  Bivariate count models having one marginal and the other conditionals being
of the Poissons form are called pseudo-Poisson distributions. Such models have
simple exible dependence structures, possess fast computation algorithms and
generate a sufficiently large number of parametric families. It has been
strongly argued that the pseudo-Poisson model will be the first choice to
consider in modelling bivariate over-dispersed data with positive correlation
and having one of the marginal equi-dispersed. Yet, before we start fitting, it
is necessary to test whether the given data is compatible with the assumed
pseudo-Poisson model. Hence, in the present note we derive and propose a few
goodness-of-fit tests for the bivariate pseudo-Poisson distribution. Also we
emphasize two tests, a lesser known test based on the supremes of the absolute
difference between the estimated probability generating function and its
empirical counterpart. A new test has been proposed based on the difference
between the estimated bivariate Fisher dispersion index and its empirical
indices. However, we also consider the potential of applying the bivariate
tests that depend on the generating function (like the Kocherlakota and
Kocherlakota and Mu~noz and Gamero tests) and the univariate goodness-of-fit
tests (like the Chi-square test) to the pseudo-Poisson data. However, for each
of the tests considered we analyse finite, large and asymptotic properties.
Nevertheless, we compare the power (bivariate classical Poisson and Com-Max
bivariate Poisson as alternatives) of each of the tests suggested and also
include examples of application to real-life data. In a nutshell we are
developing an R package which includes a test for compatibility of the data
with the bivariate pseudo-Poisson model.
