# 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 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.

## 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.