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ExtremeValue
Extreme Value Analysis
Christophe Dutang
dutangc@gmail.com
2023-11-04

Extreme values modelling and estimation are an important challenge in various domains of application, such as environment, hydrology, finance, actuarial science, just to name a few. The restriction to the analysis of extreme values may be justified since the extreme part of a sample can be of a great importance. That is, it may exhibit a larger risk potential such as high concentration of air pollutants, flood, extreme claim sizes, price shocks in the four previous topics respectively. The statistical analysis of extreme may be spread out in many packages depending on the topic of application. In this task view, we present the packages from a methodological side.

Applications of extreme value theory can be found in other task views: for financial and actuarial analysis in the r view("Finance") task view, for environmental analysis in the r view("Environmetrics") task view. General implementation of probability distributions is studied in the r view("Distributions") task view.

The maintainer gratefully acknowledges L. Belzile, E. Gilleland, P. Northrop, T. Opitz, M. Ribatet and A. Stephenson for their review papers, Kevin Jaunatre for his helpful advice and Achim Zeileis for his useful comments. If you think information is not accurate or if we have omitted a package or important information that should be mentioned here, please send an e-mail or submit an issue or pull request in the GitHub repository linked above.

Table of contents

[Univariate Extreme Value Theory]{#UnivEVT}

Several packages export the probability functions (quantile, density, distribution and random generation) for the Generalized Pareto and the Generalized Extreme Value distributions, often sticking to the classical prefixing rule (with prefixes "q", "d", "p", "r") and allowing the use of the formals such as log and lower tail, see the view r view("Distributions") for details. Several strategies can be used for the numeric evaluation of these functions in the small shape (near exponential) case. Also, some implementations allow the use of parameters in vectorized form and some can provide the derivatives w.r.t. the parameters. Nevertheless, the r pkg("nieve") package provides symbolic differentiation for two EVT probability distribution (GPD and GEV) in order to compute the log-likelihood.

[Bayesian approach]{#UniBayesian}

  • The package r pkg("extRemes", priority = "core") provides bayesian estimation.
  • The package r pkg("MCMC4Extremes") proposes some functions to perform posterior estimation for some distribution, with an emphasis to extreme value distributions.
  • The package r pkg("revdbayes") provides the Bayesian analysis of univariate extreme value models using direct random sampling from the posterior distribution, that is, without using MCMC methods.
  • The package r pkg("texmex") fit GPD models by using maximum (optionally penalised-)likelihood, or Bayesian estimation, and both classes of models may be fitted with covariates in any/all model parameters.
package function models[^1] covariates sampling[^2] prior choice generic functions
extRemes fevd 1--4,* all RWMH custom plot, summary
MCMC4Extremes ggev,gpdp 1--2,* no RWMH fixed plot, summary
revdbayes rpost 1--4 no RU custom plot, summary
texmex evm 1--2,* all IMH gaussian plot, summary, density,correlogram

[^1] model family: generalized extreme value distribution (1), generalized Pareto distribution (2), inhomogeneous Poisson process (3), order statistics/r-largest (4) or custom/other (*).

[^2] sampling: random walk Metropolis--Hastings (RWMH), exact sampling ratio-of-uniform (RU), independent Metropolis--Hastings (IMH)

[Block Maxima approach]{#UniBlockMaxima}

  • The package r pkg("climextRemes") provides functions for fitting GEV via point process fitting for extremes in climate data, providing return values, return probabilities, and return periods for stationary and nonstationary models.
  • The package r pkg("evd", priority = "core") provides functions for a wide range of univariate distributions. Modelling function allow estimation of parameters for standard univariate extreme value methods.
  • The package r pkg("evir", priority = "core") performs modelling of univariate GEV distributions by maximum likelihood fitting.
  • The package r pkg("extRemes", priority = "core") provides EVDs univariate estimation for block maxima model approache by MLE. It also incorporates a non-stationarity through the parameters of the EVDs and L-moments estimation for the stationary case for the GEV distributions. Finally, it has also Bayes estimation capabilities. A separate package r pkg("in2extRemes") provides some GUI interfaces to r pkg("extRemes", priority = "core").
  • The package r pkg("extremeStat") includes functions to fit multiple GEV distributions types available in the package r pkg("lmomco") using linear moments to estimate the parameters.
  • The package r pkg("fExtremes") provides univariate data processing and modelling. It includes clustering, block maxima identification and exploratory analysis. The estimation of stationary models for the GEV is provided by maximum likelihood and probability weighted moments.
  • The package r pkg("ismev") provides a collection of three functions to fit the GEV (diagnostic plot, MLE, likelihood profile) and follows the book of Coles (2001).
  • The package r pkg("lmom") has functions to fit probability distributions from GEV distributions to data using the low-order L-moments.
  • The package r pkg("lmomRFA") extends package r pkg("lmom") and implements all the major components for regional frequency analysis using L-moments.
  • The package r pkg("QRM") provides a function to fit GEV in Quantitative Risk Management perspective.
  • The package r pkg("Renext") provides various functions to fit the GEV distribution using an aggregated marked POT process.

Summary of GEV density functions and GEV fitting functions

package density function location scale shape fit function argdata outputS4 outputS3 outputS3par
climextRemes NA location scale shape fit_gev y NA mle NA
evd dgev loc scale shape fgev x NA estimate NA
evir dgev mu sigma xi gev data NA par.ests NA
extraDistr dgev mu sigma xi NA NA NA NA NA
extRemes devd loc scale shape fevd x NA results par
fExtremes dgev mu beta xi gevFit x fit par.ests NA
ismev NA NA NA NA gev.fit xdat NA mle NA
lmomco pdfgev xi alpha kappa NA NA NA NA NA
QRM dGEV mu sigma xi fit.GEV maxima NA par.ests NA
revdbayes dgev loc scale shape NA NA NA NA NA
SpatialExtremes dgev loc scale shape NA NA NA NA NA
texmex dgev mu sigma xi evm y NA coefficients NA
TLMoments dgev loc scale shape NA NA NA NA NA

[Extremal index estimation approach]{#UniExtremalIndex}

  • The package r pkg("evd") implements univariate estimation for extremal index estimation approach.
  • The package r pkg("evir") includes extremal index estimation.
  • The package r pkg("extRemes") also provides EVDs univariate estimation for the block maxima and poisson point process approache by MLE. It also incorporates a non-stationarity through the parameters.
  • The package r pkg("extremefit") provides modelization of exceedances over a threshold in the Pareto type tail. It computes an adaptive choice of the threshold.
  • The package r pkg("ExtremeRisks") provides risk measures such as Expectile, Value-at-Risk, for univariate independent observations and temporal dependent observations. The statistical inference is performed through parametric and non-parametric estimators. Inferential procedures such as confidence intervals, confidence regions and hypothesis testing are obtained by exploiting the asymptotic theory.
  • The package r pkg("fExtremes") provides univariate data processing and modelling. It includes extremal index estimation.
  • The package r pkg("mev") provides extremal index estimators based on interexceedance time (MLE and iteratively reweigthed least square estimators of Suveges (2007)). It provides the information matrix test statistic proposed by Suveges and Davison (2010) and MLE for the extremal index.
  • The package r pkg("ReIns") provides functions for extremal index and splicing approaches in a reinsurance perspective.
  • The package r pkg("evgam") implements a moment-based estimator of extremal index based on Ferro and Segers (2003).

[Mixture distribution or composite distribution approach]{#UniMixture}

  • The package r pkg("evmix") provides kernel density estimation and extreme value modelling. It also implements mixture extreme value models and includes help on the choice of the threshold within those models using MLE: either parametric / GPD, semi-parametric / GPD or non-parametric / GPD.

[Peak-Over-Threshold by GPD approach]{#UniPOT}

  • The package r pkg("ercv") provides a methodology to fit a generalized Pareto distribution, together with an automatic threshold selection algorithm.
  • The package r pkg("eva") provides Goodness-of-fit tests for selection of r in the r-largest order statistics and threshold selection.
  • The package r pkg("evd") includes univariate estimation for GPD approach by MLE.
  • The package r pkg("evir") performs modelling of univariate GPD by maximum likelihood fitting.
  • The package r pkg("extRemes") provides EVDs univariate estimation for GPD approach by MLE. A non-stationarity through the parameters of the EVDs and L-moments estimation for the stationnary case for the GPD distributions is also included.
  • The package r pkg("extremeStat") includes functions to fit multiple GPD distributions types available in the package r pkg("lmomco") using linear moments to estimate the parameters.
  • The package r pkg("fExtremes") includes the estimation of stationary models for the GPD by maximum likelihood and probability weighted moments.
  • The package r pkg("ismev") provides a collection of three functions to fit the GPD (diagnostic plot, MLE over a range of thresholds, likelihood profile) and follows the book of Coles (2OO1).
  • The package r pkg("lmom") includes functions to fit probability distributions from GPD to data using the low-order L-moments.
  • The package r pkg("lmomRFA") extends package r pkg("lmom") and implements all the major components for regional frequency analysis using L-moments.
  • The package r pkg("mev") provides functions to simulate data from GPD and multiple method to estimate the parameters (optimization, MLE, Bayesian methods and the method used in the r pkg("ismev") package).
  • The package r pkg("POT") provides multiple estimators of the GPD parameters (MLE, L-Moments, method of median, minimum density power divergence). L-moments diagrams and from the properties of a non-homogeneous Poisson process techniques are provided for the selection of the threshold.
  • The package r pkg("QRM") provides functions to fit and graphically assess the fit of the GPD.
  • The package r pkg("ReIns") provides a function to fit the GPD distribution as well as the extended Pareto distribution.
  • The package r pkg("Renext") provides various functions to fit and assess the GPD distribution using an aggregated marked POT process.
  • The package r pkg("SpatialExtremes", priority = "core") provides different approaches for fitting/selecting the threshold in generalized Pareto distributions. Most of them are based on minimizing the AMSE-criterion or at least by reducing the bias of the assumed GPD-model.
  • The package r pkg("texmex") fit GPD models by using maximum (optionally penalised-)likelihood, or Bayesian estimation, and both classes of models may be fitted with covariates in any/all model parameters.
  • The package r pkg("NHPoisson") provides a function to fit non-homogeneous Poisson processes for peak over threshold analysis.

Summary of GPD density functions and GPD fitting functions

package density function location scale shape fit function argdata argthres outputS4 outputS3 outputS3par
ercv NA NA NA NA fitpot data threshold NA coeff NA
eva dgpd loc scale shape gpdFit data threshold NA par.ests NA
evd dgpd loc scale shape fpot x threshold NA estimate NA
evir dgpd mu beta xi gpd data threshold NA par.ests NA
extraDistr dgpd mu sigma xi NA NA NA NA NA NA
extRemes devd loc scale shape fevd x threshold NA results par
fExtremes dgpd mu beta xi gpdFit x u fit fit par
ismev NA NA NA NA gpd.fit xdat threshold NA mle NA
lmomco pdfgpa xi alpha kappa NA NA NA NA NA NA
mev NA NA scale shape fit.gpd xdat threshold NA estimate NA
POT dgpd loc scale shape fitgpd data threshold NA fitted.values NA
QRM dGPD NA beta xi fit.GPD data threshold NA par.ests NA
ReIns dgpd mu sigma gamma GPDfit data NA NA NA NA
Renext dGPD loc scale shape fGPD x NA NA estimate NA
revdbayes dgp loc scale shape NA NA NA NA NA NA
SpatialExtremes dgpd loc scale shape gpdmle x threshold NA NA NA
tea dgpd loc scale shape gpdFit data threshold NA par.ests NA
texmex dgpd u sigma xi evm y th NA coefficients NA
TLMoments dgpd loc scale shape NA NA NA NA NA NA

[Record models:]{#UniRecord}

  • r pkg("RecordTest") studies the analysis of record-breaking events and provides non-parametric modeling/testing of a non-stationary behaviour in (extreme) records.
  • r pkg("evir") provides only a function records() for extracting records.

[Regression models:]{#UniRegression}

  • The package r pkg("VGAM") offers additive modelling for extreme value analysis. The estimation for vector generalised additive models (GAM) is performed using a backfitting algorithm and employs a penalized likelihood for the smoothing splines. It is the only package known to the authors that performs additive modelling for a range of extreme value analysis. It includes both GEV and GP distributions.
  • The package r pkg("ismev") provides a collection of functions to fit a point process with explanatory variables (diagnostic plot, MLE) and follows the book of Coles (2001).
  • The package r pkg("texmex") fit GPD models by using maximum (optionally penalised-)likelihood, or Bayesian estimation, and both classes of models may be fitted with covariates in any/all model parameters.
  • The package r pkg("evgam") provides methods for fitting various extreme value distributions with parameters of generalised additive model (GAM) form.
  • The package r pkg("GJRM") allows to fit generalized smooth/additive models (GAM like regressions) for location, scale and shape. It incorporates as margin some distributions linked to extreme value analysis and allows parametrization of location and scale for these distributions: Margin generalized Pareto, generalized Pareto II, generalized Pareto with orthogonal parametrization, discrete generalized Pareto, discrete generalized Pareto II, discrete generalized Pareto.

[Threshold selection]{#UniThreshold}

  • The package r pkg("threshr") deals with the selection of thresholds using a Bayesian leave-one-out cross-validation approach in order to compare the predictive performance resulting from a set of thresholds.
  • The package r pkg("ercv") provides a methodology to fit a generalized Pareto distribution, together with an automatic threshold selection algorithm.
  • The package r pkg("POT") provides multiple estimators of the GPD parameters (MLE, L-Moments, method of median, minimum density power divergence). L-moments diagrams and from the properties of a non-homogeneous Poisson process techniques are provided for the selection of the threshold.

[Bivariate Extreme Value Theory]{#BivEVT}

[Copula approach]{#BiCopula}

  • The package r pkg("copula") provides utilities for exploring and modelling a wide range of commonly used copulas, see also the r view("Distributions") task view (copula section).
  • The package r pkg("fCopulae") provides utilities to fit bivariate extreme copulas.

[Maxima approach]{#BiMaxima}

  • The package r pkg("evd") provides functions for multivariate distributions. Modelling function allow estimation of parameters for class of bivariate extreme value distributions. Both parametric and non-parametric estimation of bivariate EVD can be performed.
  • Nonparametric estimation of the spectral measure using a sample of pseudo-angles is available in the package r pkg("extremis") in the bivariate setting.

[Peak-Over-Threshold by GPD approach]{#BiPOT}

  • The package r pkg("evd") implements bivariate threshold modelling using censored likelihood methodology.
  • The single multivariate implementation in the package r pkg("evir") is a bivariate threshold method.
  • The package r pkg("extremefit") provides modelization of exceedances over a threshold in the Pareto type tail depending on a time covariate. It provides an adaptive choice of the threshold depending of the covariate.
  • The package r pkg("POT") provides estimators of the GPD parameters in the bivariate case.

[Tail dependence coefficient approach]{#BiTailDependence}

  • The package r pkg("RTDE") implements bivariate estimation for the tail dependence coefficient.

[Multivariate Extreme Value Theory]{#MultiEVT}

[Bayesian approach]{#MultiBayesian}

  • The package r pkg("SpatialExtremes", priority = "core") provides tools for the statistical modelling of spatial extremes using Bayesian hierarchical models (fitting, checking, selection).
  • The package r pkg("ExtremalDep") also provides function to fit a multivariate extreme value using Bayesian inference.

[Copula approach]{#MultiCopula}

  • The package r pkg("SpatialExtremes", priority = "core") provides functions to estimate a copula-based model to spatial extremes as well as model checking and selection.
  • The package r pkg("copula") provides utilities for exploring and modelling a wide range of commonly used copulas. Extreme value copulas and non-parametric estimates of extreme value copulas are implemented. See also the r view("Distributions") task view (copula section).
  • The package r pkg("SimCop") has functionalities for simulation of some bivariate extreme value distributions and the multivariate logistic model, or Gumbel copula.

[Multivariate Maxima]{#MultiMaxima}

  • The package r pkg("lmomco") is similar to the r pkg("lmom") but also implements recent advances in L-moments estimation, including L-moments for censored data, trimmed L-moments and L-moment for multivariate analysis for GEV distributions.
  • The package r pkg("SpatialExtremes", priority = "core") provides functions to fit max-stable processes to data using pairwise likelihood or spatial GEV models possibly with covariates.
  • A set of procedures for modelling parametrically and non-parametrically the dependence structure of multivariate extreme-values is provided in r pkg("ExtremalDep").
  • The r pkg("BMAmevt") package implements a Bayesian nonparametric model that uses a trans-dimensional Metropolis algorithm for fitting a Dirichlet mixture to the spectral measure based on pseudo-angles.

[Peak-Over-Threshold by GPD approach]{#MultiPOT}

  • The package r pkg("lmomco") also implements L-moments multivariate analysis for GPD distributions.
  • The package r pkg("graphicalExtremes") develops a statistical methodology for sparse multivariate extreme value models. Methods are provided for exact simulation and statistical inference for multivariate Pareto distributions on graphical structures.

[Tail dependence coefficient approach]{#MultiTailDependence}

  • The package r pkg("SpatialExtremes", priority = "core") provides functions to estimate non parametrically the extremal coefficient function as well as model checking and selection.
  • The package r pkg("ExtremeRisks") provides risk measures such as Expectile, Value-at-Risk, for multivariate independent marginals.
  • The package r pkg("tailDepFun") provides functions implementing minimal distance estimation methods for parametric tail dependence models.

[Statistical tests]{#MultiTests}

  • The r pkg("copula") package includes three tests of max-stability assumption.

[Classical graphics]{#Graphics}

Graphics for univariate extreme value analysis

Graphic name Packages Function names
Dispersion index plot POT diplot
Distribution fitting plot extremeStat distLplot
Hill plot evir hill
Hill plot evmix hillplot
Hill plot extremefit hill
Hill plot QRM hillPlot
Hill plot ReIns Hill
Hill plot ExtremeRisks HTailIndex
L-moment plot POT lmomplot
Mean residual life plot POT mrlplot
Mean residual life plot evd mrlplot
Mean residual life plot evir meplot
Mean residual life plot evmix mrlplot
Mean residual life plot ismev mrl.plot
Mean residual life plot QRM MEplot
Mean residual life plot ReIns MeanExcess
Pickand's plot evmix pickandsplot
QQ Pareto plot POT qplot
QQ Pareto plot RTDE qqparetoplot
QQ Pareto plot QRM plotFittedGPDvsEmpiricalExcesses
QQ Pareto plot ReIns ParetoQQ
QQ Exponential plot QRM QQplot
QQ Exponential plot ReIns ExpQQ
QQ Exponential plot Renext expplot
QQ Lognormal plot ReIns LognormalQQ
QQ Weibull plot ReIns WeibullQQ
QQ Weibull plot Renext weibplot
Risk measure plot QRM RMplot
Threshold choice plot evd tcplot
Threshold choice plot evmix tcplot
Threshold choice plot POT tcplot
Threshold choice plot QRM xiplot
Return level plot POT retlev
Return level plot POT Return
Return level plot Renext plot,lines

Graphics for multivariate extreme value analysis

Graphic Package Function
Angular densities plot ExtremalDep AngDensPlot
Bivariate threshold choice plot evd bvtcplot
Dependence measure (chi) plot POT chimeas
Dependence measure (chi) plot evd chiplot
Dependence diagnostic plot within time series POT tsdep.plot
Extremal index plot POT exiplot
Extremal index plot evd exiplot
(2D)map for a max-stable process SpatialExtremes map
madogram for a max-stable process SpatialExtremes madogram
madogram for a max-stable process ExtremalDep madogram
F-madogram for a max-stable process SpatialExtremes fmadogram
lambda-madogram for a max-stable process SpatialExtremes lmadogram
Multidimensional Hill plot ExtremeRisks MultiHTailIndex
Pickands' dependence function plot POT pickdep
Pickands' dependence function plot ExtremalDep bbeed
QQ-plot for the extremal coefficient SpatialExtremes qqextcoeff
Spectral density plot POT specdens

Bibliography

Review papers

  • L. Belzile, C. Dutang, P. Northrop, T. Opitz (2023), A modeler's guide to extreme value software, Extremes, r doi("10.1007/s10687-023-00475-9").
  • E. Gilleland, M. Ribatet, A. Stephenson (2013). A Software Review for Extreme Value Analysis, Extremes, 16, 103-119, r doi("10.1007/s10687-012-0155-0").
  • A.G. Stephenson, E. Gilleland (2006). Software for the analysis of extreme events: The current state and future directions. Extremes, 8, 87–109, r doi("10.1007/s10687-006-7962-0").

Classical books

  • R.-D. Reiss, M. Thomas (2007). Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields, Springer-Verlag, r doi("10.1007/978-3-7643-7399-3").
  • L. de Haan, A. Ferreira (2006). Extreme Value Theory: An Introduction, Springer-Verlag, r doi("10.1007/0-387-34471-3").
  • J. Beirlant, Y. Goegebeur, J. Teugels, J. Segers (2004). Statistics of Extremes: Theory and Applications , John Wiley & Sons, r doi("10.1002/0470012382").
  • B. Finkenstaedt, H. Rootzen (2004). Extreme Values in Finance, Telecommunications, and the Environment , Chapman & Hall/CRC, r doi("10.1201/9780203483350").
  • S. Coles (2001). An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag, r doi("10.1007/978-1-4471-3675-0").
  • P. Embrechts, C. Klueppelberg, T. Mikosch (1997). Modelling Extremal Events for Insurance and Finance, Springer-Verlag, r doi("10.1007/978-3-642-33483-2").
  • S.I. Resnick (1987). Extreme Values, Regular Variation and Point Processes, Springer-Verlag.

Scientific papers

  • Suveges and Davison (2010), Model misspecification in peaks over threshold analysis. Annals of Applied Statistics, 4(1), 203-221.
  • M. Suveges (2007). Likelihood estimation of the extremal index. Extremes, 10(1), 41-55, r doi("10.1007/s10687-007-0034-2").
  • R.L. Smith (1987). Approximations in extreme value theory. Technical report 205, Center for Stochastic Process, University of North Carolina, 1--34.