Dana Uhlig, Roman Unger: A Petrov Galerkin projection for copula density estimation

Author(s):

Dana Uhlig
Roman Unger

Title:

Dana Uhlig, Roman Unger: A Petrov Galerkin projection for copula density estimation

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 07, 2013

Mathematics Subject Classification:

62H99	[Multivariate analysis]
62H12	[Estimation]
65F22	[Ill-posedness, regularization]
45Q05	[Inverse problems]
65Y05	[Parallel computation]
65N30	[Finite elements, Rayleigh-Ritz and Galerkin methods,finite methods]

Abstract:

The reconstruction of the dependence structure of two or more random variables (d > 1) is a big issue in finance and many other applications. Looking at samples of the random vector, neither the common distribution nor the copula itself are observable. So the identification of the copula C or the copula density can be treated as an inverse problem. In the statistical literature usually kernel estimators or penalized maximum likelihood estimators are considered for the non-parametric estimation of the copula density c from given samples of the random vector.
Even though the copula C itself is unobservable we can treat the empirical copula as a noisy representation, since it is well known that the empirical copula converges for large samples to the copula and solve the d-dimensional linear integral equation for determining the copula density c.
We present a Petrov-Galerkin projection for the numerical computation of the linear integral equation and discuss the assembling algorithm of the non-sparce matrices and vectors. Furthermore we analyze the stability of the discretized linear equation.

Keywords:

Copula, Galerkin methods, Inverse Problems, Kronecker Product

Language:

English

Publication time:

07/2013