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, possibly in the case when the number of parameters is diverging. Our framework allows to manage copula-based loss functions that are potentially unbounded. Additionally, we state the weak limit of multivariate rank statistics for an arbitrary dimension and the weak convergence of empirical copula processes indexed by maps. We apply our inference method to Canonical Maximum Likelihood losses with Gaussian copulas, mixtures of copulas or conditional copulas. The theoretical results are illustrated by two numerical experiments.

翻译：本文研究了存在伪观测时稀疏M估计量的大样本性质。我们的框架涵盖了一类广泛的半参数Copula模型，其中边际分布未知，并由其经验分布替代。众所周知，相较于标准M估计，这一修正显著改变了极限分布。我们建立了稀疏惩罚M估计量的一致性和渐近正态性，并证明了其在伪观测下的渐近Oracle性质，且该结论适用于参数个数发散的情形。我们的框架允许处理基于Copula的潜在无界损失函数。此外，我们给出了任意维度多元秩统计量的弱极限形式，以及由映射索引的经验Copula过程的弱收敛性。我们将该推断方法应用于高斯Copula、Copula混合或条件Copula的典型最大似然损失。两个数值实验验证了理论结果。