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

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.^[1]

As with the univariate negative binomial distribution, if the parameter ${\displaystyle x_{0))$ 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

${\displaystyle \mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix)){\text{ with sizes )){\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix))}$

and accordingly ${\displaystyle {\boldsymbol {p))}$

${\displaystyle {\boldsymbol {p))={\begin{bmatrix}{\boldsymbol {p))^{(1)}\\{\boldsymbol {p))^{(2)}\end{bmatrix)){\text{ with sizes )){\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix))}$

and let

${\displaystyle q=1-\sum _{i}p_{i}^{(2)}=p_{0}+\sum _{i}p_{i}^{(1)))$

The marginal distribution of ${\displaystyle {\boldsymbol {X))^{(1)))$ is ${\displaystyle \mathrm {NM} (x_{0},p_{0}/q,{\boldsymbol {p))^{(1)}/q)}$. That is the marginal distribution is also negative multinomial with the ${\displaystyle {\boldsymbol {p))^{(2)))$ removed and the remaining p's properly scaled so as to add to one.

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

Conditional distributions

The conditional distribution of ${\displaystyle \mathbf {X} ^{(1)))$ given ${\displaystyle \mathbf {X} ^{(2)}=\mathbf {x} ^{(2)))$ is ${\displaystyle \mathrm {NM} (x_{0}+\sum {x_{i}^{(2))),\mathbf {p} ^{(1)})}$. That is,

${\displaystyle \Pr(\mathbf {x} ^{(1)}\mid \mathbf {x} ^{(2)},x_{0},\mathbf {p} )=\Gamma \!\left(\sum _{i=0}^{m}{x_{i))\right){\frac {(1-\sum _{i=1}^{n}{p_{i}^{(1))))^{x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)))}{\Gamma (x_{0}+\sum _{i=1}^{m-n}x_{i}^{(2)})))\prod _{i=1}^{n}{\frac {(p_{i}^{(1)})^{x_{i))}{(x_{i}^{(1)})!)).}$

Independent sums

If ${\displaystyle \mathbf {X} _{1}\sim \mathrm {NM} (r_{1},\mathbf {p} )}$ and If ${\displaystyle \mathbf {X} _{2}\sim \mathrm {NM} (r_{2},\mathbf {p} )}$ are independent, then ${\displaystyle \mathbf {X} _{1}+\mathbf {X} _{2}\sim \mathrm {NM} (r_{1}+r_{2},\mathbf {p} )}$. Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.

Aggregation

If

${\displaystyle \mathbf {X} =(X_{1},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{m}))}$

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

${\displaystyle \mathbf {X} '=(X_{1},\ldots ,X_{i}+X_{j},\ldots ,X_{m})\sim \operatorname {NM} (x_{0},(p_{1},\ldots ,p_{i}+p_{j},\ldots ,p_{m})).}$

This aggregation property may be used to derive the marginal distribution of ${\displaystyle X_{i))$ mentioned above.

Correlation matrix

The entries of the correlation matrix are

${\displaystyle \rho (X_{i},X_{i})=1.}$

${\displaystyle \rho (X_{i},X_{j})={\frac {\operatorname {cov} (X_{i},X_{j})}{\sqrt {\operatorname {var} (X_{i})\operatorname {var} (X_{j})))}={\sqrt {\frac {p_{i}p_{j)){(p_{0}+p_{i})(p_{0}+p_{j})))}.}$

Parameter estimation

Method of Moments

If we let the mean vector of the negative multinomial be

${\displaystyle {\boldsymbol {\mu ))={\frac {x_{0)){p_{0))}\mathbf {p} }$

and covariance matrix

${\displaystyle {\boldsymbol {\Sigma ))={\tfrac {x_{0)){p_{0}^{2))}\,\mathbf {p} \mathbf {p} '+{\tfrac {x_{0)){p_{0))}\,\operatorname {diag} (\mathbf {p} )}$,

then it is easy to show through properties of determinants that ${\displaystyle |{\boldsymbol {\Sigma ))|={\frac {1}{p_{0))}\prod _{i=1}^{m}{\mu _{i))}$. From this, it can be shown that

${\displaystyle x_{0}={\frac {\sum {\mu _{i))\prod {\mu _{i))}{|{\boldsymbol {\Sigma ))|-\prod {\mu _{i))))}$

and

${\displaystyle \mathbf {p} ={\frac {|{\boldsymbol {\Sigma ))|-\prod {\mu _{i))}{|{\boldsymbol {\Sigma ))|\sum {\mu _{i)))){\boldsymbol {\mu )).}$

Substituting sample moments yields the method of moments estimates

${\displaystyle {\hat {x))_{0}={\frac {(\sum _{i=1}^{m}((\bar {x_{i))})}\prod _{i=1}^{m}{\bar {x_{i)))){|\mathbf {S} |-\prod _{i=1}^{m}{\bar {x_{i))))))$

and

${\displaystyle {\hat {\mathbf {p} ))=\left({\frac {|{\boldsymbol {S))|-\prod _{i=1}^{m}((\bar {x))_{i))}{|{\boldsymbol {S))|\sum _{i=1}^{m}((\bar {x))_{i))))\right){\boldsymbol {\bar {x))))$

Related distributions

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