  In this paper the method of simulated quantiles (MSQ) of Dominicy and Veredas
(2013) and Dominick et al. (2013) is extended to a general multivariate
framework (MMSQ) and to provide a sparse estimator of the scale matrix
(sparse-MMSQ). The MSQ, like alternative likelihood-free procedures, is based
on the minimisation of the distance between appropriate statistics evaluated on
the true and synthetic data simulated from the postulated model. Those
statistics are functions of the quantiles providing an effective way to deal
with distributions that do not admit moments of any order like the
$\alpha$-Stable or the Tukey lambda distribution. The lack of a natural
ordering represents the major challenge for the extension of the method to the
multivariate framework. Here, we rely on the notion of projectional quantile
recently introduced by Hallin etal. (2010) and Kong Mizera (2012). We establish
consistency and asymptotic normality of the proposed estimator. The smoothly
clipped absolute deviation (SCAD) $\ell_1$--penalty of Fan and Li (2001) is
then introduced into the MMSQ objective function in order to achieve sparse
estimation of the scaling matrix which is the major responsible for the curse
of dimensionality problem. We extend the asymptotic theory and we show that the
sparse-MMSQ estimator enjoys the oracle properties under mild regularity
conditions. The method is illustrated and its effectiveness is tested using
several synthetic datasets simulated from the Elliptical Stable distribution
(ESD) for which alternative methods are recognised to perform poorly. The
method is then applied to build a new network-based systemic risk measurement
framework. The proposed methodology to build the network relies on a new
systemic risk measure and on a parametric test of statistical dominance.
