Introduction to extreme value theory

Q: What are the two main approaches to modeling extremes in EVT?

The two main approaches are the Block Maxima approach (modeling the maximum value in fixed time intervals) and the Peaks Over Threshold (POT) approach (modeling all values exceeding a specific limit).

Q: What are the three types of GEV distributions based on the shape parameter $ \gamma $?

The three types are Type I: Gumbel ($ \gamma = 0 $), Type II: Fréchet ($ \gamma > 0 $), and Type III: Weibull ($ \gamma < 0 $).

Q: In the Peaks Over Threshold (POT) approach, how is the "excess" defined?

The excess is defined as $ Y = X - u $, given that the original observation $ X $ exceeds the threshold $ u $.

Q: What is the Pickands-Balkema-de Haan Theorem?

It states that for a sufficiently high threshold $ u $, the distribution of excesses follows the Generalized Pareto Distribution (GPD).

Q: What is the trade-off involved in choosing the threshold $ u $ for a POT model?

A threshold that is too high results in too few data points (high variance), while a threshold that is too low causes the model to be biased because the asymptotic GPD theorem may not yet apply.

CS2 – Risk Modelling and Survival Analysis · IFoA (Institute and Faculty of Actuaries) · 5 min read

Welcome to the World of the "Unlikely": Introduction to Extreme Value Theory

Hello there! Welcome to one of the most fascinating chapters in your CS2 journey. So far in your studies, you’ve likely spent a lot of time looking at "average" outcomes—the middle of the Bell Curve. But as an actuary, you aren't just worried about the average car crash or the average life expectancy. You are worried about the catastrophes.

Extreme Value Theory (EVT) is the branch of statistics that deals with the "tails" of a distribution. It’s about the 1-in-100-year flood, the massive stock market crash, or the huge insurance claim that could bankrupt a company. This chapter teaches us how to model those rare but high-impact events. Don’t worry if the math looks intimidating at first—we’re going to break it down piece by piece!

1. Why do we need EVT?

Imagine you are building a sea wall. If you only look at the average tide, your wall will be very short, and the first big storm will wash the town away. You need to model the maximum possible tide.

Standard distributions (like the Normal distribution) often "underestimate" how often extreme events happen. They have "thin tails." EVT gives us a mathematically sound way to estimate the probability of events that we may have never even seen in our historical data yet!

Key Concept: The Two Main Approaches

In CS2, we focus on two ways to look at extremes:
1. The Block Maxima Approach: Looking at the biggest value in fixed time intervals (e.g., the highest claim each year).
2. The Peaks Over Threshold (POT) Approach: Looking at every value that "breaks the limit" (e.g., every claim over \$1 million).

2. The Block Maxima and the GEV Distribution

Let’s start with Block Maxima. Suppose we have a sequence of independent and identically distributed (i.i.d.) random variables $ X_1, X_2, ..., X_n $. We define the maximum as:
$ M_n = \max(X_1, X_2, ..., X_n) $

The Fisher-Tippett-Gnedenko Theorem (a mouthful, I know!) tells us that as $ n $ gets very large, the distribution of this maximum (after some scaling) converges to the Generalized Extreme Value (GEV) Distribution.

The GEV Formula

The cumulative distribution function (CDF) of the GEV is:
$ H_{\gamma}(x) = \exp \left( -(1 + \gamma x)^{-1/\gamma} \right) $

Where $ \gamma $ (gamma) is the shape parameter. This parameter is the "star of the show" because it determines how "heavy" the tail is.

The Three Flavors of GEV

Depending on the value of $ \gamma $, the distribution falls into one of three types:
• Type I: Gumbel ($ \gamma = 0 $): Used for "light-tailed" distributions like the Exponential or Normal. The tail decays exponentially.
• Type II: Fréchet ($ \gamma > 0 $): This is the "heavy-tailed" version. This is very important for actuaries because it models distributions like the Pareto, where huge losses are more likely.
• Type III: Weibull ($ \gamma < 0 $): This is "short-tailed" with a finite upper limit. Think of it as a "maximum possible human age"—it doesn't go to infinity.

Quick Review:
- $ \gamma > 0 $: Heavy tail (Frechet) - Watch out, big claims ahead!
- $ \gamma = 0 $: Light tail (Gumbel) - Thin tails.
- $ \gamma < 0 $: Finite tail (Weibull) - There is a ceiling.

3. Peaks Over Threshold (POT) and the GPD

While Block Maxima only looks at one value per year, the POT approach is more efficient because it uses all the data points that are "extreme."

If we pick a high threshold $ u $, we are interested in the excess values: $ Y = X - u $, given that $ X > u $.

The Pickands-Balkema-de Haan Theorem states that for a sufficiently high threshold $ u $, the distribution of these excesses follows the Generalized Pareto Distribution (GPD).

The GPD Formula

The CDF of the GPD is given by:
$ G_{\gamma, \beta}(x) = 1 - (1 + \gamma x / \beta)^{-1/\gamma} $ (for $ \gamma \neq 0 $)
$ G_{\gamma, \beta}(x) = 1 - \exp(-x/\beta) $ (for $ \gamma = 0 $)

Here, $ \beta $ is the scale parameter (how spread out the data is) and $ \gamma $ is the same shape parameter we saw in GEV!

Did you know?
There is a beautiful link between GEV and GPD. If the block maxima of a dataset follow a GEV distribution with shape parameter $ \gamma $, then the excesses over a high threshold for that same dataset will follow a GPD with the exact same $ \gamma $.

4. Practical Challenges: Choosing the Threshold

This is often where students get stuck. How high should the threshold $ u $ be?
• If $ u $ is too high: You won’t have enough data points (high variance).
• If $ u $ is too low: The theorem doesn't "kick in" yet, and your model will be biased.

The Mean Excess Plot: Actuaries use a "Mean Excess Plot" to find the sweet spot. We look for a range of $ u $ where the plot becomes linear. If the plot is upward-sloping and linear, it suggests a heavy-tailed GPD distribution.

5. Common Mistakes to Avoid

1. Mixing up GEV and GPD: Remember: GEV is for Maxima (the biggest one in a group). GPD is for Exceedances (everything above a certain line).
2. Forgetting the condition: The GPD formula only applies to the excess $ (X - u) $, not the original value $ X $.
3. Sign of $ \gamma $: Pay close attention to whether $ \gamma $ is positive or negative in exam questions, as it completely changes the "riskiness" of the tail.

6. Summary and Key Takeaways

• The Goal: EVT helps us model the very end of the tail where data is scarce.
• Block Maxima: Use the Generalized Extreme Value (GEV) distribution. It has three types: Gumbel, Frechet, and Weibull.
• Peaks Over Threshold: Use the Generalized Pareto Distribution (GPD). It’s often preferred because it uses data more efficiently.
• Shape Parameter ($ \gamma $): This is the most important value. If $ \gamma > 0 $, we are dealing with heavy tails (the "danger zone" for insurance).
• Mean Excess Plot: A tool to help us choose the right threshold for GPD.

Don't worry if this seems tricky at first! EVT is a shift in mindset from "average" thinking to "extreme" thinking. Keep practicing the formulas for $ H(x) $ and $ G(x) $, and you'll be identifying tail risks like a pro in no time!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the primary purpose of Extreme Value Theory (EVT)?

EVT is a branch of statistics used to model the "tails" of a distribution, focusing on the probability and impact of rare, high-impact extreme events.

What are the two main approaches to modeling extremes in EVT?

The two main approaches are the Block Maxima approach (modeling the maximum value in fixed time intervals) and the Peaks Over Threshold (POT) approach (modeling all values exceeding a specific limit).

What does the Fisher-Tippett-Gnedenko Theorem state?

It states that the distribution of the maximum of a sequence of i.i.d. random variables, when appropriately scaled, converges to the Generalized Extreme Value (GEV) distribution as the sample size increases.

What is the cumulative distribution function (CDF) for the Generalized Extreme Value (GEV) distribution?

$ H_{\gamma}(x) = \exp \left( -(1 + \gamma x)^{-1/\gamma} \right) $, where $ \gamma $ is the shape parameter.

What are the three types of GEV distributions based on the shape parameter $ \gamma $?

The three types are Type I: Gumbel ($ \gamma = 0 $), Type II: Fréchet ($ \gamma > 0 $), and Type III: Weibull ($ \gamma < 0 $).

In the Peaks Over Threshold (POT) approach, how is the "excess" defined?

The excess is defined as $ Y = X - u $, given that the original observation $ X $ exceeds the threshold $ u $.

What is the Pickands-Balkema-de Haan Theorem?

It states that for a sufficiently high threshold $ u $, the distribution of excesses follows the Generalized Pareto Distribution (GPD).

What is the CDF of the Generalized Pareto Distribution (GPD)?

$ G_{\gamma, \beta}(x) = 1 - (1 + \gamma x / \beta)^{-1/\gamma} $ for $ \gamma \neq 0 $, and $ G_{\gamma, \beta}(x) = 1 - \exp(-x/\beta) $ for $ \gamma = 0 $.

What is the mathematical link between the GEV and GPD distributions?

If the block maxima of a dataset follow a GEV distribution with shape parameter $ \gamma $, then the excesses over a high threshold for that same dataset will follow a GPD with the same shape parameter $ \gamma $.

What is the trade-off involved in choosing the threshold $ u $ for a POT model?

A threshold that is too high results in too few data points (high variance), while a threshold that is too low causes the model to be biased because the asymptotic GPD theorem may not yet apply.

What is a Mean Excess Plot used for in EVT?

It is a graphical tool used to select an appropriate threshold $ u $ by identifying a range where the plot becomes linear.

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