  We propose a new methodology to detect zero-inflation and overdispersion
based on the comparison of the expected sample extremes among convexly ordered
distributions. The method is very flexible and includes tests for the
proportion of structural zeros in zero-inflated models, tests to distinguish
between two ordered parametric families and a new general test to detect
overdispersion. The performance of the proposed tests is evaluated via some
simulation studies. For the well-known fetal lamb data, we conclude that the
zero-inflated Poisson model should be rejected against other more disperse
models, but we cannot reject the negative binomial model.
