Negative multinomial distribution

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Notation
Parameters — the number of failures before the experiment is stopped,
pRmm-vector of "success" probabilities,

p0 = 1 − (p1+…+pm) — the probability of a "failure".
Support
PDF
where Γ(x) is the Gamma function.
Mean
Variance
MGF
CF

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 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[edit]

Marginal distributions[edit]

If m-dimensional x is partitioned as follows

and accordingly

and let

The marginal distribution of is . That is the marginal distribution is also negative multinomial with the removed and the remaining p's properly scaled so as to add to one.

The univariate marginal is the negative binomial distribution.

Independent sums[edit]

If and If are independent, then . Similarly and conversely, it is easy to see from the characteristic function that the negative multinomial is infinitely divisible.

Aggregation[edit]

If

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

This aggregation property may be used to derive the marginal distribution of mentioned above.

Correlation matrix[edit]

The entries of the correlation matrix are

Parameter estimation[edit]

Method of Moments[edit]

If we let the mean vector of the negative multinomial be

and covariance matrix

,

then it is easy to show through properties of determinants that . From this, it can be shown that

and

Substituting sample moments yields the method of moments estimates

and

Related distributions[edit]

References[edit]

  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[edit]

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.