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Testing Generalized Pareto Tail Fits with Goodness-of-Fit Methods

Article Quant Q&A · Author: Peter Miller
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Summary

The document discusses how to assess whether returns in the tails of a distribution fit a generalized Pareto distribution (GPD), when the interior is modeled separately with a kernel distribution. It notes that a piecewise fit can differ from observed data near the cutoff points, so a goodness-of-fit test on the full distribution may conceal tail-specific behavior.

The answer recommends the Anderson-Darling test over the Kolmogorov-Smirnov test when tail fit is the main concern. KS focuses on the largest gap between empirical and hypothesized cumulative distributions and is less sensitive to tail discrepancies; Anderson-Darling weights differences across the distribution in a way that gives more emphasis to the tails. A second suggestion is to transform returns through the hypothesized GPD using the probability integral transform: under correct specification, the resulting values should be independent uniform observations, which can be assessed with independence tests. The document does not give implementation details or discuss adjustments needed when distribution parameters are estimated from the same sample.

Key ideas

  • The Kolmogorov-Smirnov test emphasizes the maximum gap between empirical and hypothesized cumulative distributions.
  • The Anderson-Darling test can be more informative when tail fit is the main concern because its weighting emphasizes tail discrepancies.
  • A piecewise kernel and generalized Pareto fit may show discrepancies near the cutoffs between regions.
  • Under a correctly specified distribution, probability integral transform values should be uniform and independent.
  • Independence tests on transformed returns provide an additional model-checking approach.

Tags

StatisticsVolatilityRisk management
Full text
# Kolmogorov-Smirnov test for Generalized Pareto Distribution


# Kolmogorov-Smirnov test for Generalized Pareto Distribution












I've fitted my data to a generalized pareto distribution as to model the returns in the tails more accurately. The interior is fitted with kernel distributions.

I would like to now test whether the original returns conform to the hypothesized distribution (i.e. generalized pareto distribution). Can I do this with the Kolmogorov-Smirnov test? I've already QQ-plots. However, I would like to conduct a statistical significance test on top. Can some one help? Kind of struggling with implementing it in Matlab.

Best

## Answer by Eric Brady (score 1)

https://quant.stackexchange.com/a/15361

I don't know if there are any additional issues that arise with using goodness off fit with a piece-wise function. When I have fit generalized pareto distributions to series like financial market returns, I have noticed that it is common to differences between the estimated distribution and observed returns at the cutoff points. This is going to be the main difference between running the goodness of fit test on the entire GPD as opposed to the fits for the tails individually, since the kernel density fit will be good.

If you have an estimate of your hypothesized distribution, I would recommend using the Anderson-Darling test instead of the KS-test. The KS-test checks for the maximum distance between the empirical distribution function and the hypothesized function, and thus is not that sensitive to the tails which is what you care about. The Anderson-Darling test integrates over the squared difference between empirical distribution and the hypothesized, and places different weights on each part of the distribution. The weighting function effectively places greater weight in the tails.

## Answer by Malick (score 1)

https://quant.stackexchange.com/a/22081

As an additional (simple) solution I would use the probability integral transform (PIT) of the returns with respect to the generalized pareto distribution. Under the null hypothesis that the distribution is correctly specified, outcomes of the PIT should be independent uniform U[0; 1] random variables. Then you can use traditional independence tests.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

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