Pickands–Balkema–de Haan theorem

From Wikipedia, the free encyclopedia
Jump to: navigation, search

The Pickands–Balkema–de Haan theorem is often called the second theorem in extreme value theory. It gives the asymptotic tail distribution of a random variable X, when the true distribution F of X is unknown. Unlike the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is the values above a threshold.

Conditional excess distribution function[edit]

If we consider an unknown distribution function F of a random variable X, we are interested in estimating the conditional distribution function F_u of the variable X above a certain threshold u. This is the so-called conditional excess distribution function, defined as

F_u(y) = P(X-u \leq y | X>u) = \frac{F(u+y)-F(u)}{1-F(u)} \,

for 0 \leq y \leq x_F-u, where x_F is either the finite or infinite right endpoint of the underlying distribution F. The function F_u describes the distribution of the excess value over a threshold u, given that the threshold is exceeded.

Statement[edit]

Let (X_1,X_2,\ldots) be a sequence of independent and identically-distributed random variables, and let F_u be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions F, and large u, F_u is well approximated by the generalized Pareto distribution. That is:

F_u(y) \rightarrow G_{k, \sigma} (y),\text{ as }u \rightarrow \infty

where

  • G_{k, \sigma} (y)= 1-(1+ky/\sigma)^{-1/k} , if k \neq 0
  • G_{k, \sigma} (y)= 1-e^{-y/\sigma} , if k = 0.

Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0.

Special cases of generalized Pareto distribution[edit]

References[edit]

  • Balkema, A., and de Haan, L. (1974). "Residual life time at great age", Annals of Probability, 2, 792–804.
  • Pickands, J. (1975). "Statistical inference using extreme order statistics", Annals of Statistics, 3, 119–131.