  We present asymptotic results for the regression-adjusted version of
approximate Bayesian computation introduced by Beaumont(2002). We show that for
an appropriate choice of the bandwidth, regression adjustment will lead to a
posterior that, asymptotically, correctly quantifies uncertainty. Furthermore,
for such a choice of bandwidth we can implement an importance sampling
algorithm to sample from the posterior whose acceptance probability tends to
unity as the data sample size increases. This compares favourably to results
for standard approximate Bayesian computation, where the only way to obtain a
posterior that correctly quantifies uncertainty is to choose a much smaller
bandwidth; one for which the acceptance probability tends to zero and hence for
which Monte Carlo error will dominate.
