Nonparametric estimation of multivariate copula using empirical bayes method

TL;DR

This paper introduces a hierarchical empirical Bayes approach using multivariate Bernstein polynomials to estimate smooth, genuine copulas in high dimensions, improving dependence measure estimation in finance and related fields.

Contribution

It develops a data-driven, hierarchical empirical Bayes estimator for multivariate copulas that is smooth, genuine, and adaptable to arbitrary dimensions, addressing limitations of existing methods.

Findings

The estimator outperforms existing nonparametric methods in simulations.

It provides more accurate estimates of dependence measures.

Application to portfolio risk demonstrates practical utility.

Abstract

In the field of finance, insurance, and system reliability, etc., it is often of interest to measure the dependence among variables by modeling a multivariate distribution using a copula. The copula models with parametric assumptions are easy to estimate but can be highly biased when such assumptions are false, while the empirical copulas are non-smooth and often not genuine copula making the inference about dependence challenging in practice. As a compromise, the empirical Bernstein copula provides a smooth estimator but the estimation of tuning parameters remains elusive. In this paper, by using the so-called empirical checkerboard copula we build a hierarchical empirical Bayes model that enables the estimation of a smooth copula function for arbitrary dimensions. The proposed estimator based on the multivariate Bernstein polynomials is itself a genuine copula and the selection of its dimension-varying degrees is data-dependent. We also show that the proposed copula estimator provides a more accurate estimate of several multivariate dependence measures which can be obtained in closed form. We investigate the asymptotic and finite-sample performance of the proposed estimator and compare it with some nonparametric estimators through simulation studies. An application to portfolio risk management is presented along with a quantification of estimation uncertainty.