# 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 $F$ 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. 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 $\sigma$ , if k = 0.
• Uniform distribution on $[0,\sigma ]$ , if k = -1.
• Pareto distribution, if k > 0.