Solving Estimating Equations With Copulas

Abstract

Thanks to their ability to capture complex dependence structures, copulas are frequently used to glue random variables into a joint model with arbitrary marginal distributions. More recently, they have been applied to solve statistical learning problems such as regression or classification. Framing such approaches as solutions of estimating equations, we generalize them in a unified framework. We can then obtain simultaneous, coherent inferences across multiple regression-like problems. We derive consistency, asymptotic normality, and validity of the bootstrap for corresponding estimators. The conditions allow for both continuous and discrete data as well as parametric, nonparametric, and semiparametric estimators of the copula and marginal distributions. The versatility of this methodology is illustrated by several theoretical examples, a simulation study, and an application to financial portfolio allocation. Supplementary materials for this article are available online.

KEYWORD:

• Bootstrap
• Nonparametric
• Quantile
• Regression
• Statistical learning

Supplementary Materials

Online supplementary material includes all mathematical proofs and auxiliary results, additional figures and results for the simulation studies in Section 5, and code and data to reproduce all results from Sections 5 and 6.

Acknowledgments

The authors thank Johannes Wiesel for the proof of Lemma 11 in the supplementary material. We are grateful to the Associate Editor and three referees for helpful comments. Part of the research was conducted while Thomas Nagler was at Delft University of Technology and Thibault Vatter was at Columbia University.

Notes

1 A formal definition is given in the supplementary material.

Additional information

Funding

Thomas Nagler was partially supported by a Spinoza grant awarded by the Netherlands Organisation of Scientific Research (NWO) and the German Federal Ministry of Education and Research (BMBF) under grant no. 01IS18036A. Thibault Vatter was partially supported by the Swiss National Science Foundation (grant 174709).