Pickands–Balkema–De Haan theorem

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

Statement Let {Anzeigestil (X_{1},X_{2},Punkte )} be a sequence of independent and identically-distributed random variables, und lass {Anzeigestil 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 {Anzeigestil F} , and large {Anzeigestil u} , {Anzeigestil F_{u}} is well approximated by the generalized Pareto distribution. Das ist: {Anzeigestil F_{u}(j)rightarrow G_{k,Sigma }(j),{Text{ wie }}urightarrow infty } wo {Anzeigestil G_{k,Sigma }(j)=1-(1+ky/sigma )^{-1/k}} , wenn {displaystyle kneq 0} {Anzeigestil G_{k,Sigma }(j)=1-e^{-y/sigma }} , wenn {displaystyle k=0.} Here σ > 0, and y ≥ 0 when k ≥ 0 und 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. Kategorien: Probability theoremsExtreme value dataTails of probability distributions

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