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Pickands–Balkema–de Haan theorem

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 for the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is in the values above a threshold.

Contents

Conditional excess distribution functionEdit

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

 

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

StatementEdit

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

 

where

  •  , if  
  •  , if  

Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0. Since a special case of the generalized Pareto distribution is a power-law, the Pickands–Balkema–de Haan theorem is sometimes used to justify the use of a power-law for modeling extreme events. Still, many important distributions, such as the normal and log-normal distributions, do not have extreme-value tails that are asymptotically power-law.

Special cases of generalized Pareto distributionEdit

Related subjectsEdit

ReferencesEdit

  • 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.