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

  • Bayesian approach
  • Block Maxima approach
  • Extremal index estimation approach
  • Mixture distribution or composite distribution approach
  • Peak-Over-Threshold by GPD approach
  • Record models
  • Regression models
  • Threshold selection

• Bivariate Extreme Value Theory

  • Copula approach
  • Maxima approach
  • Peak-Over-Threshold by GPD approach
  • Tail dependence coefficient approach

• Multivariate Extreme Value Theory

  • Bayesian approach
  • Copula approach
  • Multivariate Maxima
  • Peak-Over-Threshold by GPD approach
  • Tail dependence coefficient approach
  • Statistical tests

• Classical graphics

• Bibliography

[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.
• 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.
• The package r pkg("heavytails") provides peak over threshold analysis as well as other types of estimators.

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
heavytails	NA	NA	NA	NA.	pot_estimator	data	u	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.