## 4 EVI-Estimation under Random Censorship  
In all articles mentioned in Section 3, the available sample is complete. Let us now assume that we are under a framework of random censorship, i.e., we have access to the sample \((Z_{i},\delta_{i})\), \(1\leq i\leq n\), of independent copies of \((Z,\delta)\), in (3), but our goal is to make inference on the right tail of the unknown lifetime distribution \(F_{X}\), i.e., we want to make inference on the right tail,  
\[\overline{F}_{X}(x)=\mathbb{P}(X>x)=1-F_{X}(x),\]  
while \(F_{Y}\), the CDF of \(Y\), is considered to be a nonparametric nuisance parameter. As mentioned in [4], among others, all EVI-estimators need to then be modified in order to be consistent for the estimation of \(\xi_{X}\) in the whole domain of attraction, \(\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{X}}\big{)}\). We shall next motivate a possible and simple modification. _A Few Comments on Random Censoring_  
Let us assume that \(F_{X}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{1}}\big{)}\), with \(\xi_{1}\equiv\xi_{X}\) as the parameter we want to estimate, and that \(F_{Y}\in\mathcal{D}_{\mathcal{M}}\big{(}G_{\xi_{2}}\big{)},\xi_{1},\xi_{2}>0\) (heavy right tails). Also, instead of the potential sample \((X_{1},\ldots,X_{n})\) (non-observed), and assuming \(Y\) is independent of \(X\), we observe  
\[\bigg{(}\left( \begin{array}{c}Z_{1}\\ \delta_{1}\end{array} \right),\ldots,\left( \begin{array}{c}Z_{n}\\ \delta_{n}\end{array} \right)\bigg{)},\quad Z=\min(X,Y),\quad\delta=I_{\{X\leq Y \}}. \tag{13}\]  
**Example 1**. : _To motivate how to estimate \(\xi_{1}=\xi_{X}\), we shall simplify the problem, assuming that \(X\) and \(Y\) are indeed Pareto \((\xi_{1})\) and Pareto \((\xi_{2})\), respectively, i.e., for all \(x\geq 1\), \(F_{X}(x)=1-x^{-1/\xi_{1}}\) and \(F_{Y}(x)=1-x^{-1/\xi_{2}}\). Then,_  
\[F_{Z}(x)=\mathbb{P}\{\min(X,Y)\leq x\}=1-x^{-1/\xi_{1}}x^{-1/\xi_{2}}=1-x^{- \frac{\xi_{1}+\xi_{2}}{\xi_{1}\xi_{2}}},\]  
_i.e., \(Z\frown Pareto(\xi_{1}\xi_{2}/(\xi_{1}+\xi_{2}))\), and we can use the common semi-parametric estimators to estimate \(\xi_{Z}=\xi_{1}\xi_{2}/(\xi_{1}+\xi_{2})\). On the other hand,_  
\[ \begin{array}{rcl}c\equiv c_{z}:=\mathbb{P}(X\leq Y|Z=z)&=&\mathbb{P}(\delta =1|Z=z)=\frac{f_{X}(z)(1-F_{Y}(z))}{f_{Z}(z)}\\ &=&\frac{\frac{1}{\xi_{1}}z^{-1/\xi_{1}}z^{-1/\xi_{2}}}{\left(\frac{1}{\xi_{1 }}+\frac{1}{\xi_{2}}\right)z^{-1/\xi_{1}-1/\xi_{2}-1}}=\frac{\xi_{2}}{\xi_{1 }+\xi_{2}}.\end{array}  \tag{14}\]  
_Then, the quotient between any estimator of \(\xi=\xi_{Z}\) and an estimator of \(c=c_{Z}\) will provide an estimator of \(\xi_{X}=\xi_{1}\)._  
**Example 2**. : _Another key model in EVI is the generalized Pareto (GP) model, associated with excesses over a high threshold (see [PERSON] [31] and [PERSON] [32]), and given by \(GP_{\xi}(x):=1+\ln{\rm G}EV_{\xi}(x)\), with \({\rm G}EV_{\xi}(x)\) defined in (4). The ML-estimation of \((\xi,\sigma)\) for a model \(GP_{\xi}^{*}(x/\sigma)\), \(\sigma>0\), is numerically hardly tractable. ML estimates for the regular-case \((\xi>-1/2)\) and \(\sigma>0\), in \(GP_{\xi}(x/\sigma)\), have been studied in [PERSON] [33], where the POT approach was introduced, with POT standing for peaks-over-threshold. A survey of the POT methodology, together with several applications, can be found in [PERSON] and [PERSON] [34]. Under random censoring, let us assume that for all \(x\geq 0\), \(F_{X}(x)=1-(1+\xi_{1}x)^{-1/\xi_{1}}\) and \(F_{Y}(x)=1-(1+\xi_{2}x)^{-1/\xi_{2}}\). We then obtain_  
\[F_{Z}(z)=1-(1+\xi_{1}z)^{-1/\xi_{1}}(1+\xi_{2}z)^{-1/\xi_{2}},\]\[f_{Z}(z)=(1+\xi_{1}z)^{-1/\xi_{1}-1}(1+\xi_{2}z)^{-1/\xi_{2}}\bigg{(}1+\frac{1+ \xi_{1}z}{1+\xi_{2}z}\bigg{)},\]  
_and_  
\[