Projection Pursuit through $\Phi$-Divergence Minimisation

TL;DR

This paper extends projection pursuit methods to minimize $\

Contribution

It generalizes Touboul's relative entropy approach to $\

Findings

Introduces a $\

Provides a new test for elliptical copula fit.

Demonstrates the method's effectiveness on simulated data.

Abstract

Consider a defined density on a set of very large dimension. It is quite difficult to find an estimate of this density from a data set. However, it is possible through a projection pursuit methodology to solve this problem. Touboul's article "Projection Pursuit Through Relative Entropy Minimization", 2009, demonstrates the interest of the author's method in a very simple given case. He considers the factorization of a density through an Elliptical component and some residual density. The above Touboul's work is based on minimizing relative entropy. In the present article, our proposal will aim at extending this very methodology to the $\Phi-$divergence. Furthermore, we will also consider the case when the density to be factorized is estimated from an i.i.d. sample. We will then propose a test for the factorization of the estimated density. Applications include a new test of fit pertaining to the Elliptical copulas.