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 [math]\displaystyle{ F }[/math] of a random variable [math]\displaystyle{ X }[/math], we are interested in estimating the conditional distribution function [math]\displaystyle{ F_u }[/math] of the variable [math]\displaystyle{ X }[/math] above a certain threshold [math]\displaystyle{ u }[/math]. This is the so-called conditional excess distribution function, defined as
- [math]\displaystyle{ F_u(y) = P(X-u \leq y | X\gt u) = \frac{F(u+y)-F(u)}{1-F(u)} }[/math]
for [math]\displaystyle{ 0 \leq y \leq x_F-u }[/math], where [math]\displaystyle{ x_F }[/math] is either the finite or infinite right endpoint of the underlying distribution [math]\displaystyle{ F }[/math]. The function [math]\displaystyle{ F_u }[/math] describes the distribution of the excess value over a threshold [math]\displaystyle{ u }[/math], given that the threshold is exceeded.
Statement
Let [math]\displaystyle{ (X_1,X_2,\ldots) }[/math] be a sequence of independent and identically-distributed random variables, and let [math]\displaystyle{ F_u }[/math] be their conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution functions [math]\displaystyle{ F }[/math], and large [math]\displaystyle{ u }[/math], [math]\displaystyle{ F_u }[/math] is well approximated by the generalized Pareto distribution. That is:
- [math]\displaystyle{ F_u(y) \rightarrow G_{k, \sigma} (y),\text{ as }u \rightarrow \infty }[/math]
where
- [math]\displaystyle{ G_{k, \sigma} (y)= 1-(1+ky/\sigma)^{-1/k} }[/math], if [math]\displaystyle{ k \neq 0 }[/math]
- [math]\displaystyle{ G_{k, \sigma} (y)= 1-e^{-y/\sigma} }[/math], if [math]\displaystyle{ k = 0. }[/math]
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 [math]\displaystyle{ \sigma }[/math], if k = 0.
- Uniform distribution on [math]\displaystyle{ [0,\sigma] }[/math], if k = -1.
- Pareto distribution, if k > 0.
Related subjects
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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