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 s_obs from the data and simulating summary
statistics for different values of the parameter theta. The posterior
distribution is then approximated by an estimator of the conditional density
g(theta|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 fewer 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. To make this relationship as homoscedastic as possible, we propose to
use transformations of the summary statistics. In different examples borrowed
from the population genetics and epidemiological literature, we show the
potential of the methods with adjustment and of the transformations of the
summary statistics. Supplemental materials containing the details of the proofs
are available online