## 5 Monte-Carlo Simulation  
We have considered \(X\in\mathcal{D}_{\mathcal{M}}\big{(}EV_{\xi_{1}}\big{)}\), with \(\xi_{1}\equiv\xi_{X}=0.25\) censored by \(Y\in\mathcal{D}_{\mathcal{M}}\big{(}EV_{\xi_{2}}\big{)}\), with \(\xi_{2}=\xi_{1}p/(1-p)\), \(p=0.35(0.10)0.95\). Then, \(Z\in\mathcal{D}_{\mathcal{M}}\big{(}EV_{\xi}\big{)}\), with \(\xi\equiv\xi_{Z}=\xi_{1}\xi_{2}/(\xi_{1}+\xi_{2})\). Consequently, we allow for a percentage of censoring in the right tail equal to values that change from \(65\%\) to \(5\%\). For heavy right tails, we have performed a large-scale Monte Carlo simulation for different parents. Here, we consider the following:  
* \(X\) and \(Y\) both come from Frechet models (the CDF of a Frechet(\(\xi\)) model is \(F(x)=\exp\Bigl{(}-x^{-1/\xi}\Bigr{)}\), \(x\geq 0\), \(\xi>0\) (\(\rho=-1\) and \(\beta=1/2\))). * \(X\) and \(Y\) both come from Burr models (the CDF of a Burr(\(\xi,\rho\)) model is \(F(x)=1-\ \bigl{(}1+x^{-\rho/\xi}\bigr{)}^{1/\rho}\), \(x\geq 0\), \(\xi>0\), \(\rho<0\) and \(\beta=1\)). * \(X\) and \(Y\) both come from GP models (the CDF of a GP(\(\xi\)) model is given in Example 2 (\(\rho=-\xi\) and \(\beta=1\))). For simplicity, the GM EVI-estimators in (9), (A1) and (A2) will be denoted by PM, MO and L, respectively, and the RB GM EVI-estimators in (A3), (A4) and (A5) by RBPM, RBMO and RBL, respectively. In Figures 1 and 2, for a sample size \(n=1000\), we can see the mean values and the root-mean-squared errors (RMSEs) of some of the estimators under study when we consider  
\[X\!\sim\!\text{Frechet}(\xi_{1}\!\equiv\xi_{X}=0.25)\ \text{ censored by}\ \ Y\!\sim\!\text{Frechet}(\xi_{2}=0.75).\]  
We then obtain for the observed \(Z\) a tail index \(\xi\equiv\xi_{Z}=0.1875\). The censoring in the right tail is thus equal to \(25\%\). Even for sample sizes as small as \(n=100\), from the same parents, the results are still interesting, as can be seen in Figures 3 and 4. This generally happens if the censoring in the right tail is not too heavy (smaller than or equal to 25%, for example). Figure 1: Sample paths of \(\xi_{X}(k)\) (**on the left**) and of \(\xi_{Z}(k)\) (**on the right**) for samples of size 1000 from the Fréchet parents under study. Figure 4: RMSEs of \(\xi_{X}(k)\) (**on the left**) and of \(\xi_{Z}(k)\) (**on the right**) for samples of size 100 from the Fréchet parents under study. Figure 2: RMSEs of \(\xi_{X}(k)\) (**on the left**) and of \(\xi_{Z}(k)\) (**on the right**) for samples of size 1000 from the Fréchet parents under study. The sample paths and MSE of \(\delta_{k,n}\) are presented in Figure 5. From this figure, we can infer that, for both sample sizes, the bias remains very small, up to 70% of the sample size. Figures 6 and 7 depict, for a sample size of \(n=1000\), the mean values and RMSEs of the estimators under study when we consider  
\[X\sim\text{Burr}(\xi_{1}\equiv\xi_{X}=0.25,\rho=-1)\,\text{ censored by }\,Y\sim\text{Burr}(\xi_{2}=0.464,\rho=-1),\]  
and, in this case, we obtain for the observed \(Z\) a tail index \(\xi\equiv\xi_{Z}=0.1625\). The censoring in the right tail is thus equal to 35%. For the Burr models, the estimates \(\hat{c}_{k,n}\) in (16) exhibit patterns similar to the ones presented in Figure 5 and are presented in Figure 8. As expected, when the censoring in the right tail increases, the \(k\)-region where the bias remains very small decreases, and in this case, it is up to 55% of the sample size. Figure 5: Sample paths of \(\hat{c}_{k,n}\) based on 1000 replicates of samples of size 100 (**on the left**) and samples of size 1000 (**on the right**) from the [PERSON] parents under study. Figure 6: Sample paths of \(\hat{c}_{X}(k)\) (**on the left**) and of \(\hat{c}_{Z}(k)\) (**on the right**) for samples of size 1000 from the [PERSON]\((\xi,\rho=-1)\) parents under study. Figure 7: RMSEs of \(\hat{c}_{X}(k)\) (**on the left**) and of \(\hat{c}_{Z}(k)\) (**on the right**) for samples of size 1000 from the [PERSON]\((\xi,\rho=-1)\) parents under study.