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.
Funding Statement
J.D. Fermanian was supported by the labex Ecodec (reference project ANR-11-LABEX-0047) and B. Poignard by the Japanese Society for the Promotion of Science (Grant 22K13377).
Acknowledgments
The authors are grateful to two anonymous referees, an Associate Editor and the Editor for their insightful comments.
Citation
Jean-David Fermanian. Benjamin Poignard. "Sparse M-estimators in semi-parametric copula models." Bernoulli 30 (3) 2475 - 2500, August 2024. https://doi.org/10.3150/23-BEJ1681
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