Pickands–Balkema–De Haan theorem

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.

For an unknown distribution function ${\displaystyle F}$ of a random variable ${\displaystyle X}$, the Pickands–Balkema–De Haan theorem describes the conditional distribution function ${\displaystyle F_{u}}$ of the variable ${\displaystyle X}$ above a certain threshold ${\displaystyle u}$. This is the so-called conditional excess distribution function, defined as

${\displaystyle F_{u}(y)=P(X-u\leq y|X>u)={\frac {F(u+y)-F(u)}{1-F(u)}}}$

for ${\displaystyle 0\leq y\leq x_{F}-u}$, where ${\displaystyle x_{F}}$ is either the finite or infinite right endpoint of the underlying distribution ${\displaystyle F}$. The function ${\displaystyle F_{u}}$ describes the distribution of the excess value over a threshold ${\displaystyle u}$, given that the threshold is exceeded.

Let ${\displaystyle F_{u}}$ be the conditional excess distribution function. Pickands,^[1] Balkema and De Haan^[2] posed that for a large class of underlying distribution functions ${\displaystyle F}$, and large ${\displaystyle u}$, ${\displaystyle F_{u}}$ is well approximated by the generalized Pareto distribution, in the following sense. Suppose that there exist functions ${\displaystyle a(u),b(u)}$, with ${\displaystyle a(u)>0}$ such that ${\displaystyle F_{u}(a(u)y+b(u))}$ as ${\displaystyle u\rightarrow \infty }$ converge to a non-degenerate distribution, then such limit is equal to the generalized Pareto distribution:

${\displaystyle F_{u}(a(u)y+b(u))\rightarrow G_{k,\sigma }(y),{\text{ as }}u\rightarrow \infty }$,

where

• ${\displaystyle G_{k,\sigma }(y)=1-(1+ky/\sigma )^{-1/k}}$, if ${\displaystyle k\neq 0}$
• ${\displaystyle G_{k,\sigma }(y)=1-e^{-y/\sigma }}$, if ${\displaystyle k=0.}$

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

• Exponential distribution with mean ${\displaystyle \sigma }$, if k = 0,
• Uniform distribution on ${\displaystyle [0,\sigma ]}$, if k = -1,
• Pareto distribution, if k > 0.

The class of underlying distribution functions ${\displaystyle F}$ are related to the class of the distribution functions ${\displaystyle F}$ 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]

• Stable distribution