更改

其中<math>X_{(i,n)}</math>是<math>X_1, \dots, X_n</math>的第<math>i</math>次序统计量。该估计量依概率收敛于<math>\xi</math>，并且在基于高阶的正则变化性质的情况下，是限制<math>k(n) \to \infty  </math>的渐近正态<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>。一致性和渐近正态性适用于一大类相关序列和异类序列<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>，而不管是否观测到<math>X_t</math>，或者来自大量模型和估计量（包括错误指定的模型和具有相关误差的模型）计算出的残差或筛选数据。<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>的第<math>i</math>次序统计量。该估计量随着<math>k(n) \to \infty  </math>依概率收敛于<math>\xi</math>，并且在高阶的正则变化性质的限制条件下，是渐近正态的。<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>。一致性和渐近正态性适用于一大类相关序列和异类序列<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>，不管<math>X_t</math>是否被观测到，或者是来自大量模型和估计量（包括错误指定的模型和具有相关误差的模型）计算出的残差或筛选数据。<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>

===尾部指数的比率估计量 ===

===尾部指数的比率估计量 ===