Pickands–Balkema–De Haan theorem

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The Pickands–Balkema–De Haan theorem gives the asymptotic tail distribution of a random variable, when its true distribution is unknown. It is often called the second theorem in extreme value theory. Unlike the first theorem (the Fisher–Tippett–Gnedenko theorem), which concerns the maximum of a sample, the Pickands–Balkema–De Haan theorem describes the values above a threshold.

The theorem owes its name to mathematicians James Pickands, Guus Balkema, and Laurens de Haan.

Conditional excess distribution function[edit]

For an unknown distribution function of a random variable , the Pickands–Balkema–De Haan theorem describes 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.


Let be the conditional excess distribution function. Pickands,[1] Balkema and De Haan[2] posed that for a large class of underlying distribution functions , and large , is well approximated by the generalized Pareto distribution, in the following sense. Suppose that there exist functions , with such that as converge to a non-degenerate distribution, then such limit is equal to the generalized Pareto distribution:



  • , if
  • , if

Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0. These special cases are also known as

The class of underlying distribution functions are related to the class of the distribution functions satisfying the Fisher–Tippett–Gnedenko theorem.[3]

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.

The theorem has been extended to include a wider range of distributions.[4][5] While the extended versions cover, for example the normal and log-normal distributions, still continuous distributions exist that are not covered.[6]

See also[edit]


  1. ^ Iii, James Pickands (1975-01-01). "Statistical Inference Using Extreme Order Statistics". The Annals of Statistics. 3 (1). doi:10.1214/aos/1176343003. ISSN 0090-5364.
  2. ^ Balkema, A. A.; de Haan, L. (1974-10-01). "Residual Life Time at Great Age". The Annals of Probability. 2 (5). doi:10.1214/aop/1176996548. ISSN 0091-1798.
  3. ^ Balkema, A. A.; de Haan, L. (1974-10-01). "Residual Life Time at Great Age". The Annals of Probability. 2 (5). doi:10.1214/aop/1176996548. ISSN 0091-1798.
  4. ^ Papastathopoulos, Ioannis; Tawn, Jonathan A. (2013). "Extended Generalised Pareto Models for Tail Estimation". Journal of Statistical Planning and Inference. 143 (1): 131–143. arXiv:1111.6899. doi:10.1016/j.jspi.2012.07.001. S2CID 88512480.
  5. ^ Lee, Seyoon; Kim, Joseph H. T. (2019-04-18). "Exponentiated generalized Pareto distribution: Properties and applications towards extreme value theory". Communications in Statistics - Theory and Methods. 48 (8): 2014–2038. arXiv:1708.01686. doi:10.1080/03610926.2018.1441418. ISSN 0361-0926. S2CID 88514574.
  6. ^ Smith, Richard L.; Weissman, Ishay. Extreme Values (PDF) (draft 2/27/2020 ed.).