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.
The theorem owes its name to mathematicians James Pickands, Guus Balkema, and Laurens de Haan.
the Pickands–Balkema–De Haan theorem describes the conditional distribution function
This is the so-called conditional excess distribution function, defined as for
is either the finite or infinite right endpoint of the underlying distribution
describes the distribution of the excess value over a threshold
be the conditional excess distribution function.
Pickands,[1] Balkema and De Haan[2] posed that for a large class of underlying distribution functions
is well approximated by the generalized Pareto distribution, in the following sense.
These special cases are also known as The class of underlying distribution functions
[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.