Nonparametric estimation of copulas and copula densities by orthogonal projections

TL;DR

This paper introduces a novel nonparametric copula and density estimator based on orthogonal polynomial projections, demonstrating optimality and superior performance through simulations and real data application.

Contribution

It proposes a new orthogonal projection-based copula estimator with proven asymptotic optimality and a method for selecting smoothing parameters.

Findings

Estimator shows strong performance in simulations

Method achieves minimax and maxiset optimality

Real data application demonstrates practical usefulness

Abstract

In this paper we study nonparametric estimators of copulas and copula densities. We first focus our study on a density copula estimator based on a polynomial orthogonal projection of the joint density. A new copula estimator is then deduced. Its asymptotic properties are studied: we provide a large functional class for which this construction is optimal in the minimax and maxiset sense and we propose a method selection for the smoothing parameter. An intensive simulation study shows the very good performance of both copulas and copula densities estimators which we compare to a large panel of competitors. A real dataset in actuarial science illustrates this approach.