Peaks Over Threshold: Fitting the GPD

The Peaks-Over-Threshold (POT) Approach

The intuition behind the POT method is straightforward. Rather than asking "what is the largest loss per year?", we ask "what does the distribution of losses look like, given that a large loss has occurred?".

The process involves two key concepts:

1. Threshold $u$: We select a high value, or threshold, that separates "normal" losses from "extreme" ones.
2. Exceedances and Excesses:
   • The observations that are larger than the threshold $u$ are called exceedances.
   • The amount by which an exceedance surpasses the threshold is called the excess. If a loss $X$ exceeds $u$, the excess is $Y=X-u$.

The following video provides an excellent introduction to this idea, contrasting it with the block maxima method we learned previously.

The Generalized Pareto Distribution (GPD)

Just as the Fisher-Tippett-Gnedenko theorem provided a limiting distribution (the GEV) for block maxima, a similar keystone result exists for excess losses. The Pickands–Balkema–De Haan theorem states that for a sufficiently high threshold $u$, the distribution of the excess losses $Y=X-u$ can be approximated by the Generalized Pareto Distribution (GPD).

This is a powerful result. It means that for a vast range of underlying loss distributions, the behavior of their tails, once they cross a high threshold, follows a universal pattern described by the GPD.

The GPD is typically defined by two parameters:

• Scale parameter, $\beta >0$: Analogous to the scale parameter in other distributions.
• Shape parameter, $\xi$: (Sometimes denoted $\gamma$). As with the GEV, this is the crucial parameter that dictates the tail's shape and heaviness.

The resource below provides the formal definition and properties of the GPD.

Applying the POT Method: A Two-Step Process

Fitting a GPD model to data is a practical, two-step procedure.

Step 1: Choosing the Threshold $u$

This is the most critical and subjective step in the analysis. The choice of $u$ involves a fundamental bias-variance trade-off:

• If $u$ is too low: We include too many non-extreme data points. The GPD approximation will be poor (high bias), even though we have many data points to estimate the parameters (low variance).
• If $u$ is too high: The GPD approximation will be excellent (low bias), but we will have very few data points, leading to highly uncertain parameter estimates (high variance).

The primary tool for navigating this trade-off is the Mean Excess Plot. The theoretical mean excess loss function is defined as $e(u)=E[X-u|X>u]$, which is the expected size of an excess, given that the loss has exceeded the threshold $u$.

A key property of the GPD, derived in the previous reading, is that its mean excess loss function is linear in the threshold. We can exploit this by plotting the empirical mean excess for our data against a range of possible thresholds. The ideal threshold $u$ is the lowest value above which the empirical plot becomes approximately linear.

Step 2: Estimating GPD Parameters $\xi$ and $\beta$

Once a suitable threshold $u$ has been selected, the process is straightforward:

1. Create a new dataset consisting of the excesses $y_j=x_j-u$ for all original data points $x_j$ that are greater than $u$.
2. Fit the two-parameter GPD to this new dataset of excesses.

The most common fitting method is Maximum Likelihood Estimation (MLE). The log-likelihood function is constructed based on the GPD density function and the sample of excesses, and then maximized numerically to find the optimal estimates for $\xi$ and $\beta$. While the technical details of the optimization are complex, the principle is the same as for any other MLE problem you have encountered.

A Practical Walkthrough: The Danish Fire Insurance Data

Let's see how these steps come together in a real-world example. We will analyze a classic EVT dataset: the Danish fire insurance losses. This dataset contains large, infrequent fire loss claims, making it a perfect candidate for the POT method.

The following video segment provides a complete, narrated analysis of this dataset.

This example encapsulates the entire process: visual data inspection, using the mean excess plot to choose a threshold, fitting the GPD, and interpreting the results within a practical risk management context.

Conclusion

Today, we have added the Peaks-Over-Threshold method to our toolkit, providing a more data-efficient and often more robust alternative to the block maxima approach. By focusing on all losses that exceed a high threshold, we can make better use of our available data to understand the tail of a distribution.

Key Takeaways:

• The Peaks-Over-Threshold (POT) method models excess losses above a high threshold $u$, making it more data-efficient than the Block Maxima method.
• The Pickands–Balkema–De Haan theorem states that the distribution of these excess losses converges to the Generalized Pareto Distribution (GPD).
• The GPD's tail behavior is governed by the shape parameter $\xi$, with $\xi >0$ being the most relevant case for heavy-tailed insurance and financial loss data.
• Fitting a GPD is a two-step process:
  1. Choose a threshold $u$ by identifying where the empirical mean excess plot becomes linear.
  2. Fit the GPD parameters ($\xi ,\beta$) to the sample of excesses using Maximum Likelihood Estimation (MLE).

Preview of the Next Lesson

We have now learned how to fit both GEV and GPD models to extreme data. But what is the ultimate goal of this modeling? In risk management, it is to quantify risk. In our next lesson, we will put our fitted GPD model to work. We will learn how to use the estimated parameters $\xi$ and $\beta$ to calculate crucial risk measures for the extreme tail of our distribution, such as high-quantile Value-at-Risk (VaR) and Expected Shortfall (ES). This will allow us to answer critical business questions like, "What is our estimated 1-in-200 year loss?".