Negative polynomial distribution

negative multinomial distribution

The joint probability distribution (cf. also Joint distribution) of random variables $X _ {1} \dots X _ {k}$ that take non-negative integer values $m = 0, 1, \dots$ defined by the formula

$$\tag{* } {\mathsf P} \{ X _ {1} = m _ {1}, \dots, X _ {k} = m _ {k} \} =$$

$$= \ \frac{\Gamma ( r+ m _ {1} + \dots + m _ {k} ) }{\Gamma ( r) m _ {1} ! \cdots m _ {k} ! } p _ {0} ^ {r} p _ {1} ^ {m _ {1} } \cdots p _ {k} ^ {m _ {k} } ,$$

where $r > 0$ and $p _ {0} \dots p _ {k}$( $0 < p _ {i} < 1$, $i = 0, \dots, k$; $p _ {0} + \dots + p _ {k} = 1$) are parameters. A negative multinomial distribution is a multi-dimensional discrete distribution — a distribution of a random vector $( X _ {1}, \dots, X _ {k} )$ with non-negative integer components.

The generating function of the negative polynomial distribution with parameters $r, p _ {0} \dots p _ {k}$ has the form

$$P( z _ {1} \dots z _ {k} ) = p _ {0} ^ {r} \left ( 1 - \sum _ { i= 1} ^ { k } z _ {i} p _ {i} \right ) ^ {-r} .$$

A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial $k+ 1$ different outcomes with labels $0 \dots k$ are possible, having probabilities $p _ {0} \dots p _ {k}$, respectively. The trials continue up to the $r$-th appearance of the outcome with label 0 (here $r$ is an integer). If $X _ {i}$ is the number of appearances of the outcome with label $i$, $i = 1 \dots k$, during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels $1, \dots, k$, equal, respectively, $m _ {1} \dots m _ {k}$ times, up to the $r$-th appearance of the outcome 0. A negative multinomial distribution in this sense is a generalization of a negative binomial distribution, coinciding with it when $k= 1$.

If a random vector $( X _ {0} \dots X _ {k} )$ has, conditionally on $n$, a multinomial distribution with parameters $n > 1$, $p _ {0} \dots p _ {k}$ and if the parameter $n$ is itself a random variable having a negative binomial distribution with parameters $r > 0$, $0 < \pi < 1$, then the marginal distribution of the vector $( X _ {1} \dots X _ {k} )$, given the condition $X _ {0} = r$, is the negative multinomial distribution with parameters $r$, $p _ {0} ( 1- \pi ) \dots p _ {k} ( 1- \pi )$.

References

[a1]	J. Neyman, "Proceedings of the international symposium on discrete distributions" , Montreal (1963)