A Class of Location-Invariant Extreme Value Index Estimators

PU Yuyao; CHEN Shouquan

doi:10.13718/j.cnki.xsxb.2023.04.010

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In this paper, we propose a class of location-invariant heavy-tailed extreme value estimators $ \hat{\gamma}_n\left(k_0, k, r\right)=\frac{\frac{1}{k_0} \sum\nolimits_{i=1}^{k 0}\left(\frac{1}{r}\left(\left(\frac{X_{n-i, n}-X_{n-k, n}}{X_{n-k 0, n}-X_{n-k, n}}\right)^r-1\right)\right)}{1+\frac{r}{k_0} \sum\nolimits_{i=1}^{k 0}\left(\frac{1}{r}\left(\left(\frac{X_{n-i, n}-X_{n-k, n}}{X_{n-k 0, n}-X_{n-k, n}}\right)^r-1\right)\right)} $ where \lt i \gt γ \lt /i \gt \gt 0, \lt i \gt k \lt /i \gt \lt sub \gt 0 \lt /sub \gt is a positive integer less than \lt i \gt k \lt /i \gt . The weak conjunction and asymptotic normality of this location invariant extreme value estimator are obtained, and the optimal choice of \lt i \gt k \lt /i \gt \lt sub \gt 0 \lt /sub \gt is obtained according to its asymptotic expansion.

Corresponding author: CHEN Shouquan

Received Date: 02/03/2022

Available Online: 20/04/2023

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Abstract: In this paper, we propose a class of location-invariant heavy-tailed extreme value estimators $ \hat{\gamma}_n\left(k_0, k, r\right)=\frac{\frac{1}{k_0} \sum\nolimits_{i=1}^{k 0}\left(\frac{1}{r}\left(\left(\frac{X_{n-i, n}-X_{n-k, n}}{X_{n-k 0, n}-X_{n-k, n}}\right)^r-1\right)\right)}{1+\frac{r}{k_0} \sum\nolimits_{i=1}^{k 0}\left(\frac{1}{r}\left(\left(\frac{X_{n-i, n}-X_{n-k, n}}{X_{n-k 0, n}-X_{n-k, n}}\right)^r-1\right)\right)} $ where \lt i \gt γ \lt /i \gt \gt 0, \lt i \gt k \lt /i \gt \lt sub \gt 0 \lt /sub \gt is a positive integer less than \lt i \gt k \lt /i \gt . The weak conjunction and asymptotic normality of this location invariant extreme value estimator are obtained, and the optimal choice of \lt i \gt k \lt /i \gt \lt sub \gt 0 \lt /sub \gt is obtained according to its asymptotic expansion.

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设{X_n，n≥1}是一列独立同分布的随机变量序列，其共同的分布函数为F(x)，X_1，n≤…≤X_n，n为X₁，…，X_n的顺序统计量. 如果F属于极值吸引场^[1]：

其中$ \gamma \in \mathbb{R}, 1+\gamma x \geqslant 0$. 这意味着，如果存在规范化常数$a_n>0, b_n \in \mathbb{R} $，使得当$n \rightarrow \infty $，对所有$x \in \mathbb{R} $，都有

则可以记为F∈D(G_γ) ^[2]，这等价于$ U(t):=F^{\leftarrow}\left(1-\left(\frac{1}{t}\right)\right)$是指数为γ的正则变化函数. 本文主要讨论极值指数γ>0的分布函数，即分布函数为重尾分布函数. 对所有的x>0，

对于极值指数的研究，当γ>0时，文献[3]最早提出了著名的Hill估计量. 也有一些基于不同思想的估计量：将样本分块，在每个块中取两个最大值的比率^[4-6]，然后将线性函数f(x)=x而不是对数函数应用于这些比率. 文献[7]将函数族f_r(t)引入到次序统计量中，得到估计量

其中

借助文献[8]得到广义Hill估计量

本文主要讨论$ \hat{\gamma}_n(k, r)$的位置不变估计量^[9-11]，参照文献[12]中的方法，其对应的位置不变估计量的形式为

本文在二阶条件^[13]下证明其渐近性质.

1. 主要结果

在下文中，设F∈D(G_γ)，γ>0，假设存在一个函数A(t)>0，有如下二阶条件成立

由条件(2)可得

其中，对任意的中间序列k，满足

定理1  若对γ>0有二阶条件(7)成立，且当$n \rightarrow \infty $时，中间序列k满足条件(9)，则对γr < 1，有

定理2  假设二阶条件(7)成立，当$n \rightarrow \infty $，中间序列k₀和k满足$k \rightarrow \infty, k_0 \rightarrow \infty \text {, 且 } \frac{k_0}{k} \rightarrow 0 $，则对$r <\frac{1}{2 \gamma} $，渐近分布表达式为

其中

此外，当$n \rightarrow \infty $时，如果存在$ \lambda_1 \in \mathbb{R}, \lambda_2 \in \mathbb{R}$使得$\sqrt{k_0}\left(\frac{k_0}{k}\right)^\gamma \rightarrow \lambda_1, \quad \sqrt{k_0} A\left(\frac{n}{k}\right) \rightarrow \lambda_2 $，那么有

定理3  假设二阶条件(7)成立，A(t)~ct^ρ，其中ρ < 0，c≠0，令

中间序列$ k_0^{\text {opt }}$是使得$ AMSE\left(\hat{\gamma}_n\left(k_0, k, r\right)\right)$最小的k₀：

(ⅰ) 如果$ \gamma \leqslant-\rho \text {, 则 } k_0^{\mathrm{opt}}=k_0^{(1)}$；

(ⅱ) 如果γ≥-ρ，

(a) 若$ k \ll n^{-\rho(2 \gamma+1) /(\gamma(-2 \rho+1))} \text {, 则 } k_o^{\mathrm{opt}}=k_0^{(1)}$；

(b) $ k \gg n^{-\rho(2 \gamma+1) /(\gamma(-2 \rho+1))}$，若c < 0, 则$k_0^{\mathrm{opt}}=k_0^{(3)} $，若c>0, 则$k^{\mathrm{opt}}=k_0^{(2)} $；

(c) 若$ k \sim D n^{-\rho(2 \gamma+1) /(\gamma(-2 \rho+1))}, D \neq 0$，那么$k_0^{\text {opt }} \sim D_1 n^{\frac{-2 \rho}{-2 \rho+1}}, D_1 \equiv D_1(\gamma, \rho, r, c, D) $满足

其中

2. 定理的证明

设{Y_n，n≥1}是分布函数为$ F_Y(y)=1-\frac{1}{y}$，y≥1的独立同分布的随机变量序列，Y_1，n≤…≤Y_n，n为Y₁，…，Y_n的顺序统计量，对于任意中间序列k，由于$ \left\{X_i\right\}_{i=1}^n \stackrel{d}{=}\left\{U\left(Y_i\right)\right\}_{i=1}^n \text { 且 } \frac{n}{k} Y_{n-k, n} \stackrel{p}{\rightarrow} 1, $，由Renyi’s表达式^[14]可得

此外，对于序列{Y_n，n≥1}，有

定理1的证明

最后一步由大数定律可得. 又因为

所以，

其中$ r <\frac{1}{\gamma}$.

定理2的证明 首先定义

利用泰勒展式，有

成立. 因此可得

由此可得

那么

这就得到$ \hat{\gamma}_n\left(k_0, k, r\right)$的渐近分布表达式，由此可得$ \hat{\gamma}_n\left(k_0, k, r\right)$的渐近正态性(14). 证毕.

定理3的证明 类似文献[15]中的定理2.3可得.

Reference (15)

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[1]	JENKINSON A F. The Frequency Distribution of the Annual Maximum (or Minimum) Values of Meteorological Elements [J]. Quarterly Journal of the Royal Meteorological Society, 1955, 81(348): 158-171. doi: 10.1002/qj.49708134804 CrossRef Google Scholar
[2]	RESNICK S. Extreme Values, Regular Variation, and Point Processes [M]. Colorado: Springer, 1987. Google Scholar
[3]	HILL B M. A Simple General Approach to Inference about the Tail of a Distribution [J]. The Annals of Statistics, 1975, 3(5): 1163-1174. Google Scholar
[4]	DE HAAN L, FERREIRA A. Extreme Value Theory [M]. New York: Springer, 2006. Google Scholar
[5]	王嫣然, 彭作祥. 基于分块的重尾指数估计量的推广[J]. 西南师范大学学报(自然科学版), 2019, 44(1): 25-28. Google Scholar
[6]	胡爽, 彭作祥. 基于分块思想的Pickands型估计量[J]. 西南大学学报(自然科学版), 2019, 41(5): 53-58. Google Scholar
[7]	PAULAUSKAS V. A New Estimator for a Tail Index [J]. Acta Applicandae Mathematica, 2003, 79(1-2): 55-67. Google Scholar
[8]	PAULAUSKAS V, VAIČIULIS M. On an Improvement of Hill and some other Estimators [J]. Lithuanian Mathematical Journal, 2013, 53(3): 336-355. doi: 10.1007/s10986-013-9212-x CrossRef Google Scholar
[9]	ALVES M I F. A Location Invariant Hill-Type Estimator [J]. Extremes, 2001, 4(3): 199-217. doi: 10.1023/A:1015226104400 CrossRef Google Scholar
[10]	LING C X, PENG Z X, NADARAJAH S. A Location Invariant Moment-Type Estimator. I [J]. Theory of Probability and Mathematical Statistics, 2008, 76: 23-31. doi: 10.1090/S0094-9000-08-00728-X CrossRef Google Scholar
[11]	FRAGA ALVES M I, GOMES M I, HAAN L, et al. Mixed Moment Estimator and Location Invariant Alternatives [J]. Extremes, 2009, 12(2): 149-185. doi: 10.1007/s10687-008-0073-3 CrossRef Google Scholar
[12]	LI J N, PENG Z X, NADARAJAH S. Asymptotic Normality of Location Invariant Heavy Tail Index Estimator [J]. Extremes, 2010, 13(3): 269-290. doi: 10.1007/s10687-009-0088-4 CrossRef Google Scholar
[13]	DE HAAN L, STADTMÜLLER U. Generalized Regular Variation of Second Order [J]. Journal of the Australian Mathematical Society Series A Pure Mathematics and Statistics, 1996, 61(3): 381-395. Google Scholar
[14]	RÉNYI A. On the Theory of Order Statistics [J]. Acta Mathematica Academiae Scientiarum Hungarica, 1953, 4(3-4): 191-231. doi: 10.1007/BF02127580 CrossRef Google Scholar
[15]	LI J N, PENG Z X, NADARAJAH S. A Class of Unbiased Location Invariant Hill-Type Estimators for Heavy Tailed Distributions [J]. Electronic Journal of Statistics, 2008(2): 829-847. Google Scholar

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A Class of Location-Invariant Extreme Value Index Estimators

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