

The conditional distribution of $\mathbf^$ given $\mathbf^=\mathbf^$ is $\mathrm(x_0+\sum,\mathbf^)$. That is,

:$\Pr(\mathbf^\mid \mathbf^, x_0, \mathbf )= \Gamma\!\left(\sum_^m\right)\frac\prod_^n.$

Independent sums

If $\mathbf_1 \sim \mathrm(r_1, \mathbf)$ and If $\mathbf_2 \sim \mathrm(r_2, \mathbf)$ are independent, then
$\mathbf_1+\mathbf_2 \sim \mathrm(r_1+r_2, \mathbf)$. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.

Aggregation

If

:$\mathbf = (X_1, \ldots, X_m)\sim\operatorname(x_0, (p_1,\ldots,p_m))$

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

:$\mathbf' = (X_1, \ldots, X_i + X_j, \ldots, X_m)\sim\operatorname (x_0, (p_1, \ldots, p_i + p_j, \ldots, p_m)).$

This aggregation property may be used to derive the marginal distribution of $X_i$ mentioned above.

Correlation matrix

The entries of the correlation matrix are

:$\rho(X_i,X_i) = 1.$

:$\rho(X_i,X_j) = \frac = \sqrt.$

Parameter estimation

Method of Moments

If we let the mean vector of the negative multinomial be

$\boldsymbol=\frac\mathbf$

and covariance matrix

$\boldsymbol=\tfrac\,\mathbf\mathbf' + \tfrac\,\operatorname(\mathbf)$,

then it is easy to show through properties of determinants that
$, \boldsymbol, =\frac\prod_^m$. From this, it can be shown that

:$x_0=\frac$

and
:$\mathbf= \frac\boldsymbol.$

Substituting sample moments yields the method of moments estimates
:$\hat_0=\frac$

and
:$\hat=\left(\frac\right)\boldsymbol$

Related distributions

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

References

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

Further reading

{{DEFAULTSORT:Negative Multinomial Distribution
Factorial and binomial topics
Multivariate discrete distributions