Rectified 24-cell honeycomb

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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

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

Fu(y)=P(Xuy|X>u)=F(u+y)F(u)1F(u)

for 0yxFu, where xF is either the finite or infinite right endpoint of the underlying distribution F. The function Fu describes the distribution of the excess value over a threshold u, given that the threshold is exceeded.

Statement

Let (X1,X2,) be a sequence of independent and identically-distributed random variables, and let Fu 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, Fu is well approximated by the generalized Pareto distribution. That is:

Fu(y)Gk,σ(y), as u

where

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

Special cases of generalized Pareto distribution

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References

  • 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.