Let $X_{1,n} \leq .... \leq X_{n,n}$ be the order statistics associated with a sample $X_{1}, ...., X_{n}$ whose pertaining distribution function (\textit{df}) is $F$. We are concerned with the functional asymptotic behaviour of the sequence of stochastic processes

 

\begin{equation}
T_{n}(f,s)=\sum_{j=1}^{j=k}f(j)\left( \log X_{n-j+1,n}-\log
X_{n-j,n}\right)^{s} ,  \label{fme}
\end{equation}

 

indexed by some classes $\mathcal{F}$ of functions $f:\mathbb{N}%^{\ast}\longmapsto \mathbb{R}_{+}$ and $s \in ]0,+\infty[$ and where $k=k(n)$ satisfies

 

\begin{equation*}
1\leq k\leq n,k/n\rightarrow 0\text{ as }n\rightarrow \infty .
\end{equation*}

 

We show that this is a stochastic process whose margins generate estimators of the extreme value index when $F$ is in the extreme domain of attraction. We focus in this paper on its finite-dimension asymptotic law and provide a class of new estimators of the extreme value index whose performances are compared to analogous ones. The results are next particularized for one explicit class $\mathcal{F}$.