  Several kernel based testing procedures are proposed to solve the problem of
model selection in the presence of parameter estimation in a family of
candidate models. Extending the two sample test of Gretton et al. (2006), we
first provide a way of testing whether some data is drawn from a given
parametric model (model specification). Second, we provide a test statistic to
decide whether two parametric models are equally valid to describe some data
(model comparison), in the spirit of Vuong (1989). All our tests are
asymptotically standard normal under the null, even when the true underlying
distribution belongs to the competing parametric families.Some simulations
illustrate the performance of our tests in terms of power and level.
