The Sparse Multivariate Method of Simulated Quantiles

Abstract

In this paper the method of simulated quantiles (MSQ) of and is extended to a general multivariate framework (MMSQ) and to provide sparse estimation of the scaling 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 α-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 et al. (2010b) and Kong and Mizera (2012). We establish consistency and asymptotic normality of the proposed estimator. The smoothly clipped absolute deviation (SCAD) 1 -penalty of Fan and Li (2001) is then introduced into the MMSQ objective function in order to