Sparse M-estimators in semi-parametric copula models

We study the large sample properties of sparse M-estimators in the presence of pseudo-observations. Our framework covers a broad class of semi-parametric copula models, for which the marginal distributions are unknown and replaced by their empirical counterparts. It is well known that the latter modification significantly alters the limiting laws compared to usual M-estimation. We establish the consistency and the asymptotic normality of our sparse penalized M-estimator and we prove the asymptotic oracle property with pseudo-observations, including the case when the number of parameters is diverging. Our framework allows to manage copula based loss functions that are potentially unbounded. As additional results, we state the weak limit of multivariate rank statistics and the weak convergence of the empirical copula process indexed by such maps. We apply our inference method to copula vine models and copula regressions. The numerical results emphasize the relevance of this methodology in the context of model misspecifications.