While likelihood-based inference and its variants provide a statistically
efficient and widely applicable approach to parametric inference, their
application to models involving intractable likelihoods poses challenges. In
this work, we study a class of minimum distance estimators for intractable
generative models, that is, statistical models for which the likelihood is
intractable, but simulation is cheap. The distance considered, maximum mean
discrepancy (MMD), is defined through the embedding of probability measures
into a reproducing kernel Hilbert space. We study the theoretical properties of
these estimators, showing that they are consistent, asymptotically normal and
robust to model misspecification. A main advantage of these estimators is the
flexibility offered by the choice of kernel, which can be used to trade-off
statistical efficiency and robustness. On the algorithmic side, we study the
geometry induced by MMD on the parameter space and use this to introduce a
novel natural gradient descent-like algorithm for efficient implementation of
these estimators. We illustrate the relevance of our theoretical results on
several classes of models including a discrete-time latent Markov process and
two multivariate stochastic differential equation models.