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第269行： + + + + + + − 对于<math>(X_n , n \geq 1)</math>的独立且相同密度函数<math>F \in D(H(\xi))</math>的随机序列，广义极值密度<math> H </math>的最大吸引域，其中<math>\xi \in \mathbb{R}</math>。如果<math>\lim_{n\to\infty} k(n) = \infty </math>和<math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>，则Pickands尾部指数估计为+
→Pickand's tail-index estimator Pickand的尾指数估算器
With <math>(X_n , n \geq 1)</math> a random sequence of independent and same density function <math>F \in D(H(\xi))</math>, the Maximum Attraction Domain<ref name=Pickands>{{cite journal|last=Pickands III|first=James|title=Statistical Inference Using Extreme Order Statistics|journal=The Annals of Statistics|date=Jan 1975|volume=3|issue=1|pages=119–131|jstor=2958083|doi=10.1214/aos/1176343003|doi-access=free}}</ref> of the generalized extreme value density <math> H </math>, where <math>\xi \in \mathbb{R}</math>. If <math>\lim_{n\to\infty} k(n) = \infty </math> and <math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>, then the ''Pickands'' tail-index estimation is<ref name="Embrechts"/><ref name="Pickands"/>
With <math>(X_n , n \geq 1)</math> a random sequence of independent and same density function <math>F \in D(H(\xi))</math>, the Maximum Attraction Domain<ref name=Pickands>{{cite journal|last=Pickands III|first=James|title=Statistical Inference Using Extreme Order Statistics|journal=The Annals of Statistics|date=Jan 1975|volume=3|issue=1|pages=119–131|jstor=2958083|doi=10.1214/aos/1176343003|doi-access=free}}</ref> of the generalized extreme value density <math> H </math>, where <math>\xi \in \mathbb{R}</math>. If <math>\lim_{n\to\infty} k(n) = \infty </math> and <math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>, then the ''Pickands'' tail-index estimation is<ref name="Embrechts"/><ref name="Pickands"/>
对于<math>(X_n , n \geq 1)</math>的独立且相同密度函数<math>F \in D(H(\xi))</math>的随机序列，广义极值密度<math> H </math>的最大吸引域，其中<math>\xi \in \mathbb{R}</math>。如果<math>\lim_{n\to\infty} k(n) = \infty </math>和<math>\lim_{n\to\infty} \frac{k(n)}{n}= 0</math>，则Pickands尾部指数估计为
:<math>
:<math>
\xi^\text{Pickands}_{(k(n),n)} =\frac{1}{\ln 2} \ln \left( \frac{X_{(n-k(n)+1,n)} - X_{(n-2k(n)+1,n)}}{X_{(n-2k(n)+1,n)} - X_{(n-4k(n)+1,n)}}\right)
\xi^\text{Pickands}_{(k(n),n)} =\frac{1}{\ln 2} \ln \left( \frac{X_{(n-k(n)+1,n)} - X_{(n-2k(n)+1,n)}}{X_{(n-2k(n)+1,n)} - X_{(n-4k(n)+1,n)}}\right)
</math>
</math>
where <math>X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)</math>. This estimator converges in probability to <math>\xi</math>.
where <math>X_{(n-k(n)+1,n)}=\max \left(X_{n-k(n)+1},\ldots ,X_{n}\right)</math>. This estimator converges in probability to <math>\xi</math>.
其中<math> X _ {（n-k（n）+ 1，n）} = \ max \ left（X_ {n-k（n）+1}，\ ldots，X_ {n} \ right）</ math>。 此估计量的概率收敛到<math> \ xi </ math>。
=== Hill's tail-index estimator ===
=== Hill's tail-index estimator ===