Extremes

Lesson

Introduction
Theory
Extreme value theory
Multifractal phase transition
To go further
Exercises
Conclusion

Hill estimator

The Hill estimator is given as follows:

$\hat{\xi} = \frac{1}{k-1}\sum \limits_{i=1}^{k-1} (ln X_{1:n} - ln X_{k:n}) \quad for \quad k \geq 2$

with the highest statistical order and is distribution tail index. This estimator is involved in the construction of the Hill-plot graph : representation according to the statistical order . The Hill-plot allows us to choose a threshold for a construction of GPD model.

The Hill-plot is a dual utility tool:

Estimation of the index of the tail of the distribution

Estimation of the threshold.

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