一类双指标泛函Hill过程及其应用
A double-indexed functional Hill process and applications
摘要：
令$X_{1,n} \leq .... \leq X_{n,n}$为与样本$X_{1}, ...., X_{n}$相关的顺序统计量，该样本的相关分布函数（% \textit{df}）为$F$。我们关心的功能渐近行为的随机过程序列\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}的一些类$\mathcal{F}$的功能$f:\mathbb{N}% ^{\ast}\longmapsto \mathbb{R}_{+}$和$s \in ]0,+\infty[$和$k=k(n)$满足\begin{equation*} 1\leq k\leq n,k/n\rightarrow 0\text{as}n\rightarrow \infty . \end{equation*} \noindent我们表明，这是一个随机过程的利润率产生的极值指标的估计时，$F$是在极端域的吸引力。本文着重研究了它的有限维渐近律，并给出了一类新的极值指数估计，其性能与类似的估计进行了比较。结果接下来针对一个显式类$\mathcal{F}$进行详细说明。
英文摘要：
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*} \noindent 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 $\mathcal{F}$.
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