Pickands–Balkema–De Haan theorem

Pickands–Balkema–De Haan theorem   (Redirected from Pickands–Balkema–de Haan theorem) Jump to navigation Jump to search 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.

Contents 1 Conditional excess distribution function 2 Statement 3 Special cases of generalized Pareto distribution 4 Related subjects 5 References Conditional excess distribution function If we consider an unknown distribution function {displaystyle F} of a random variable {displaystyle X} , we are interested in estimating the conditional distribution function {displaystyle F_{u}} of the variable {displaystyle X} above a certain threshold {displaystyle u} . This is the so-called conditional excess distribution function, defined as {displaystyle F_{u}(y)=P(X-uleq y|X>u)={frac {F(u+y)-F(u)}{1-F(u)}}} for {displaystyle 0leq yleq x_{F}-u} , where {displaystyle x_{F}} is either the finite or infinite right endpoint of the underlying distribution {displaystyle F} . The function {displaystyle F_{u}} describes the distribution of the excess value over a threshold {displaystyle u} , given that the threshold is exceeded.

Statement Let {displaystyle (X_{1},X_{2},ldots )} be a sequence of independent and identically-distributed random variables, and let {displaystyle 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 {displaystyle F} , and large {displaystyle u} , {displaystyle F_{u}} is well approximated by the generalized Pareto distribution. That is: {displaystyle F_{u}(y)rightarrow G_{k,sigma }(y),{text{ as }}urightarrow infty } where {displaystyle G_{k,sigma }(y)=1-(1+ky/sigma )^{-1/k}} , if {displaystyle kneq 0} {displaystyle G_{k,sigma }(y)=1-e^{-y/sigma }} , if {displaystyle 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 {displaystyle sigma } , if k = 0. Uniform distribution on {displaystyle [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. Categories: Probability theoremsExtreme value dataTails of probability distributions