Nonparametric Estimation of Copulae with Penalized Hierarchical Mixture Densities

• Activity Number: 122
• Type: Topic Contributed
• Date/Time: Monday, August 2, 2010 : 8:30 AM to 10:20 AM
• Sponsor: Section on Nonparametric Statistics
• Author(s): Goeran Kauermann*+
• Companies: University Bielefeld
• Keywords: copulae ; density estimation ; penalized splines

Abstract

Nonparametric density estimation is generally well developed, but the methods face two problems when being adapted to nonparametric copula density estimation. 1) Due to the curse of dimensionality multivariate density estimation becomes generally problematic. 2) Copula densities require that marginal distributions are uniform.

We propose an estimation routine which aims to solve both problems. 1) The density is estimated with a penalized mixture of hierarchical B-spline-densities. For the multivariate scenario instead of constructing a full tensor product of marginal B-spline-densities we make use sparse grids, i.e only tensor products up to a given hierarchical order are included. 2) The condition that marginal distributions are uniform is easily included as linear constraints on the spline coefficients. The copula density is then easily estimated with quadratic programming.