Rectified 24-cell honeycomb

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 the first theorem (the Fisher–Tippett–Gnedenko theorem) in extreme value theory, the interest here is the values above a threshold.

Conditional excess distribution function

If we consider an unknown distribution function $F$ of a random variable $X$, we are interested in estimating the conditional distribution function $F_{u}$ of the variable $X$ above a certain threshold $u$. This is the so-called conditional excess distribution function, defined as

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

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

Statement

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

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

where

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

Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0.

Special cases of generalized Pareto distribution

• Exponential distribution with mean $\sigma$, if k = 0.
• Uniform distribution on $[0,\sigma ]$, if k = 1.
• Pareto distribution, if k < 0.

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.