Negative multinomial distribution

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.

Notation	${\textrm {NM}}(x_{0},\,\mathbf {p} )$
Parameters	$x_{0}>0$ — 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".
Support	$x_{i}\in \{0,1,2,\ldots \},1\leq i\leq m$
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	${\tfrac {x_{0}}{p_{0}}}\,\mathbf {p}$
Variance	${\tfrac {x_{0}}{p_{0}^{2}}}\,\mathbf {pp} '+{\tfrac {x_{0}}{p_{0}}}\,\operatorname {diag} (\mathbf {p} )$
MGF	${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{t_{j}}}}{\bigg )}^{\!x_{0}}$
CF	${\bigg (}{\frac {p_{0}}{1-\sum _{j=1}^{m}p_{j}e^{it_{j}}}}{\bigg )}^{\!x_{0}}$

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 $$\mathbf {X} ={\begin{bmatrix}\mathbf {X} ^{(1)}\\\mathbf {X} ^{(2)}\end{bmatrix}}{\text{ with sizes }}{\begin{bmatrix}n\times 1\\(m-n)\times 1\end{bmatrix}}$$ 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 $${\hat {\mathbf {p} }}=\left({\frac {|{\boldsymbol {S}}|-\prod _{i=1}^{m}{{\bar {x}}_{i}}}{|{\boldsymbol {S}}|\sum _{i=1}^{m}{{\bar {x}}_{i}}}}\right){\boldsymbol {\bar {x}}}$$

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.

Further reading

Johnson, Norman L.; Kotz, Samuel; Balakrishnan, N. (1997). "Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions". Discrete Multivariate Distributions. Wiley. ISBN 978-0-471-12844-1.