The Hill Estimator

Python implementation of the Hill Estimator and its corresponding Hill Plot within a specified zoomed window, avaliable in the following jypitor notebook

hill_estimator.ipynb

Very short intuition behind code

Given the sorted order statistics of a sample, the Hill estimator is defined by

where k is the number of upper order statistics in the sample used for the estimation and n is the number of observed sample values. With increasing k the Hill estimator is stabilizing, and the question is which k gives a reasonable estimation of . One might say that a good approximation of lies within a region where the Hill plot is stable.

For more information about the Hill Estimator, and in general about heavy-tailed time series analysis, I recommend taking a look at this PhD Thesis [1].

How to run existing code

• Step 1: Open hill_estimator.ipynb.
• Step 2: Load time series data, here S&P 500 Stock Index data, and split into gains and losses.
• Step 3: Run code from utils/functions.py and plot the Hill plot within some specified zoomed window range.

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Get access to some time series data, apply the algorithm for the Hill estimator, and try out different zoomed windows and see what fit the data the best. Cheers!

References

[1] Xiaolei Xie. Analysis of Heavy-Tailed Time Series. PhD Thesis, School of the Faculty of Science, University of Copenhagen (2017). Link