Negative multinomial 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)}}. ==Parameter estimation==

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 ==References==