Negative multinomial distribution

Notation	[math]\displaystyle{ \textrm{NM}(x_0,\,\mathbf{p}) }[/math]
Parameters	[math]\displaystyle{ x_0\gt 0 }[/math] — the number of failures before the experiment is stopped, [math]\displaystyle{ \mathbf{p} }[/math] ∈ Rm — m-vector of "success" probabilities, p0 = 1 − (p1+…+pm) — the probability of a "failure".
Support	[math]\displaystyle{ x_i \in \{0,1,2,\ldots\}, 1\leq i\leq m }[/math]
pmf	[math]\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!}}, }[/math] where Γ(x) is the Gamma function.
Mean	[math]\displaystyle{ \tfrac{x_0}{p_0}\,\mathbf{p} }[/math]
Variance	[math]\displaystyle{ \tfrac{x_0}{p_0^2}\,\mathbf{pp}' + \tfrac{x_0}{p_0}\,\operatorname{diag}(\mathbf{p}) }[/math]
MGF	[math]\displaystyle{ \bigg(\frac{p_0}{1 - \sum_{j=1}^m p_j e^{t_j}}\bigg)^{\!x_0} }[/math]
CF	[math]\displaystyle{ \bigg(\frac{p_0}{1 - \sum_{j=1}^m p_j e^{it_j}}\bigg)^{\!x_0} }[/math]

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 [math]\displaystyle{ x_0 }[/math] 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 [math]\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} }[/math] and accordingly [math]\displaystyle{ \boldsymbol{p} }[/math] [math]\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} }[/math] and let [math]\displaystyle{ q = 1-\sum_i p_i^{(2)} = p_0+\sum_i p_i^{(1)} }[/math]

The marginal distribution of [math]\displaystyle{ \boldsymbol X^{(1)} }[/math] is [math]\displaystyle{ \mathrm{NM}(x_0,p_0/q, \boldsymbol p^{(1)}/q ) }[/math]. That is the marginal distribution is also negative multinomial with the [math]\displaystyle{ \boldsymbol p^{(2)} }[/math] removed and the remaining p's properly scaled so as to add to one.

The univariate marginal [math]\displaystyle{ m=1 }[/math] is said to have a negative binomial distribution.

Conditional distributions

The conditional distribution of [math]\displaystyle{ \mathbf{X}^{(1)} }[/math] given [math]\displaystyle{ \mathbf{X}^{(2)}=\mathbf{x}^{(2)} }[/math] is [math]\displaystyle{ \mathrm{NM}(x_0+\sum{x_i^{(2)}},\mathbf{p}^{(1)}) }[/math]. That is, [math]\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)})!}}. }[/math]

Independent sums

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

Aggregation

If [math]\displaystyle{ \mathbf{X} = (X_1, \ldots, X_m)\sim\operatorname{NM}(x_0, (p_1,\ldots,p_m)) }[/math] then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum, [math]\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)). }[/math]

This aggregation property may be used to derive the marginal distribution of [math]\displaystyle{ X_i }[/math] mentioned above.

Correlation matrix

The entries of the correlation matrix are [math]\displaystyle{ \rho(X_i,X_i) = 1. }[/math] [math]\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)}}. }[/math]

Parameter estimation

Method of Moments

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

Substituting sample moments yields the method of moments estimates [math]\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}}} }[/math] and [math]\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}} }[/math]

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.