Abstract:  We deal with non-parametric estimation of multivariate Archimedean copulas. The method uses Geometrically Designed splines (GeD splines) to represent the cumulative distribution function of a random variable W, obtained through the probability integral transform of an Archimedean copula. Sufficient conditions for the GeD spline estimator to posses the properties of the underlying theoretical cumulative distribution function of W, are given. The latter conditions allow for defining a three-step estimation procedure for solving the resulting non-linear regression problem with linear inequality constraints. In the proposed procedure, finding the number and location of the knots and the coefficients of the unconstrained GeD spline estimator and solving the constraint least-squares optimisation problem, are separated. The resulting spline estimator of the cumulative distribution is used to recover the generator and the related Archimedean copula by solving an ordinary differential equation. The proposed method is numerically efficient. In addition, as opposed to most other methods for Archimedean copula estimation, the method is truly multivariate which means that is works efficiently for dimensions d>2. This allows us to apply it with large data sets. Applications in testing goodness-of-fit are also discussed.

This is joint work with Dimitrina Dimitrova and Vladimir Kaishev.