Negative polynomial distribution

negative multinomial distribution

The joint probability distribution (cf. also Joint distribution) of random variables that take non-negative integer values defined by the formula

(*)

where and (, ; ) are parameters. A negative multinomial distribution is a multi-dimensional discrete distribution — a distribution of a random vector with non-negative integer components.

The generating function of the negative polynomial distribution with parameters has the form

A negative multinomial distribution arises in the following multinomial scheme. Successive independent trials are carried out, and in each trial different outcomes with labels are possible, having probabilities , respectively. The trials continue up to the -th appearance of the outcome with label 0 (here is an integer). If is the number of appearances of the outcome with label , , during the trials, then formula (*) expresses the probability of the appearance of outcomes with labels , equal, respectively, times, up to the -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 .

If a random vector has, conditionally on , a multinomial distribution with parameters , and if the parameter is itself a random variable having a negative binomial distribution with parameters , , then the marginal distribution of the vector , given the condition , is the negative multinomial distribution with parameters , .

References

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