Harman Patil (Editor)

Pickands–Balkema–de Haan theorem

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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 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 function

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 ≤ y | X > u ) = F ( u + y ) − F ( u ) 1 − F ( u )

for 0 ≤ y ≤ 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

Let ( X 1 , X 2 , … ) 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 ) → G k , σ ( y ) ,  as  u → ∞

where

  • G k , σ ( y ) = 1 − ( 1 + k y / σ ) − 1 / k , if k ≠ 0
  • G k , σ ( y ) = 1 − e − y / σ , if k = 0.

    • 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 distribution

  • Exponential distribution with mean σ , if k = 0.
  • Uniform distribution on [ 0 , σ ] , if k = -1.
  • Pareto distribution, if k < 0.

    • Stable distribution

    References

    Pickands–Balkema–de Haan theorem Wikipedia
    (Text) CC BY-SA