## 6 Simulations  
Let us examine three simulations and an application to real datasets. The first simulation studies the elliptical copula and the second studies the independent copula. In each simulation, our program will aim at creating a sequence of densities \((g^{(j)})\), \(j=1,..,d\) such that \(g^{(0)}=g,\)\(g^{(j)}=g^{(j-1)}f_{a_{j}}/[g^{(j-1)}]_{a_{j}}\) and \(D_{\phi}(g^{(d)},f)=0,\) where \(D_{\phi}\) is a divergence--see Appendix B for its definition--and \(a_{j}=arg\inf_{b}D_{\phi}(g^{(j-1)}f_{b}/g_{b}^{(j-1)},f),\) for all \(j=1,...,d\). We will therefore perform the tests introduced at theorems 5.1 and 5.2. Finally, the third simulation compares the optimisations obtained, when we execute the process with, each time, a new \(\phi\)-divergence. \(\mathcal{S}imulation\) **6.1**. : We are in dimension \(2\)(=\(d\)), and we use the \(\chi^{2}\) divergence to perform our optimisations. Let us consider a sample of \(50\)(=\(n\)) values of a random variable \(X\) with a density law \(f\) defined by :  
\[f(x)=c_{\rho}(F_{Gumbel}(x_{1}),F_{Exponential}(x_{2})).Gumbel(x_{1}).Exponential (x_{2})\]  
where \(c\) is the Gaussian copula with correlation coefficient \(\rho=0.5\), and where the Gumbel distribution parameters are \(-1\) and \(1\) and the exponential density parameter is \(2\). Let us generate then a Gaussian random variable \(Y\) with a density--that we will name as \(g\)--presenting the same mean and variance as \(f\). We theoretically obtain \(k=2\) and \((a_{1},a_{2})=((1,0),(0,1))\). To get this result, we perform the following test:  
\[(H_{0}):\ (a_{1},a_{2})=((1,0),(0,1))\text{ versus }(H_{1}):\ (a_{1},a_{2})\
eq((1,0),(0,1))\]  
Then, theorem 5.1 enables us to verify \((H_{0})\) by the following 0.9(=\(\alpha\)) level confidence ellipsoid  
\[\mathcal{E}_{2}=\{b\in\mathbb{R}^{2};\ (Var_{\mathbf{P}}(M(b,b)))^{(-1/2)} \mathbb{P}_{n}M(b,b)\leq q_{\alpha}^{\mathcal{N}(0,1)}/\sqrt{n}\simeq 0,2533/7.0710=0.03582\}\]  
Results of this optimisation can be found in Table 3 and Figure 1.  
\(\mathcal{S}imulation\) **6.2. ** We are in dimension \(2\)(=\(d\)), and we use the \(\chi^{2}\) divergence to perform our optimisations. Let us consider a sample of \(50\)(=\(n\)) values of a random variable \(X\) with a density law \(f\) defined by  
\(f(x)=Gumbel(x_{1}).Exponential(x_{2})\)  
where the Gumbel distribution parameters are \(-1\) and \(1\) and the exponential density parameter is \(2\). Let \(g\) be an instrumental product of univariate Gaussian densities with \(diag(Var(X_{1}),...,Var(X_{d}))\) as covariance matrix and with the same mean as \(f\). We theoretically obtain \(k=2\) and \((a_{1},a_{2})=((1,0),(0,1))\). To get this result, we perform the following test:  
\((H_{0}):\ (a_{1},a_{2})=((1,0),(0,1))\) _versus_\((H_{1}):\ (a_{1},a_{2})\
eq((1,0),(0,1))\) \begin{table}
\begin{tabular}{l l} \hline \hline Our Algorithm & \\ \hline \multirow{2}{*}{Projection Study 0:} & minimum : 0.445199 \\ \cline{2-3}  & at point : (1.0171,0.0055) \\ \cline{2-3}  & _P_-Value : 0.94579 \\ \hline Test: & \(H_{1}\) : \(a_{1}\)\(\
ot\in\)\(\mathcal{E}_{1}\) : True \\ \hline \multirow{2}{*}{Projection Study 1:} & minimum : 0.009628 \\ \cline{2-3}  & at point : (0.0048,0.9197) \\ \cline{2-3}  & _P_-Value : 0.99801 \\ \hline Test: & \(H_{0}\) : \(a_{2}\)\(\in\)\(\mathcal{E}_{2}\) : True \\ \hline \(\chi^{2}\)(Kernel Estimation of \(g^{(2)}\), \(g^{(2)}\)) & 3.57809 \\ \hline \end{tabular} Therefore, we can conclude that \(H_{0}\) is verified.  
\end{table} 
Table 3: Simulation 1: Numerical results of the optimisation. Then, theorem 5.2 enables us to verify \((H_{0})\) by the following 0.9(=\(\alpha\)) level confidence ellipsoid  
\[\mathcal{E}_{2}=\{b\in\mathbb{R}^{2};\;(Var_{\mathbf{P}}(M(b,b)))^{(-1/2)} \mathbb{P}_{n}M(b,b)\leq q_{\alpha}^{\mathcal{N}(0,1)}/\sqrt{n}\simeq 0.03582203\}\]  
Results of this optimisation can be found in Table 4 and Figure 2.  
Therefore, we can conclude that \(f=\Pi_{i=1}^{d}f_{i}\). \(\mathcal{S}imulation\) **6.3** (On the choice of a \(\phi\)-divergence).