Nonparametric Copula Density Estimation Using a Petrov–Galerkin Projection

Abstract

Nonparametrical copula density estimation is a meaningful tool for analyzing the dependence structure of a random vector from given samples. Usually kernel estimators or penalized maximum likelihood estimators are considered. We propose solving the Volterra integral equation $$\begin{aligned} \int \limits _0^{u_1} \cdots \int \limits _0^{u_d} \mathrm{c}(s_1,\ldots , s_d) d s_1 \cdots d s_d = \mathrm{C}(u_1, \ldots , u_d) \end{aligned}$$ to find the copula density $$\mathrm{c}(u_1, \ldots , u_d) = \frac{\partial ^d \mathrm{C}}{\partial u_1 \cdots \partial u_d}$$ of the given copula $$\mathrm{C}$$ . In the statistical framework, the copula $$\mathrm{C}$$ is not available and we replace it by the empirical copula of the pseudo samples, which converges to the unobservable copula $$\mathrm{C}$$ for large samples. Hence, we can treat the copula density estimation from given samples as an inverse problem and consider the instability of the inverse operator, which has an important impact if the input data of the operator equation are noisy. The well-known curse of high dimensions usually results in huge nonsparse linear equations after discretizing the operator equation. We present a Petrov–Galerkin projection for the numerical computation of the linear integral equation. A special choice of test and ansatz functions leads to a very special structure of the linear equations, such that we are able to estimate the copula density also in higher dimensions.