## 1 Introduction  
The need to describe the dependency between two or more random variables triggered the concept of copulas. We consider a joint cumulative distribution function (cdf) \(F\) on \(\mathbb{R}^{d}\) and its cdf margins \(F_{1}\), \(F_{2}\),...,\(F_{d}\). A copula \(C\) is a function such that \(F=C(F_{1},F_{2},...,F_{d})\). [PERSON] [1] is the first to lay the foundations of this new theory. Several parametric families of copulas have been defined, namely elliptical, archimedean, periodic copulas _etc._, see [PERSON] [2] and [PERSON] [3] as well as Appendix A for an overview of these families. Finding criteria to determine the best copula for a given problem can only be achieved through a goodness-of-fit (GOF) approach. So far several GOF copula approaches have been proposed in the literature, e.g., [PERSON] [4], [PERSON] and [PERSON] [5], [PERSON] [6], [PERSON] and [PERSON] [7], [PERSON] [PERSON] [8], [PERSON] and [PERSON] [9], [PERSON] and [PERSON] [10], [PERSON] and [PERSON] [11], [PERSON] [12], [PERSON] and [PERSON] [13], among others. However, the field is still at an embryonic stage which explains the current shortage in recommendations. In univariate distributions, the GOF assessment can be performed using for instance the well-known Kolmogorov test. In the multivariate field, there are fewer alternatives. A simple way to build GOF approaches for multivariate random variables is to consider multi-dimensional chi-square approaches, as in for example [PERSON] [14]. However, these approaches present feasibility issues for high dimensional problems due to the curse of dimensionality. In order to solve this, we recall some facts from the theory of projection pursuit.  
The objective of projection pursuit is to generate one or several projections providing as much information as possible about the structure of the dataset regardless of its size. Once a structure has been isolated, the corresponding data are transformed through a Gaussianization. Through a recursive approach, this process is iterated to find another structure in the remaining data, until no further structure can be evidenced in the data left at the end. [PERSON] [15] and [PERSON] [16] count among the first authors who introduced this type of approaches for evidencing structures. They each describe, with many examples, how to evidence such a structure and consequently how to estimate the density of such data through two different methodologies each. Their work is based on maximizing Kullback-Leibler divergence. In the present article, we will introduce a new projection pursuit methodology based on the minimisation of any \(\phi\)-divergence greater than the \(L^{1}\)-distance (\(\phi\)-PP). We will show that this algorithm presents the extra advantage of being robust and fast from a numerical standpoint. Its key rationale lies in the fact that it allows not only to carry out GOF tests for elliptical and independent copulas but also to determine the axis system upon which these very copulas are based. The exact expression of these copulas in the basis constituted by these axes can therefore be derived.  
This paper is organised as follows: Section 2 contains preliminary definitions and properties. In Section 3, we present in details the \(\phi\)-projection pursuit algorithm. In Section 4, we present our first results. In Section 5, we introduce our tests. In Section 6, we provide three simulations pertaining to the two major situations described herein and we will study a real case.