it is possible to consider $(\omega^{2}_1,...,\omega^{2}_{d_s})=(1,...,1)$, however in general the variability of the statistics is unknown and, depending on the type of data and the underlying model, these can have very different magnitude. An inappropriate choice for the elements in $\Omega$ affects the accuracy of the ABC inference negatively, with $J_{\delta}(\cdot)$ being dominated by the most variable statistic so that $\delta$ will bound the distance with respect to such statistic, not the remaining ones. To our knowledge the only systematic study on the weighting of summary statistics in ABC is [@prangle-adaptiveABC]. When using summary statistics in our experiments, before starting the data cloning procedure we run a pilot study using ABC-MCMC (i.e. $K=1$) with $(\omega_1,...,\omega_{d_s})=(1,...,1)$ and collect the values of the accepted $S(z)$. At the end of the pilot run we compute (after some appropriate burnin) the mean or the median absolute deviation MAD for each coordinate of the accepted $S(z)$ and define $(\omega_1,...,\omega_{d_s}):=(\mathrm{MAD}_1,...,\mathrm{MAD}_{d_s})$. Then we plug the obtained $\Omega$ to weight the summary statistics into data-cloning ABC (introduced in sections \[sec:abc-dc\]–\[sec:dynamic-abcdc\]) or in a further run of ABC-MCMC, for comparison purposes. See [@prangle-adaptiveABC] for a thorough study on this approach.

For ease of reading we produce two ABC-DC algorithms: 