Introduction to Extreme Value Theory (EVT)

Extreme Value Theory (EVT) is a statistical approach used to analyze extreme events in datasets. Unlike traditional statistics that focus on the entire distribution, EVT specifically models the tail behavior of a distribution, which is essential in fields like finance, meteorology, and autonomous driving safety. EVT is a mathematical theory specifically designed for modeling and analyzing extreme values in data, deriving their statistical distribution, and making reasonable predictions about future extreme events.

Generalized Pareto Distribution (GPD)

The Generalized Pareto Distribution (GPD) is a core component of EVT. It models the distribution of exceedances $ Y $ beyond a given threshold $ u $:

\[Y = X - u \quad \text{for} \quad X > u\]

GPD Probability Distribution Function

The cumulative distribution function (CDF) of GPD is given by:

\[F(y) = 1 - \left(1 + \xi \frac{y}{\sigma} \right)^{-\frac{1}{\xi}}, \quad y > 0\]

where:

  • $ \xi $ (Shape parameter): Determines the tail heaviness.
    • $ \xi > 0 $ → Heavy-tailed (e.g., Pareto distribution).
    • $ \xi = 0 $ → Exponential distribution (light tail).
    • $ \xi < 0 $ → Short-tailed (bounded distribution, e.g., Weibull).
  • $ \sigma $ (Scale parameter): Defines the spread of exceedances.
  • $ y $ is the exceedance ($ X - u $), meaning values above the threshold.

The shape parameter $ \xi $ is crucial as it defines the tail behavior of the extreme values:

  1. Heavy-tailed ($ \xi > 0 $):
    • The tail is long and heavy.
    • Represents catastrophic or extreme events with large exceedances.
    • Common in financial risks, insurance claims, and traffic accidents.
  2. Exponential decay ($ \xi = 0 $):
    • The distribution resembles an exponential distribution.
    • Frequently appears in queueing theory and reliability analysis.
  3. Short-tailed ($ \xi < 0 $):
    • The distribution has a finite maximum value.
    • Often seen in natural processes with strict physical limits (e.g., rainfall amounts).

Why GPD for POT?

According to Pickands-Balkema-de Haan Theorem:

\[\lim_{u \to u_{max}} P(Y \leq y | X > u) = G(y; \xi, \sigma)\]

This means that for a sufficiently high threshold $ u $, the exceedances approximately follow a GPD.

Choosing the correct threshold $ u $ is crucial:

  • Too low: Includes non-extreme values, making the GPD model inaccurate.
  • Too high: Reduces the sample size, leading to unstable parameter estimation.

Relationship Between GPD and POT Threshold $ u $

Steps in the POT Method

  1. Choose an appropriate threshold $ u $.
  2. Extract all values $ X > u $.
  3. Compute the exceedances $ Y = X - u $.
  4. Fit the Generalized Pareto Distribution (GPD) to $ Y $.
  5. Estimate the shape parameter $ \xi $ and scale parameter $ \sigma $.

Threshold Selection Methods

Some common methods for selecting $ u $:

  • Mean Residual Life (MRL) Plot: Identifies the point where exceedance mean remains linear.
  • Parameter Stability Analysis: Checks the consistency of GPD parameter estimates over different thresholds.

AIC and BIC in GPD Model Selection

When fitting the GPD model using Maximum Likelihood Estimation (MLE), we evaluate model quality using AIC and BIC.

Akaike Information Criterion (AIC)

\(AIC = 2k - 2\ln L\) where:

  • $ k $ is the number of parameters (2 for GPD: $ \xi, \sigma $).
  • $ L $ is the log-likelihood function.

Bayesian Information Criterion (BIC)

\(BIC = k \ln n - 2\ln L\) where:

  • $ n $ is the sample size.

Both AIC and BIC are used to compare models:

  • Lower values indicate better models.
  • AIC prefers a good fit, while BIC penalizes excessive parameters more strictly.

In the code:

log_likelihood = -neg_log_likelihood(result.x)
num_params = 2  # shape and scale parameters
n = len(exceedances)
aic = 2 * num_params - 2 * log_likelihood
bic = np.log(n) * num_params - 2 * log_likelihood