Sparse M-estimators in semi-parametric copula models

TL;DR

This paper develops and analyzes sparse M-estimators for semi-parametric copula models with unknown marginals, establishing their asymptotic properties and applying them to various copula-based inference problems.

Contribution

It introduces a comprehensive framework for sparse M-estimation in semi-parametric copula models with pseudo-observations, including asymptotic properties and oracle properties.

Findings

Proved consistency and asymptotic normality of the estimators.

Established the asymptotic oracle property with diverging parameters.

Demonstrated the framework's applicability through numerical experiments.

Abstract

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.