  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.
