论文标题
限制极端样本规范的法律
Limit laws for the norms of extremal samples
论文作者
论文摘要
令表示$ s_n（p）= k_n^{ - 1} \ sum_ {i = 1}^{k_n} \ left（\ log（x_ {x_ {n+1-i，n} / n} / x_ {n-k_n，n，n，n}） \infty$ and $k_n / n \to 0$, and $X_{1,n} \leq \ldots \leq X_{n,n}$ is the order statistics of iid random variables with regularly varying upper tail.估计器$ \wideHatγ（n）=（s_n（p）/γ（p+1））^{1/p} $是山丘估计器的扩展。我们调查了$ s_n（p）$和$ \wideHatγ（n）$的渐近属性，均为固定的$ p> 0 $，对于$ p = p_n \ to \ infty $。在适当的假设下，我们证明了强大的一致性和渐近正态性。应用于真实数据，我们发现，对于更大的$ p $，估算器对$ k_n $的变化的敏感性不如山丘估计器。
Let denote $S_n(p) = k_n^{-1} \sum_{i=1}^{k_n} \left( \log (X_{n+1-i,n} / X_{n-k_n, n}) \right)^p$, where $p > 0$, $k_n \leq n$ is a sequence of integers such that $k_n \to \infty$ and $k_n / n \to 0$, and $X_{1,n} \leq \ldots \leq X_{n,n}$ is the order statistics of iid random variables with regularly varying upper tail. The estimator $\widehat γ(n) = (S_n(p)/Γ(p+1))^{1/p}$ is an extension of the Hill estimator. We investigate the asymptotic properties of $S_n(p)$ and $\widehat γ(n)$ both for fixed $p > 0$ and for $p = p_n \to \infty$. We prove strong consistency and asymptotic normality under appropriate assumptions. 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.
