Bernstein Polynomials and n-Copulas

TL;DR

This paper explores Bernstein polynomial approximations of copulas, demonstrating they preserve copula properties and providing a stochastic interpretation, thus advancing the mathematical tools for copula analysis.

Contribution

It establishes that Bernstein approximations of copulas are themselves copulas and offers a stochastic interpretation, enhancing understanding of copula approximation methods.

Findings

Bernstein approximations of copulas are copulas.

Provides a stochastic interpretation for Bernstein copula approximations.

Derives basic results for Bernstein approximation in multiple variables.

Abstract

We give derivations of some basic results for the Bernstein approximation in $n$ variables that are useful in investigating copulas. It is shown that Bernstein approximations of copulas are again copulas. We exhibit a stochastic interpretation for the Bernstein approximation of a copula.