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→Hill's tail-index estimator
Let <math>(X_t , t \geq 1)</math> be a sequence of independent and identically distributed random variables with distribution function <math>F \in D(H(\xi))</math>, the maximum domain of attraction of the [[generalized extreme value distribution]] <math> H </math>, where <math>\xi \in \mathbb{R}</math>. The sample path is <math>{X_t: 1 \leq t \leq n}</math> where <math>n</math> is the sample size. If
Let <math>(X_t , t \geq 1)</math> be a sequence of independent and identically distributed random variables with distribution function <math>F \in D(H(\xi))</math>, the maximum domain of attraction of the [[generalized extreme value distribution]] <math> H </math>, where <math>\xi \in \mathbb{R}</math>. The sample path is <math>{X_t: 1 \leq t \leq n}</math> where <math>n</math> is the sample size. If
<math>\{k(n)\}</math> is an intermediate order sequence, i.e. <math>k(n) \in \{1,\ldots,n-1\}, </math>, <math>k(n) \to \infty</math> and <math>k(n)/n \to 0</math>, then the Hill tail-index estimator is<ref>Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.</ref>
<math>\{k(n)\}</math> is an intermediate order sequence, i.e. <math>k(n) \in \{1,\ldots,n-1\}, </math>, <math>k(n) \to \infty</math> and <math>k(n)/n \to 0</math>, then the Hill tail-index estimator is<ref>Hill B.M. (1975) A simple general approach to inference about the tail of a distribution. Ann. Stat., v. 3, 1163–1174.</ref>
令<math>(X_t , t \geq 1)</math>为具有分布函数<math>F \in D(H(\xi))</math>独立且均匀分布的随机变量序列，其分布函数为广义极值分布<math> H </math>的最大吸引域，其中<math>\xi \in \mathbb{R}</math>。样本路径为<math>{X_t: 1 \leq t \leq n}</math>，其中<math>n</math>为样本大小。 如果<math>\{k(n)\}</math>是中间阶数序列，即<math>k(n) \in \{1,\ldots,n-1\}, </math>，<math>k(n) \to \infty</math>和<math>k(n)/n \to 0</math>，则希尔尾指数估计器为：
: <math>
: <math>
\xi^\text{Hill}_{(k(n),n)} = \left(\frac 1 {k(n)} \sum_{i=n-k(n)+1}^n \ln(X_{(i,n)}) - \ln (X_{(n-k(n)+1,n)})\right)^{-1},
\xi^\text{Hill}_{(k(n),n)} = \left(\frac 1 {k(n)} \sum_{i=n-k(n)+1}^n \ln(X_{(i,n)}) - \ln (X_{(n-k(n)+1,n)})\right)^{-1},
</math>
</math>
where <math>X_{(i,n)}</math> is the <math>i</math>-th [[order statistic]] of <math>X_1, \dots, X_n</math>.
where <math>X_{(i,n)}</math> is the <math>i</math>-th [[order statistic]] of <math>X_1, \dots, X_n</math>.
This estimator converges in probability to <math>\xi</math>, and is asymptotically normal provided <math>k(n) \to \infty </math> is restricted based on a higher order regular variation property<ref>Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.</ref>
This estimator converges in probability to <math>\xi</math>, and is asymptotically normal provided <math>k(n) \to \infty </math> is restricted based on a higher order regular variation property<ref>Hall, P.(1982) On some estimates of an exponent of regular variation. J. R. Stat. Soc. Ser. B., v. 44, 37–42.</ref>
.<ref>Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.</ref> Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,<ref>Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.</ref><ref>Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.</ref> irrespective of whether <math>X_t</math> is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.<ref>Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.</ref><ref>Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.</ref><ref>Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.</ref>
.<ref>Haeusler, E. and J. L. Teugels (1985) On asymptotic normality of Hill's estimator for the exponent of regular variation. Ann. Stat., v. 13, 743–756.</ref> Consistency and asymptotic normality extend to a large class of dependent and heterogeneous sequences,<ref>Hsing, T. (1991) On tail index estimation using dependent data. Ann. Stat., v. 19, 1547–1569.</ref><ref>Hill, J. (2010) On tail index estimation for dependent, heterogeneous data. Econometric Th., v. 26, 1398–1436.</ref> irrespective of whether <math>X_t</math> is observed, or a computed residual or filtered data from a large class of models and estimators, including mis-specified models and models with errors that are dependent.<ref>Resnick, S. and Starica, C. (1997). Asymptotic behavior of Hill’s estimator for autoregressive data. Comm. Statist. Stochastic Models 13, 703–721.</ref><ref>Ling, S. and Peng, L. (2004). Hill’s estimator for the tail index of an ARMA model. J. Statist. Plann. Inference 123, 279–293.</ref><ref>Hill, J. B. (2015). Tail index estimation for a filtered dependent time series. Stat. Sin. 25, 609–630.</ref>
其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的i阶统计量。该估计量收敛于<math>\xi</math>的概率，并且当<math>k(n) \to \infty </math>基于较高阶的正则变化性质受到限制时，它是渐近正态的。一致性和渐近正态性适用于一大类相关序列和异类序列，它与是否观测到<math>X_t</math>无关，也无关于是否从大量模型和估计量（包括错误指定的模型和具有相关误差的模型）中计算出的残差或滤波数据。
=== Ratio estimator of the tail-index ===
=== Ratio estimator of the tail-index ===