Denote $S_n\left(p\right)=k_n^{-1}{\sum }_{i=1}^{k_n}{\left(log(X_{n+1-i,n}∕X_{n-k_n,n})\right)}^p$, where $p>0$, $k_n\le n$ is a sequence of integers such that $k_n\to \infty$ and $k_n∕n\to 0$, and $X_{1,n}\le \cdots \le X_{n,n}$ are the order statistics of iid random variables $X_1,\dots ,X_n$ with regularly varying upper tail of index $1∕\gamma$. The estimator $\widehat{\gamma }\left(n\right)={(S_n\left(p\right)∕\Gamma (p+1))}^{1∕p}$ is an extension of the Hill estimator. We investigate the asymptotic properties of $S_n\left(p\right)$ and $\widehat{\gamma }\left(n\right)$ both for fixed $p>0$ and for $p=p_n\to \infty$. We prove consistency for $\widehat{\gamma }\left(n\right)$ and limit theorem for $\widehat{\gamma }\left(n\right)-\gamma$ under appropriate assumptions. We obtain both Gaussian and non-Gaussian (stable) limit depending on the growth rate of the power sequence $p_n$. Applied to real data we find that for larger $p$ the estimator is less sensitive to the change in $k_n$ than the Hill estimator.

表示 ${\mathrm{S。}}_n\left(磷\right)={克}_n^{——1}{\sum }_{\mathrm{一世}=1}^{{克}_n}{\left(日志(X_{n+1——\mathrm{一世},n}∕X_{n——{克}_n,n})\right)}^{磷}$， 在哪里 $磷>0$, ${克}_n\le n$ 是一个整数序列，使得 ${克}_n\to \infty$ 其他 ${克}_n∕n\to 0$， 和 $X_{1,n}\le \cdots \le X_{n,n}$ 是 iid 随机变量的阶次统计 $X_1,...,X_n$ 索引的上尾有规律地变化 $1∕\gamma$. 估算器$\widehat{\gamma }\left(n\right)={({\mathrm{S。}}_n\left(磷\right)∕\Gamma (磷+1))}^{1∕磷}$是希尔估计量的扩展。我们研究了渐近性质${\mathrm{S。}}_n\left(磷\right)$ 其他 $\widehat{\gamma }\left(n\right)$ 两者都是固定的 $磷>0$ 并为 $磷={磷}_n\to \infty$. 我们证明一致性$\widehat{\gamma }\left(n\right)$ 和极限定理 $\widehat{\gamma }\left(n\right)——\gamma$在适当的假设下。我们根据幂序列的增长率获得高斯和非高斯（稳定）极限${磷}_n$. 应用于实际数据，我们发现对于更大的$磷$ 估计量对变化不太敏感 ${克}_n$ 比 Hill 估计量。