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.

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.

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