A weighted likelihood ratio of two related negative hypergeometric distributions

In this paper we consider some related negative hypergeometric distributions arising from the problem of sampling without replacement from an urn containing balls of different colours and in different proportions but stopping only after some specific number of balls of different colours have been obtained. With the aid of some simple recurrence relations and identities we obtain in the case of two colours the moments for the maximum negative hypergeometric distribution, the minimum negative hypergeometric distribution, the likelihood ratio negative hypergeometric distribution and consequently the likelihood proportional negative hypergeometric distribution. To the extent that the sampling scheme is applicable to modelling data as illustrated with a biological example and, in fact, many situations of estimating Bernoulli parameters for binary traits within a finite population, these are important first-step results.