  Model misspecification can create significant challenges for the
implementation of probabilistic models, and this has led to development of a
range of robust methods which directly account for this issue. However, whether
these more involved methods are required will depend on whether the model is
really misspecified, and there is a lack of generally applicable methods to
answer this question. In this paper, we propose one such method. More
precisely, we propose kernel-based hypothesis tests for the challenging
composite testing problem, where we are interested in whether the data comes
from any distribution in some parametric family. Our tests make use of minimum
distance estimators based on the maximum mean discrepancy and the kernel Stein
discrepancy. They are widely applicable, including whenever the density of the
parametric model is known up to normalisation constant, or if the model takes
the form of a simulator. As our main result, we show that we are able to
estimate the parameter and conduct our test on the same data (without data
splitting), while maintaining a correct test level. Our approach is illustrated
on a range of problems, including testing for goodness-of-fit of an
unnormalised non-parametric density model, and an intractable generative model
of a biological cellular network.
