Multivariate geometric distributions

Families of multivariate geometric distributions with flexible correlations can be constructed by applying inverse sampling to a sequence of multinomial trials, and counting outcomes in possibly overlapping categories. Further multivariate families can be obtained by considering other stopping rules, with the possibility of different stopping roles for different counts, A simple characterisation is given for stopping rules which produce joint distributions with marginals having the same form as that of the number of trials. The inverse sampling approach provides a unified treatment of diverse results presented by earlier authors, including Goldberg (1934), Bates and Meyman (1952), Edwards and Gurland (1961), Hawkes (1972), Paulson and Uppulori (1972) and Griffiths and Milne (1987). It also provides a basis for investigating the range of possible correlations for a given set of marginal parameters. In the case of more than two joint geometric or negative binomial variables, a convenient matrix formulation is provided.