Choosing Extreme Value Thresholds with Hill Plots

Summary

The document describes the threshold choice in extreme value theory as a bias–variance trade-off. A lower threshold provides more exceedances and can make estimates smoother, but may include observations from the distribution’s center and bias tail estimates. A higher threshold focuses more narrowly on extremes but leaves fewer observations and can increase estimation variance.

The suggested practical method is to calculate the estimate across a range of thresholds and inspect a plot for a region where results begin to stabilize. For the Hill estimator, this diagnostic is called a Hill plot; related variants exist for other extreme value methods. The document offers no data, worked example, or criterion for how much stability is sufficient, so the plot is a diagnostic rather than a guaranteed way to identify an optimal threshold. It also leaves open how to weigh bias against variance for a particular application.

Key ideas

• Lower thresholds increase the number of exceedances but may introduce bias by including non-extreme observations.
• Higher thresholds focus on the tail but can increase estimation variance because fewer observations remain.
• Plot estimates across thresholds and look for a region of convergence.
• The Hill plot is a threshold-stability diagnostic for the Hill estimator.

Tags

# What is a good choice of threshold for Value at Risk?


# What is a good choice of threshold for Value at Risk?












As far as I know, there is usually a betwixt in choosing the right value for a threshold. A trade off between bias and variance has to be encountered.

If a low threshold is chosen, the number of observations (exceedances) increase and the estimation becomes more smooth.

However, low threshold also introduces some observations from the center of the distribution and the estimation becomes biased.

On the other hand, a relatively high threshold eliminates values that would have been part of the extrema hence a higher variance in the estimations.

How do I know the value of a threshold chosen is fit to produce the best results? Moreover, If I am to do a trade off between , what would be worth? Overlook bias and ensure minimal variance or forsake variance and combat bias? Which is better? Are there any texts that I can refer to?

## Answer by Ami44 (score 2, accepted)

https://quant.stackexchange.com/a/29987

The basic method is to plot the result against different thresholds and use the one where it starts to converge. If you use the Hill-estimator it's called Hill-plot. But a lot of variants exist in EVT.

Also see: https://www.ine.pt/revstat/pdf/rs120102.pdf

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.