2013/10/22_Noel Veraverbeke (University of Hasselt)

Bernstein estimator for a copula and its density

2013/10/22 – 12:00 h | Noel Veraverbeke, University of Hasselt

Abstract

Copulas are functions that couple the multivariate distribution function H(x; y) of a random vector (X; Y) to its one-dimensional marginals F(x) and G(y). According to Sklar’s theorem, there exists a bivariate function C, called copula, such that H(x; y) = C(F(x);G(y)). Several papers deal with the estimation of C, based on a random sample of size n from (X; Y). In this talk we discuss the asymptotic properties of the so called Bernstein estimation method.

This nonparametric smoothing method approximates C(u; v) by a polynomial of degree m in (u; v). Asymptotics are considered as n andmtend to infinity. The estimator for the copula C leads in a very natural way to an estimator for the corresponding density c of C. Asymptotic normality is obtained and optimal order of the degree m is discussed. Compared to the existing results our theorem does not assume known marginals. Joint work with Paul Janssen and Jan Swanepoel.