We are interested in the SPSS implementation of bivariate Poisson distributions, which allow the modelling of (negative and positive) correlated marginal Poisson distributions. Two different constructions of the bivariate distribution are of special interest:
"trivariate reduction"-type:
the bivariate distribution is constructed via three independet random variables (RV) Z1~Pois(lambda1), Z2~Pois(lambda2) and Z3~Pois(lambda3). Then the RV X=Z1+Z3 and Y=Z2+Z3 follow jointly a bivariate Poisson distribution BivPoiss(lambda1,lambda2,lambda3).
the theoretical foundations are described in very detail in
Johnson, N., Kotz, S. and Balakrishnan, N. (1997) Discrete Multivariate Distributions. New York: Wiley. (pages126ff)
the implemtation in the open source statistical software R is illustrated in
Karlis, D et al, JOURNAL OF STATISTICAL SOFTWARE 14, 10 (2005) (
http://www.jstatsoft.org/v14/a10/)
"multiplicative"-type:
the bivariate dristribution is constructed from marginals with a multiplicative factor, leading to BivPois(lambda1,lambda2, alpha) where alpha describes the multiplicative coupling.
the theoretical foundations are give in
J. Lakshminarayana, S.N.N. Pandit & K. Srinivasa Rao (1999): On a bivariate poisson distribution, Communications in Statistics - Theory and Methods, 28:2, 267-276 (
http://www.tandfonline.com/doi/abs/10.1080/03610929908832297#tabModule)
a very recent publication extends this model to take into account possible over- and underdispersion in the marginal distributions, which will allow a much broader application of the SPSS implementation.
Famoye, F.: A new bivariate generalized Poisson distribution; Statistica Neerlandica 64, 1; (2010)(
http://dx.doi.org/10.1111/j.1467-9574.2009.00446.x)