  Approximate Bayesian computation (ABC) has become an essential part of the
Bayesian toolbox for addressing problems in which the likelihood is
prohibitively expensive or entirely unknown, making it intractable. ABC defines
a pseudo-posterior by comparing observed data with simulated data,
traditionally based on some summary statistics, the elicitation of which is
regarded as a key difficulty. Recently, using data discrepancy measures has
been proposed in order to bypass the construction of summary statistics. Here
we propose to use the importance-sampling ABC (IS-ABC) algorithm relying on the
so-called two-sample energy statistic. We establish a new asymptotic result for
the case where both the observed sample size and the simulated data sample size
increase to infinity, which highlights to what extent the data discrepancy
measure impacts the asymptotic pseudo-posterior. The result holds in the broad
setting of IS-ABC methodologies, thus generalizing previous results that have
been established only for rejection ABC algorithms. Furthermore, we propose a
consistent V-statistic estimator of the energy statistic, under which we show
that the large sample result holds, and prove that the rejection ABC algorithm,
based on the energy statistic, generates pseudo-posterior distributions that
achieves convergence to the correct limits, when implemented with rejection
thresholds that converge to zero, in the finite sample setting. Our proposed
energy statistic based ABC algorithm is demonstrated on a variety of models,
including a Gaussian mixture, a moving-average model of order two, a bivariate
beta and a multivariate $g$-and-$k$ distribution. We find that our proposed
method compares well with alternative discrepancy measures.
