## 4 EVI-Estimation under Random Censorship
\begin{array}{rcl}c\equiv c_{z}:&=&\mathbb{P}(X\leq Y|Z=z)\\ &=&\frac{(1+\xi_{1}z)^{-\frac{1}{\xi_{1}}-1}(1+\xi_{2}z)^{-\frac{1}{\xi_{2}}-1 }}{(1+\xi_{1}z)^{-\frac{1}{\xi_{1}}-1}(1+\xi_{2}z)^{-\frac{1}{\xi_{2}}-1}\big{(} 1+\frac{1+\xi_{1}z}{1+\xi_{2}z}\big{)}}\\ &=&\frac{1}{1+\frac{1+\xi_{2}z}{1+\xi_{2}z}}\underset{z\to 0}{\longrightarrow} \frac{\xi_{2}}{\xi_{1}+\xi_{2}}.\end{array} \]  
The abovementioned limiting result, at the end of Example 2, holds generally, for any \(F_{X}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{1}}\big{)}\), \(F_{Y}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{2}}\big{)}\). Indeed, under the above general setting, in (13), and as can be seen in [4], it is possible to prove that  
\[F_{X}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{1}}\big{)},\ F_{Y}\in\mathcal{D }_{\mathcal{M}}\big{(}G_{\xi_{2}}\big{)}\ \Longrightarrow\ F_{Z=\min(X,Y)}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi} \big{)},\ \text{with}\ \ \xi=\frac{\xi_{1}\xi_{2}}{\xi_{1}+\xi_{2}}. \tag{15}\]  
Note that  
\[ \begin{array}{rcl}F_{X}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{1}}\big{)},\ F_{Y}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{2}}\big{)},\ \ \ \xi_{1},\xi_{2}>0\ \ \Longleftrightarrow\\ 1-F_{X}(x)=x^{-1/\xi_{1}}\mathbb{L}_{X}(x)\ \ \ \text{and}\ \ \ 1-F_{Y}(y)=y^{-1/\xi_{2}}\mathbb{L}_{Y}(x)\end{array} \]  
are Pareto-type right tails, already mentioned in (2). Since the CDF of \(Z=\min(X,Y)\) is \(F_{Z}(z)=1-(1-F_{X}(z))(1-F_{Y}(z))\), we obtain  
\[1-F_{Z}(z)=z^{-(1/\xi_{1}+1/\xi_{2})}\mathbb{L}_{X}(z)\mathbb{L}_{Y}(z)=:z^{- \frac{\xi_{1}+\xi_{2}}{\xi_{1}\xi_{2}}}\,L_{Z}(z),\]  
Consequently, any functional devised for the semi-parametric estimation of the tail index in complete samples associated with the observed sample \((Z_{1},Z_{2},\ldots,Z_{n})\) and based on the \(k+1\) upper OS in the sample, generically denoted as \(\xi_{k,n,Z}^{\bigstar\star}\), will converge towards \(\xi_{Z}=\xi_{1}\xi_{2}/(\xi_{1}+\xi_{2})\) in (15) for intermediate \(k\), i.e., whenever (12) holds. Moreover, and for the same intermediate sequences \(k\),  
\[\frac{\xi_{k,n,Z}^{\bigstar\star}}{c}\ \underset{n\to\infty}{\longrightarrow}\ \ \xi_{1}\equiv\xi_{X}.\]  
Consequently, we only need to find an estimator of \(c=c_{Z}\) in (14) with \(1-c\), the percentage of censoring in the right tail of \(F_{X}\). A possible semi-parametric consistent estimator of \(c\) (for intermediate \(k\)) is  
\[\hat{c}_{k,n}:=\frac{1}{k}\sum_{i=1}^{k}\delta_{[n-i+1]}, \tag{16}\]  
where \(\delta_{[n-i+1]},\ 1\leq i\leq n\) are the induced or concomitant OS associated with the ascending ordering of \((Z_{1},Z_{2},\ldots,Z_{n})\), i.e., \(\delta_{[n-i+1]}\) is the concomitant value of \(\delta\) associated with \(Z_{n-i+1:n}\), \(1\leq i\leq n\). An obvious semi-parametric estimator of \(\xi_{X}\), based on the observed \((Z,\delta)\), is thus  
\[\hat{\xi}_{k,n,X|Z,\delta}^{\bigstar\star}=\frac{k\ \hat{\xi}_{k,n,Z}^{\bigstar \star}}{\sum_{i=1}^{k}\delta_{[n-i+1]}}, \tag{17}\]  
an estimator already studied in [6; 7], for a few estimation procedures. The final estimate associated with \(\hat{\xi}_{k,n,X|Z,\delta}^{\bigstar\star}\) will be computed using Algorithm 4.5 in [29], which we next summarize:  
(i) Given the observed sample \((Z_{1},\ldots,Z_{n})\) and \((\delta_{1},\ldots,\delta_{n})\), compute, for \(k=1,\ldots,n-1\), the observed values of \(\hat{\xi}_{k,n,X|Z,\delta}^{\bigstar\star}\), in (17). 2. Obtain \(j_{0}\), the minimum value of \(j\), a non-negative integer, such that the rounded values, to \(j\) decimal places, of the estimates in (i) are distinct. Define \(a_{k}^{(\xi)}(j)=round\Big{(}\frac{\xi^{\bigstar}}{\xi_{k,n,X|Z,\delta},j}\Big{)}\), \(k=1,2,...,n-1\), the rounded values of \(\xi_{k,n,X|Z,\delta}^{\bigstar}\) to \(j\) decimal places;
3. Consider the sets of \(k\) values associated with equal consecutive values of \(a_{k}^{(\xi)}(j)\), obtained in (ii). Set \(k_{min}^{(\xi)}\) and \(k_{max}^{(\xi)}\) the minimum and maximum values, respectively, of the set with the largest range.