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)