Pickands–Balkema–de Haan theorem

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.

Conditional excess distribution function

If we consider an unknown distribution function ${\displaystyle F}$ of a random variable ${\displaystyle X}$, we are interested in estimating 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.

Statement

Let ${\displaystyle (X_{1},X_{2},\ldots )}$ be a sequence of independent and identically-distributed random variables, and let ${\displaystyle F_{u}}$ be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions ${\displaystyle F}$, and large ${\displaystyle u}$, ${\displaystyle F_{u}}$ is well approximated by the generalized Pareto distribution. That is:

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

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

Related subjects

Stable distribution

References

• Balkema, A., and de Haan, L. (1974). "Residual life time at great age", Annals of Probability, 2, 792–804.
• Pickands, J. (1975). "Statistical inference using extreme order statistics", Annals of Statistics, 3, 119–131.