Extreme Value Theory

Key issues in extreme value theory:

  1. [Data Sparsity] there are very few observations in the tail of the distribution;
  2. [Extrapolation] estimates are often required beyond the largest observed data value;

Standard data analysis/model-fitting techniques work well where the data have greatest density, but can be severely biased in estimating tail probabilities.

Limit of Maxima

Definition: Sample Maximum

\[ M_n = \text{max}(X_1, \cdots , X_n) \]

If support of a population is bounded above, the maximum order statistics converges to the supremum of the support. Symbolically,

\[ \forall x \text{ s.t. } F(x)<1: \lim P(M_n \leq x) = 0 \]

Linear Rescaling

Under proper linear rescaling, some sequences of maxima have non-degenerate limiting distributions. Symbolically, \[ \exists \{ a_m, b_m: a_m > 0 \}, \text{ s.t. } (M_m - b_m) / a_m \Rightarrow H \]

Examples:

  1. Exponential(1): \( a_n = 1, b_n = \ln n, H = \exp\{-\exp\{-x\}\} \)
  2. Pareto-type tail, \( 1 - F(x) \approx c x^{-a} \): \( a_n = (nc)^{1/a}, b_n = 0, H = \exp\{-x^{-a}\} \)
  3. Finite upper endpoint, \( 1 - F(x) \approx c (w-x)^a, x \rightarrow w- \): \( a_n = (nc)^{-1/a}, b_n = w, H = \exp\{-(-x)^a\} \)
  4. Normal(0,1): \( a_n = 1 / b_n, b_n = Q(n), H = \exp\{-\exp\{-x\}\} \), where Q(n) is tail quantile function.

Exceptions:

  1. Poisson distribution
  2. \( F(x) = 1 − 1 / \ln x \) (x>e)

Extreme Value Distribution

Max-stable Distribution

Defintion: A distribution H is said to be max-stable if for sequences \( \{a_k\} \) and \( \{b_k\} \), \( H_k(x) = H(a_k x + b_k), k \in \mathbb{N} \).

Lemma: If rescaled maxima have a non-degenerate limit distribution, the limiting distribution must be max-stable.

Extreme Value Theorem

The extreme value theorem says, if linearly rescaled maxima have a limiting distribution, then that limit will be a member of the generalized extreme value (GEV) family.

Theorem: The maximum of a sample of iid random variables after proper renormalization can only converge in distribution to one of three possible distributions, the Gumbel distribution, the Fréchet distribution, or the Weibull distribution, also known as type I, II and III extreme value distributions.

Credit for the extreme value theorem, aka Fisher–Tippett–Gnedenko theorem, is given to Gnedenko (1948).

GEV disribution

Generalised Extreme Value (GEV) distribution:

\[ H(x) = \exp \left\{ 􏰇− 􏰁\left( 1 + \frac{\xi (x − \mu)}{\psi} \right)_{+}^{􏰂−1/\xi} 􏰈\right\} \]

  • Type I: Gumbel distribution, ξ = 0
  • Type II: Fréchet distribution, ξ > 0
  • Type III: Weibull distribution, ξ < 0

Sufficient Condition

von Mises Conditions

Definition: For a sufficiently smooth distribution F with upper terminal \(x_F\) , define the reciprocal hazard function as \( r(x)= \frac{1−F(x)}{f(x)} \)

Definition: Tail quantile function \( Q(y) = F^{−1}(1 − 1/y), y \in [1,∞] \)

von Mises conditions: If \( \xi = \lim_{x \to x_F} r'(x) \) exists, denote \(b_m = Q(m), a_m = r(b_m)\), then \( \frac{M_m − b_m}{a_m} \) converge to a GEV distribution with shape parameter \( \xi \).

Proof:

Penultimate Approximation

Taking \( \xi_m = r'(b_m) \) may give a better approximation to the distribution of \( \frac{M_m − b_m}{a_m} \) for finite \(m\) than does using the limiting approximation.

Convergence Rate

There has been a good deal of work on the speed of convergence of \(M_m\) to the limiting regime, which depends on the underlying distribution F. For example, maxima of Gaussian variables convergence very slow.

Direct use of the GEV rather than the three types separately allows for flexible modelling, and ducks the question of which type is most appropriate — the data decide.

Diagnostics

Gumbel Plots

QQ Plots

Reference

  1. Anthony Davison. Lecture slides on Statistics of Extremes, Autumn 2011. http://stat.epfl.ch
  2. Richard L. Smith, 2003. Statistics of Extremes, With Applications in Environment, Insurance and Finance.