Negative multinomial distribution explained

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

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,, each occurring with non-negative probabilities respectively. If sampling proceeded until n observations were made, then 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 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$$\backslash mathbf=\backslash begin\; \backslash mathbf^\; \backslash \backslash \; \backslash mathbf^\backslash end$$

\text\begin n \times 1 \\ (m-n) \times 1 \endand accordingly

\boldsymbol{p} $$\backslash boldsymbol\; p=\backslash begin\; \backslash boldsymbol\; p^\; \backslash \backslash \; \backslash boldsymbol\; p^\backslash end\backslash text\backslash begin\; n\; \backslash times\; 1\; \backslash \backslash \; (m-n)\; \backslash times\; 1\; \backslash end$$and let$$q\; =\; 1-\backslash sum\_i\; p\_i^\; =\; p\_0+\backslash sum\_i\; p\_i^$$

The marginal distribution of

\boldsymbol X ^{(1)} is 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 ps 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

X ^{(1)} given X ^{(2)} = x ^{(2)} is $\backslash mathrm(x\_0+\backslash sum,\backslash mathbf^)$. That is,$$\backslash Pr(\backslash mathbf^\backslash mid\; \backslash mathbf^,\; x\_0,\; \backslash mathbf)=\; \backslash Gamma\backslash !\backslash left(\backslash sum\_^m\backslash right)\backslash frac\backslash prod\_^n.$$

Independent sums

If

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

Aggregation

If$$\backslash mathbf\; =\; (X\_1,\; \backslash ldots,\; X\_m)\backslash sim\backslash operatorname(x\_0,\; (p\_1,\backslash ldots,p\_m))$$then, if the random variables with subscripts i and j are dropped from the vector and replaced by their sum,$$\backslash mathbf\text{'}\; =\; (X\_1,\; \backslash ldots,\; X\_i\; +\; X\_j,\; \backslash ldots,\; X\_m)\backslash sim\backslash operatorname\; (x\_0,\; (p\_1,\; \backslash ldots,\; p\_i\; +\; p\_j,\; \backslash ldots,\; p\_m)).$$

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$$\backslash rho(X\_i,X\_i)\; =\; 1.$$$$\backslash rho(X\_i,X\_j)\; =\; \backslash frac\; =\; \backslash sqrt.$$

Parameter estimation

Method of Moments

If we let the mean vector of the negative multinomial be$$\backslash boldsymbol=\backslash frac\backslash mathbf$$and covariance matrix$$\backslash boldsymbol=\backslash tfrac\backslash ,\backslash mathbf\backslash mathbf\text{'}\; +\; \backslash tfrac\backslash ,\backslash operatorname(\backslash mathbf),$$then it is easy to show through properties of determinants that $|\backslash boldsymbol|\; =\; \backslash frac\backslash prod\_^m$. From this, it can be shown that$$x\_0=\backslash frac$$

-\prod

and$$\backslash mathbf=\; \backslash frac$$

-\prod

\sum

\boldsymbol.

Substituting sample moments yields the method of moments estimates$$\backslash hat\_0=\backslash frac$$

-\prod_^

and$$\backslash hat=\backslash left(\backslash frac$$

-\prod_^

\sum_^

\right)\boldsymbol

Related distributions

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

References

1. Le Gall, F. The modes of a negative multinomial distribution, Statistics & Probability Letters, Volume 76, Issue 6, 15 March 2006, Pages 619-624, ISSN 0167-7152, 10.1016/j.spl.2005.09.009.

Waller LA and Zelterman D. (1997). Log-linear modeling with the negative multi-nomial distribution. Biometrics 53: 971–82.

Further reading

Book: Johnson . Norman L. . Kotz . Samuel . Balakrishnan . N.. Discrete Multivariate Distributions . Chapter 36: Negative Multinomial and Other Multinomial-Related Distributions . 1997 . Wiley . 978-0-471-12844-1.