Pickands–Balkema–De Haan theorem

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