Pickands–Balkema–de Haan theorem
Encyclopedia
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.
of a random variable 
, we are interested in estimating the distribution function 
of variable 
above a certain threshold 
. The distribution function Fu is so called the conditional excess distribution function and is defined as
for
, where 
is either the finite or infinite right endpoint of the underlying distribution 
. The function 
describes the distribution of the excess value over the threshold 
, given that 
is exceeded.
be a sequence of independent and identically-distributed random variables, let 
be the conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution function F, for u large, Fu is well approximated by the generalized Pareto distribution. That is:
where
Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0.
Extreme value theory
Extreme value theory is a branch of statistics dealing with the extreme deviations from the median of probability distributions. The general theory sets out to assess the type of probability distributions generated by processes...
. It gives the asymptotic tail distribution of a random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
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 of a random variable , we are interested in estimating the distribution function of variable above a certain threshold . The distribution function Fu is so called the conditional excess distribution function and is defined as
for
, where is either the finite or infinite right endpoint of the underlying distribution . The function describes the distribution of the excess value over the threshold , given that is exceeded.Statement
Let be a sequence of independent and identically-distributed random variables, let be the conditional excess distribution function. Pickands (1975), Balkema and de Haan (1974) posed that for a large class of underlying distribution function F, for u large, Fu is well approximated by the generalized Pareto distribution. That is:
where
, if 
, if 
Here σ > 0, and y ≥ 0 when k ≥ 0 and 0 ≤ y ≤ −σ/k when k < 0.
Special cases of generalized Pareto distribution
- Exponential distributionExponential distributionIn probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
with meanExpected valueIn probability theory, the expected value of a random variable is the weighted average of all possible values that this random variable can take on...
, if k = 0. - Triangular distribution, if k = 1/2.
- Uniform distribution on
, if k = 1. - Pareto distribution, if k < 0.