# Publications: Thoery & Methods

Here are references to some articles that describe significant developments in theory and methods for extreme values. A short summary of each article is given in blue.

Fisher RA and Tippett LHC (1928) Limiting forms of the frequency distributions of the largest or smallest member of a sample. *Proceedings of the Cambridge Philosophical Society*, **24**, 180-190.

The first statement of the extremal limit theorem, which says that the
maximum or minimum of a sample can have one of only three `types' of
probability distribution as the sample size increases to infinity.

Gnedenko BV (1943) Sur la distribution limite du terme maximum d'une serie aleatoire. *Annals of Mathematics*, **44**, 423-453.

A rigorous mathematical treatment of the extremal limit theorem.

Jenkinson AF (1955) The frequency distribution of the annual maximum
(or minimum) values of meteorological elements. *Quarterly Journal
of the Royal Meteorological Scoiety*, **87**, 158-171.

The first demonstration that the three types of extreme-value
distribution can be written in a single, parametric form: the
generalised extreme-value (GEV) distribution.

Pickands III J (1971) The two-dimensional Poisson process and extremal
processes. *Journal of Applied Probability*, **8**, 745-756.

A change of focus from maxima to threshold exceedances, which are
shown to occur according to a Poisson process. This will lead to statistical models for threshold exceedances, enabling more data than just sample maxima to be used to make inferences about extremal behaviour.

Leadbetter MR (1974) On extreme values in stationary sequences. *Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete*, **28**, 289-303.

Most previous work had been for samples of independent random
variables. This paper shows that the extremal limit theorem holds for sequences of dependent random variables if the long-range dependence at extreme levels is not too strong.

*Annals of Statistics*,

**3**, 119-131.

Threshold exceedances follow the generalised Pareto (GP) distribution in the limit as the threshold increases.

de Haan L and Resnick SI (1977) Limit theory for multivariate sample extremes. *Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete*, **40**, 317-337.

An extension from univariate to multivariate random variables. The
limiting, `multivariate extreme-value' distribution of the `componentwise maximum' has GEV components and must satisfy certain other properties, but is not completely characterised by a finite number of parameters.

Leadbetter MR (1983) Extremes and local dependence in stationary sequences. *Zeitschrift fur Wahrscheinlichkeitstheorie und Verwandte Gebiete*, **65**, 291-306.

The impact of serial dependence on the GEV limit distribution can be encapsulated by a scalar constant, the extremal index.

de Haan L (1985) Extremes in higher dimensions. *Proceedings of the 45th Session of the International Statistics Institute*, paper 26.3.

Multivariate random variables that are extreme in at least one
component occur according to a multivariate Poisson process in the limit, with an intensity that is related to the multivariate extreme-value distribution.

Hsing T, Husler J and Leadbetter MR (1988) On the exceedance point process for a stationary sequence. *Probability Theory and Related Fields*, **78**, 97-112.

The times at which a stationary process exceeds a high threshold occur in clusters and form a marked Poisson process in the limit.

Tawn JA (1988) Bivariate extreme value theory - models and estimation. *Biometrika*, **77**, 245-253.

Since bivariate extreme-value distributions are not characterised by a finite number of parameters, parametric modelling must restrict attention to a sub-class of models. This paper describes some models and their estimation.

Davison AC and Smith RL (1990) Models for exceedances over high thresholds (with discussion). *Journal of the Royal Statistical Society*, **52**, 393-442.

Statistical methods, including the introduction of covariates, for
modelling threshold exceedances with the generalised Pareto distribution: the `peaks-over-threshold' approach.

Leadbetter MR (1991) On a basis for `Peaks over Threshold' modeling. *Statistics and Probability Letters*, **12**, 357-362.

The probabilistic justification for the peaks-over-threshold model.

Coles SG and Tawn JA (1991) Modelling extreme multivariate events. *Journal of the Royal Statistical Society*, **53**, 377-392.

A method for constructing parametric models for multivariate
extreme-value distributions by transformation, and their estimation using the limiting Poisson process model. See also Joe H, Smith RL and Weissman I (1992) Bivariate threshold methods for extremes. *Journal of the Royal Statistical Society B*, **54**, 171-183.

Smith RL, Tawn JA and Coles SG (1997) Markov chain models for threshold exceedances. *Biometrika*, **84**, 249-268.

Markov chains with extreme-value transition distributions can be used to model the time evolution within clusters of extreme values.

Ferro CAT and Segers J (2003) Inference for clusters of extreme values. *Journal of the Royal Statistical Society B*, **65**, 545-556.

If cluster evolution is not modelled then inference requires extremes to be `declustered'. Grouping extremes into clusters had been a fairly arbitrary procedure, but this paper gives an asymptotically justified method and shows that the extremal index can be estimated without declustering.

Heffernan J and Tawn JA (2004) A conditional approach for multivariate extreme values (with discussion). *Journal of the Royal Statistical Society B*, **66**, 497-546.

A model for the extremes of multivariate processes that seems more
natural than the classical, componentwise maximum approach.

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