## 1 Introduction and Preliminaries  
The primary focus of statistical extreme value theory (EVT) is the estimation of parameters related to extreme events. In EVT, a common assumption for any univariate dataset \((X_{1},\ldots,X_{n})\) is that the data form a complete sample of size \(n\), which may be independent, identically distributed, or weakly dependent and stationary, drawn from an unknown cumulative distribution function (CDF) \(F=F_{X}\). A key parameter of interest is the unknown extreme value index (EVI), denoted by \(\xi\equiv\xi\chi\), which will be defined later. The estimation of the EVI forms the foundation for estimating all other extreme value parameters. An extended literature review is provided in Appendix A, including a few books on this topic. When examining extreme events, particularly focusing on large values in the right tail, several key parameters are of interest:  
* The probability of exceedance for a high threshold \(x_{H}\), defined as \(p_{H}:=\mathbb{P}(X>x_{H})\). * A high quantile corresponding to a probability \(1-q\), where \(q\) is small, which is located at or beyond the range of the observed data. It is given by \[\chi_{1-q}:=\inf\{x:F(x)\geq 1-q\}=F^{-1}(1-q),\quad\text{where}\;\;q<1/n,\] (1) with \(F^{-1}\) representing the generalized inverse function of \(F\). The right endpoint of the underlying distribution \(F\), denoted \(x^{F}:=\sup\{x:F(x)<1\}\). In the analysis of lifetime or reliability data, observations are often censored, frequently in a random manner. In this context, we will focus specifically on estimating the extreme value index (EVI), \(\xi_{X}\), under conditions of random censoring and for a heavy Pareto-type regularly varying right tail function:  
\[\overline{F}_{X}(x):=\mathbb{P}(X>x)=1-F_{X}(x)=x^{-1/\xi_{X}}\mathbb{L}_{X}(x), \tag{2}\]  
where \(\mathbb{L}(\cdot)\) is a slowly varying function at infinity, i.e., such that \(\lim_{t\to\infty}\mathbb{L}(tx)/\mathbb{L}(t)=1\), for all \(x>0\) (see, among others, [PERSON], [PERSON] and [PERSON] [1], for details on regular and slow variation). A brief reference to the subject can be found in [PERSON] and Thomas [2], First Edition (see Section 6.1 of [2]). We further mention a few initial articles on the topic ([PERSON], [PERSON], [PERSON] and [PERSON] [3], [PERSON], [PERSON] and [PERSON] [4], [PERSON], [PERSON] and [PERSON] [5], and [PERSON] and [PERSON] [6; 7]). Additional references can be found in Appendix A. In the context of random censoring, we consider a random variable \(Y\) such that, instead of observing \(X\) directly, we only observe  
\[Z=X\wedge Y\quad\text{and}\quad\delta=\mathbb{I}_{\{X\leq Y\}}, \tag{3}\]  
where \(\mathbb{I}_{A}\) denotes the indicator function of the set \(A\). The indicator variable \(\delta\) indicates whether \(X\) has been censored. Within this framework, semi-parametric EVI-estimators require only minor adjustments to ensure consistency in estimating \(\xi_{X}\) across parametric, semi-parametric, or even nonparametric settings. We shall give special attention to such estimation for heavy tails, i.e., for \(\xi_{X}>0\), making use of several generalized means (GM) EVI-estimators (see [PERSON] and [PERSON] [8], [PERSON], [PERSON] [9] and [PERSON], [PERSON], [PERSON] and [PERSON] [10], among others), which generalize the classical Hill EVI-estimator (Hill [11]), and of one of the most simple minimum-variance reduced-bias (MVRB) estimators--the one in the work of [PERSON], [PERSON], and [PERSON] [12]. We further consider RB-GM EVI-estimators, which will be detailed in Appendix B. For additional GM EVI-estimators, see [PERSON] and [PERSON] study [13] and the recent overview article by [PERSON] [14]. After a brief introduction in Section 2 regarding the notion of EVI and the most relevant first- and second-order conditions in statistical EVT, under a semi-parametric framework, we advance, in Section 3, with the functional definition of some of the already-mentioned EVI-estimators for complete samples and a heavy right tail, i.e., a positive EVI (see also, Appendix B).