  Approximate Bayesian computation (ABC) is a widely used inference method in
Bayesian statistics to bypass the point-wise computation of the likelihood. In
this paper we develop theoretical bounds for the distance between the
statistics used in ABC. We show that some versions of ABC are inherently robust
to misspecification. The bounds are given in the form of oracle inequalities
for a finite sample size. The dependence on the dimension of the parameter
space and the number of statistics is made explicit. The results are shown to
be amenable to oracle inequalities in parameter space. We apply our theoretical
results to given prior distributions and data generating processes, including a
non-parametric regression model. In a second part of the paper, we propose a
sequential Monte Carlo (SMC) to sample from the pseudo-posterior, improving
upon the state of the art samplers.
