Negative multinomial distribution

Notation	$\text{NM}(x_0,𝐩)$
Parameters	$x_0>0$ — the number of failures before the experiment is stopped, $𝐩$ ∈ Rm — m-vector of "success" probabilities, p0 = 1 − (p1+…+pm) — the probability of a "failure".
Support	$x_i\in \{0,1,2,\dots \},1\le i\le m$
PMF	$\Gamma \left(\sum _{i=0}^m{x}_i\right)\frac{p_0^{x_0}}{\Gamma (x_0)}\prod _{i=1}^m\frac{p_i^{x_i}}{x_i!},$ where Γ(x) is the Gamma function.
Mean	$\frac{x_0}{p_0}𝐩$
Variance	$\frac{x_0}{p_0^2}{𝐩𝐩}^{\prime }+\frac{x_0}{p_0}\mathrm{diag}(𝐩)$
MGF	$(\frac{p_0}{1-\sum _{j=1}^m{p}_j{e}^{t_j}}{)}^{x_0}$
CF	$(\frac{p_0}{1-\sum _{j=1}^m{p}_j{e}^{i{t}_j}}{)}^{x_0}$

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x₀, p)) to more than two outcomes.

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, {X₀,...,X_m}, each occurring with non-negative probabilities {p₀,...,p_m} respectively. If sampling proceeded until n observations were made, then {X₀,...,X_m} would have been multinomially distributed. However, if the experiment is stopped once X₀ reaches the predetermined value x₀ (assuming x₀ is a positive integer), then the distribution of the m-tuple {X₁,...,X_m} is negative multinomial. These variables are not multinomially distributed because their sum X₁+...+X_m is not fixed, being a draw from a negative binomial distribution.

Properties

Marginal distributions

If m-dimensional x is partitioned as follows $$𝐗=\left[\begin{array}{c}{𝐗}^{(1)}\\ {𝐗}^{(2)}\end{array}\right]\text{ with sizes }\left[\begin{array}{c}n\times 1\\ (m-n)\times 1\end{array}\right]$$ and accordingly $𝒑$ $$𝒑=\left[\begin{array}{c}{𝒑}^{(1)}\\ {𝒑}^{(2)}\end{array}\right]\text{ with sizes }\left[\begin{array}{c}n\times 1\\ (m-n)\times 1\end{array}\right]$$ and let $$q=1-\sum _i{p}_i^{(2)}=p_0+\sum _i{p}_i^{(1)}$$

The marginal distribution of ${𝑿}^{(1)}$ is $\mathrm{N}\mathrm{M}(x_0,p_0/q,{𝒑}^{(1)}/q)$. That is the marginal distribution is also negative multinomial with the ${𝒑}^{(2)}$ removed and the remaining p's 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 ${𝐗}^{(1)}$ given ${𝐗}^{(2)}={𝐱}^{(2)}$ is $\mathrm{N}\mathrm{M}(x_0+\sum x_i^{(2)},{𝐩}^{(1)})$. That is, $$\mathrm{Pr}({𝐱}^{(1)}\mid {𝐱}^{(2)},x_0,𝐩)=\Gamma \left(\sum _{i=0}^m{x}_i\right)\frac{(1-\sum _{i=1}^n{p}_i^{(1)}{)}^{x_0+\sum _{i=1}^{m-n}{x}_i^{(2)}}}{\Gamma (x_0+\sum _{i=1}^{m-n}{x}_i^{(2)})}\prod _{i=1}^n\frac{(p_i^{(1)}{)}^{x_i}}{(x_i^{(1)})!}.$$

Independent sums

If ${𝐗}_1\sim \mathrm{N}\mathrm{M}(r_1,𝐩)$ and If ${𝐗}_2\sim \mathrm{N}\mathrm{M}(r_2,𝐩)$ are independent, then ${𝐗}_1+{𝐗}_2\sim \mathrm{N}\mathrm{M}(r_1+r_2,𝐩)$. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.

Aggregation

If $$𝐗=(X_1,\dots ,X_m)\sim \mathrm{NM}(x_0,(p_1,\dots ,p_m))$$ then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum, $${𝐗}^{\prime }=(X_1,\dots ,X_i+X_j,\dots ,X_m)\sim \mathrm{NM}(x_0,(p_1,\dots ,p_i+p_j,\dots ,p_m)).$$

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 $$\rho (X_i,X_i)=1.$$ $$\rho (X_i,X_j)=\frac{\mathrm{cov}(X_i,X_j)}{\sqrt{\mathrm{var}(X_i)\mathrm{var}(X_j)}}=\sqrt{\frac{p_i{p}_j}{(p_0+p_i)(p_0+p_j)}}.$$

Parameter estimation

Method of Moments

If we let the mean vector of the negative multinomial be $$𝝁=\frac{x_0}{p_0}𝐩$$ and covariance matrix $$𝜮=\frac{x_0}{p_0^2}𝐩{𝐩}^{\prime }+\frac{x_0}{p_0}\mathrm{diag}(𝐩),$$ then it is easy to show through properties of determinants that $|𝜮|=\frac{1}{p_0}{\prod }_{i=1}^m{\mu }_i$. From this, it can be shown that $$x_0=\frac{\sum {\mu }_i\prod {\mu }_i}{|𝜮|-\prod {\mu }_i}$$ and $$𝐩=\frac{|𝜮|-\prod {\mu }_i}{|𝜮|\sum {\mu }_i}𝝁.$$

Substituting sample moments yields the method of moments estimates $${\hat{x}}_0=\frac{(\sum _{i=1}^m\overline{x_i})\prod _{i=1}^m\overline{x_i}}{|𝐒|-\prod _{i=1}^m\overline{x_i}}$$ and $$\widehat{𝐩}=\left(\frac{|𝑺|-\prod _{i=1}^m{\overline{x}}_i}{|𝑺|\sum _{i=1}^m{\overline{x}}_i}\right)\overline{𝒙}$$

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.