## 6 Simulations
: In this paragraph, we perform our algorithm several times. We first use several \(\phi\)-divergences (see Appendix B for their definitions and their notations). We then perform a sensitivity analysis by varying the number \(n\) of simulated variables. Finally we introduce outliers. At present, we consider a sample of \(n\) values of a random variable \(X\) with a density \(f\) defined by \(f(x)=Laplace(x_{1}).Gumbel(x_{0})\) \begin{table}
\begin{tabular}{l l} \hline \hline Our Algorithm & \multicolumn{1}{c}{} \\ \hline \multirow{3}{*}{Projection Study 0 :} & minimum : 0.057833 \\ \cline{2-3}  & at point : (0.9890,0.1009) \\ \cline{2-3}  & _P_-Value : 0.955651 \\ \hline Test : & \(H_{1}\) : \(a_{1}\
ot\in\mathcal{E}_{1}\) : True \\ \hline \multirow{3}{*}{Projection Study 1 :} & minimum : 0.02611 \\ \cline{2-3}  & at point : (\(-\)0.1105,0.9290) \\ \cline{2-3}  & _P_-Value : 0.921101 \\ \hline Test : & \(H_{0}\) : \(a_{2}\in\mathcal{E}_{2}\) : True \\ \hline \(\chi^{2}\)(Kernel Estimation of \(g^{(2)}\), \(g^{(2)}\)) & 1.25945 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Simulation 2: Numerical results of the optimisation. Figure 2: Graph of the independent copula estimate. where the Gumbel distribution parameters are \((1,2)\) and where the Laplace distribution parameters are \(4\) and \(3\). In theory, we get \(a_{1}=(0,1)\) and \(a_{2}=(1,0)\). Then, following the procedure of the first simulation, we get \begin{tabular}{l c c c c c} \hline \hline \(n=50\) & Outliers = 0 & Time & Outliers = 2 & Time \\ \hline Relative Entropy & \((0.10,0.83)\)\((1.13,0.11)\) & 30 mn & \((0.1,0.8)\)\((0.80,0.024)\) & 43 mn \\ \(\chi^{2}\)-divergence & \((0,0.8)\)\((1.021,0.09)\) & 22 mn & \((0.12,0.79)\)\((0.867,-0.104)\) & 31 mn \\ Hellinger distance & \((0.1,0.9)\)\((0.91,0.15)\) & 35 mn & \((0.1,0.85)\)\((0.81,0.14)\) & 46 mn \\ \hline \hline \(n=100\) & Outliers = 0 & Time & Outliers = 5 & Time \\ \hline Relative Entropy & \((0.09,0.89)\)\((1.102,0.089)\) & 50 mn & \((0.1,0.88)\)\((1.15,0.144)\) & 60 mn \\ \(\chi^{2}\)-divergence & \((0,0.9)\)\((0.97,-0.1)\) & 43 mn & \((-0.1,0.9)\)\((0.87,0.201)\) & 52 mn \\ Hellinger distance & \((0.1,0.91)\)\((0.93,-0.11)\) & 57 mn & \((-0.05,1.1)\)\((0.79,0.122)\) & 62 mn \\ \hline \hline \(n=500\) & Outliers = 0 & Time & Outliers = 25 & Time \\ \hline Relative Entropy & \((0,1.07)\)\((1.1,-0.05)\) & 107 mn & \((0.13,0.75)\)\((0.79,0.122)\) & 121 mn \\ \(\chi^{2}\)-divergence & \((0,0.95)\)\((1.12,-0.02)\) & 91 mn & \((0.15,0.814\)\((0.922,0.147)\) & 103 mn \\ Hellinger distance & \((-0.01,0.95)\)\((1.01,-0.073)\) & 100 mn & \((-0.17,1.3)\)\((0.973,0.206)\) & 126 mn \\ \hline \hline \end{tabular}   
\(\mathcal{R}emark\) **6.1**. :  
(1) _We have worked with a calculator presenting the following characteristics :_  
_- Processor : Mobile AMD 3000+,_  
_- Memory RAM : 512 DDR,_  
_- Windows XP._  
(2) _Our method, which uses the_ \(\chi^{2}\) _as_ \(\phi\)_-divergence, is faster and its performance is as good if not better than any other divergence method._  
_This results from the fact that the projection index (or criteria) of_ \(\chi^{2}\) _is a second degree polynomial. It is consequently easier and faster to assess. Moreover, these simulations illustrate the robustness of our method._