Negative multinomial distribution


In probability theory and statistics, the negative multinomial distribution is a generalization of the negative binomial distribution to more than two outcomes.
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, then the distribution of the m-tuple 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

Marginal distributions

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

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

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

The entries of the correlation matrix are

Parameter estimation

Method of Moments

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

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