Dana Uhlig, Roman Unger: The Petrov-Galerkin projection for copula density estimation isn't counting

Author(s):

Dana Uhlig
Roman Unger

Title:

Dana Uhlig, Roman Unger: The Petrov-Galerkin projection for copula density estimation isn't counting

Electronic source:

application/pdf

Preprint series:

Technische Universität Chemnitz, Fakultät für Mathematik (Germany). Preprint 03, 2014

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:

Non-parametric copula density estimation in the d-dimensional case is a big challenge in particular if the dimension d of the problem increases. In Preprint 2013-07 we proposed to solve the d-dimensional Volterra integral equation \int_0^u c(s)ds = C(u) for a given copula C.
In the statistical framework the copula C is unobservable and hence we solved the linear integral equation for the empirical copula. For the numerical computation we used a Petrov-Galerkin projection for the approximated piecewise constant function c_h(u)=Sum_{i=1}^N c_i \phi_i(u).
Other than might be expected, the coefficient vector c does not count the number of samples in the elements of the discretized grid, even the approximated solution c_h is a piecewise constant function on the elements.
We will establish that solving the Volterra integral equation by a Petrov-Galerkin projection is not simple counting.

Keywords:

Copula, Galerkin methods, Inverse Problems, Kronecker Product

Language:

English

Publication time:

02/2014