Negative multinomial distribution

In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution (NB(x0,&thinsp;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,...,Xm}, each occurring with non-negative probabilities {p0,...,pm} respectively. If sampling proceeded until n observations were made, then {X0,...,Xm} 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,...,Xm} is negative multinomial. These variables are not multinomially distributed because their sum X1+...+Xm is not fixed, being a draw from a negative binomial distribution.

Marginal distributions
If m-dimensional x is partitioned as follows $$ \mathbf{X} = \begin{bmatrix} \mathbf{X}^{(1)} \\ \mathbf{X}^{(2)} \end{bmatrix}

\text{ with sizes }\begin{bmatrix} n \times 1 \\ (m-n) \times 1 \end{bmatrix}$$ and accordingly $$\boldsymbol{p}$$ $$ \boldsymbol p = \begin{bmatrix} \boldsymbol p^{(1)} \\ \boldsymbol p^{(2)} \end{bmatrix} \text{ with sizes }\begin{bmatrix} n \times 1 \\ (m-n) \times 1 \end{bmatrix}$$ and let $$q = 1-\sum_i p_i^{(2)} = p_0+\sum_i p_i^{(1)}$$

The marginal distribution of $$\boldsymbol X^{(1)}$$ is $$\mathrm{NM}(x_0,p_0/q, \boldsymbol p^{(1)}/q )$$. That is the marginal distribution is also negative multinomial with the $$\boldsymbol p^{(2)}$$ 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 $$\mathbf{X}^{(1)}$$ given $$\mathbf{X}^{(2)}=\mathbf{x}^{(2)}$$ is $\mathrm{NM}(x_0+\sum{x_i^{(2)}},\mathbf{p}^{(1)}) $. That is, $$ \Pr(\mathbf{x}^{(1)}\mid \mathbf{x}^{(2)}, x_0, \mathbf{p} )= \Gamma\!\left(\sum_{i=0}^m{x_i}\right)\frac{(1-\sum_{i=1}^n{p_i^{(1)}})^{x_0+\sum_{i=1}^{m-n}x_i^{(2)}}}{\Gamma(x_0+\sum_{i=1}^{m-n}x_i^{(2)})}\prod_{i=1}^n{\frac{(p_i^{(1)})^{x_i}}{(x_i^{(1)})!}}. $$

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

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

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

Method of Moments
If we let the mean vector of the negative multinomial be $$\boldsymbol{\mu}=\frac{x_0}{p_0}\mathbf{p}$$ and covariance matrix $$\boldsymbol{\Sigma}=\tfrac{x_0}{p_0^2}\,\mathbf{p}\mathbf{p}' + \tfrac{x_0}{p_0}\,\operatorname{diag}(\mathbf{p}),$$ then it is easy to show through properties of determinants that $ |\boldsymbol{\Sigma}| = \frac{1}{p_0}\prod_{i=1}^m{\mu_i}$. From this, it can be shown that $$x_0=\frac{\sum{\mu_i}\prod{\mu_i}}{|\boldsymbol{\Sigma}|-\prod{\mu_i}}$$ and $$ \mathbf{p}= \frac{|\boldsymbol{\Sigma}|-\prod{\mu_i}}{|\boldsymbol{\Sigma}|\sum{\mu_i}}\boldsymbol{\mu}. $$

Substituting sample moments yields the method of moments estimates $$\hat{x}_0=\frac{(\sum_{i=1}^{m}{\bar{x_i})}\prod_{i=1}^{m}{\bar{x_i}}}{|\mathbf{S}|-\prod_{i=1}^{m}{\bar{x_i}}}$$ and $$ \hat{\mathbf{p}}=\left(\frac{|\boldsymbol{S}|-\prod_{i=1}^{m}{\bar{x}_i}}{|\boldsymbol{S}|\sum_{i=1}^{m}{\bar{x}_i}}\right)\boldsymbol{\bar{x}} $$

Related distributions

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