(using draws from the ABC rejection algorithm proposed in [@pritchard1999population]), then the maximizer of the non-parametric density is found numerically. In our approach we do not require any kernel estimation procedure (which is onerous unless the dimension of $\theta$ is low), nor direct optimization procedures. An approach similar to the one by Rubio and Johansen is in [@grazian-liseo(2015)].

In conclusion, we use draws produced by algorithm \[alg:abc-dc-dynam\] under $(\delta,K)\equiv(\delta_{r_p},K_{s_q})$ and for these draws we compute their sample mean $\hat{\theta}_{\delta,K}$ to obtain an approximate maximum likelihood estimate (MLE) of $\theta$. In principle, if we were able to construct sufficient summaries $S(\cdot)$ for $\theta$, and by letting $\delta\rightarrow 0$ and then $K\rightarrow\infty$, we would have $\hat{\theta}_{MLE}\equiv \hat{\theta}_{\delta,K}$ and $\hat{\Sigma}_{MLE} \equiv K\cdot\hat{\Sigma}_{\delta,K}$, where $\hat{\theta}_{MLE}$ is the MLE of $\theta$ and $\Sigma_{MLE}$ is the covariance of the MLE (i.e. the inverse of the Fisher information based on $y$) computed using the sample covariance $\hat{\Sigma}_{\delta,K}$ derived under $(\delta,K)\equiv(\delta_{r_p},K_{s_q})$. The reasoning behind these results, when $S(\cdot)$ is sufficient, is that (i) for an ABC approximation to a posterior it holds that in distribution $\lim_{\delta\rightarrow 0}\pi_{\delta}(\theta|y)=\pi(\theta|y)$, and that (b) data-cloning implies that in distribution $\lim_{K\rightarrow\infty}\pi(\theta|y^{(K)})=N(\hat{\theta}_{MLE},\hat{\Sigma}_{MLE})$ ([@jacquier-johannes-polson], [@lele2010estimability]). Therefore by first taking the limit for $\delta$ and then 