# Negative multinomial distribution

Notation ${\textrm {NM}}(x_{0},\,\mathbf {p} )$ $x_{0}>0$ — the number of failures before the experiment is stopped, $\mathbf {p}$ ∈ Rm — m-vector of "success" probabilities,p0 = 1 − (p1+…+pm) — the probability of a "failure". $x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m$ $\Gamma \!\left(\sum _{i=0}^{m}{x_{i}}\right){\frac {p_{0}^{x_{0}}}{\Gamma (x_{0})}}\prod _{i=1}^{m}{\frac {p_{i}^{x_{i}}}{x_{i}!}},$ where Γ(x) is the Gamma function. ${\tfrac {x_{0}}{p_{0}}}\,\mathbf {p}$ ${\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {pp} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )$ ${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{t_{j}}}}{\bigg )}^{\!x_{0}}$ ${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{it_{j}}}}{\bigg )}^{\!x_{0}}$ 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,...,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.

## Properties

### 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 ${\textstyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2)}},\mathbf {p} ^{(1)})}$ . 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 ${\textstyle |{\boldsymbol {\Sigma }}|={\frac {1}{p_{0}}}\prod _{i=1}^{m}{\mu _{i}}}$ . 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
${\hat {\mathbf {p} }}=\left({\frac {|{\boldsymbol {S}}|-\prod _{i=1}^{m}{{\bar {x}}_{i}}}{|{\boldsymbol {S}}|\sum _{i=1}^{m}{{\bar {x}}_{i}}}}\right){\boldsymbol {\bar {x}}}$ 