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

Distribution free MMD tests for model selection with estimated parameters