Approximate Bayesian Computation is a family of likelihood-free inference
techniques that are well-suited to models defined in terms of a stochastic
generating mechanism. In a nutshell, Approximate Bayesian Computation proceeds
by computing summary statistics ${\bf s}_{obs}$ from the data and simulating
synthetic summary statistics for different values of the parameter $\Theta$.
The posterior distribution is then approximated with an estimator of the
conditional density $g(\Theta|{\bf s}_{obs})$. In this paper, we derive the
asymptotic bias and variance of the standard estimators of the posterior
distribution which are based on rejection sampling and linear adjustment.
Additionally, we introduce an original estimator of the posterior distribution
based on quadratic adjustment and we show that its bias contains a smaller
number of terms than the estimator with linear adjustment. Although we find
that the estimators with adjustment are not universally superior to the
estimator based on rejection sampling, we find that they can achieve better
performance when there is a nearly homoscedastic relationship between the
summary statistics and the parameter of interest. Last, we present model
selection in Approximate Bayesian Computation. As for parameter estimation, the
asymptotic results raise the importance of the curse of dimensionality in
Approximate Bayesian Computation.