## 3 The Algorithm  
### The Model  
Let \(f\) be a density on \(\mathbb{R}^{d}\). We assume there exists \(d\) non-null linearly independent vectors \(a_{j}\), with \(1\leq j\leq d\), of \(\mathbb{R}^{d}\), such that  
\[f(x)=n(a_{j+1}^{\top}x,...,a_{d}^{\top}x)h(a_{1}^{\top}x,...,a_{j}^{\top}x) \tag{3.1}\] \begin{table}
\begin{tabular}{r l} \hline \hline \(0\). & We define \(g\), a density with same mean and variance as \(f\) and we set \(g^{(0)}=g\). \\ \hline \(i-1\). & We perform the goodness-of-fit test \(D_{\phi}(g^{(i-1)},f)=0\): \\  & \(\bullet\) Should this test be passed, we derive \(f\) from \\  & \(f=g\Pi_{i=1}^{j}\frac{f_{a_{i}}}{g_{a_{i}}^{(i-1)}}\) \\  & And the algorithm stops. \\  & \(\bullet\) Should this test not be verified, and should we look to approximate \(f\), \\  & when we get to the \(d^{th}\) iteration of the algorithm, we derive \(f\) from \\  & \(f=g\Pi_{i=1}^{d}\frac{f_{a_{i}}}{g_{a_{i}}^{(i-1)}}\) \\  & Otherwise, let us define a vector \(a_{i}\) and a density \(g^{(i)}\) by \\  & \(a_{i}\;=\;arg\inf_{a\in\mathbb{R}^{d}}\;D_{\phi}\left(g^{(i-1)}\frac{f_{a}}{g_ {a}^{(i-1)}},f\right),\;\text{and}\;g^{(i)}=g^{(i-1)}\frac{f_{a_{i}}}{g_{a_{i} }^{(i-1)}}\) \\ \hline \(i\). & Then we replace \(g^{(i-1)}\) with \(g^{(i)}\) and go back to \(i-1\). \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Proposal. with \(j<d\), \(n\) being an elliptical density on \(\mathbb{R}^{d-j}\) and with \(h\) being a density on \(\mathbb{R}^{j}\), which does not belong to the same family as \(n\). Let \(X=(X_{1},...,X_{d})\) be a vector with \(f\) as density. We define \(g\) as an elliptical distribution with the same mean and variance as \(f\). For simplicity, let us assume that the family \(\{a_{j}\}_{1\leq j\leq d}\) is the canonical basis of \(\mathbb{R}^{d}\):  
The very definition of \(f\) implies that \((X_{j+1},...,X_{d})\) is independent from \((X_{1},...,X_{j})\). Hence, the density of \((X_{j+1},...,X_{d})\) given \((X_{1},...,X_{j})\) is \(n\). Let us assume that \(D_{\phi}(g^{(j)},f)=0,\) for some \(j\leq d\). We then get \(\frac{f(x)}{f_{a_{1}}f_{a_{2}}...f_{a_{j}}}=\frac{g(x)}{g_{a_{1}}^{(1-1)}g_{a_ {2}}^{(2-1)}...g_{a_{j}}^{(j-1)}}\), since, by induction, we have \(g^{(j)}(x)=g(x)\frac{f_{a_{1}}}{g_{a_{1}}^{(1-1)}}\frac{f_{a_{2}}}{g_{a_{2}}^{ (2-1)}}...\frac{f_{a_{j}}}{g_{a_{j}}^{(j-1)}}\). Consequently, lemma C.1 and the fact that the conditional densities with elliptical distributions are also elliptical, as well as the above relationship, lead us to infer that \(n(a_{j+1}^{\top}x,.,a_{d}^{\top}x)=f(./a_{i}^{\top}x,1\leq i\leq j)=g(./a_{i}^{ \top}x,1\leq i\leq j).\) In other words, \(f\) coincides with \(g\) on the complement of the vector subspace generated by the family \(\{a_{i}\}_{i=1,...,j}\). Now, if the family \(\{a_{j}\}_{1\leq j\leq d}\) is no longer the canonical basis of \(\mathbb{R}^{d}\), then this family is again a basis of \(\mathbb{R}^{d}\). Hence, lemma C.2 implies that  
\[g(./a_{1}^{\top}x,...,a_{j}^{\top}x)=n(a_{j+1}^{\top}x,...,a_{d}^{\top}x)=f(. /a_{1}^{\top}x,...,a_{j}^{\top}x) \tag{3.2}\]  
which is equivalent to \(D_{\phi}(g^{(j)},f)=0\), since by induction \(g^{(j)}=g\frac{f_{a_{1}}}{g_{a_{1}}^{(1-1)}}\frac{f_{a_{2}}}{g_{a_{2}}^{(2-1) }}...\frac{f_{a_{j}}}{g_{a_{j}}^{(j-1)}}\). The end of our algorithm implies that \(f\) coincides with \(g\) on the complement of the vector subspace generated by the family \(\{a_{i}\}_{i=1,...,j}\). Therefore, the nullity of the \(\phi\)-divergence provides us with information on the density structure. In summary, the following proposition clarifies our choice of \(g\) which depends on the family of distribution one wants to find in \(f\) :  
**Proposition 3.1**. : _With the above notations, \(D_{\phi}(g^{(j)},f)=0\) is equivalent to_  
\[g(./a_{1}^{\top}x,...,a_{j}^{\top}x)=f(./a_{1}^{\top}x,...,a_{j}^{\top}x)\]  
More generally, the above proposition defines the co-support of \(f\) as the vector space generated by the vectors \(a_{1},...,a_{j}\). **Definition 3.1**. : _Let \(f\) be a density on \(\mathbb{R}^{d}\).