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
Finance
task view,
for environmental analysis in the
Environmetrics
task view.
General implementation of probability distributions is studied
in the
Distributions
task view.
The maintainers gratefully acknowledge
E. Gilleland, M. Ribatet and A. Stephenson for their review for extreme value analysis
packages (2013) 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 let us know.

Block Maxima approach:

The package
evd
provides functions for a wide range of univariate distributions. Modelling function allow estimation of parameters for standard univariate extreme value methods.

The package
evdbayes
provides the Bayesian analysis of univariate extreme value models using MCMC methods. It uses likelihood to estimate the parameters of the GEV distributions.

The package
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
evir
performs modelling of univariate GEV distributions by maximum likelihood fitting.

The package
extRemes
provides EVDs univariate estimation for block maxima model approache by MLE. It also incorporates a nonstationarity through the parameters of the EVDs and Lmoments estimation for the stationary case for the GEV distributions. Finally, it has also Bayes estimation capabilities.
A separate package
in2extRemes
provides some GUI interfaces to
extRemes.

The package
extremeStat
includes functions to fit multiple GEV distributions types available in the package
lmomco
using linear moments to estimate the parameters.

The package
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
lmom
has functions to fit probability distributions from GEV distributions to data using the loworder Lmoments.

The package
lmomRFA
extends package
lmom
and implements all the major components for regional frequency analysis using Lmoments.

The package
texmex
provides a univariate extreme value modeling approach for GEV distributions by bootstrap, MCMC simulations and maximum likelihood for parameter estimation.

The package
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
mev
has a function using the Smith (1987) penultimate approximation for block maxima approach.

The package
Renext
provides various functions to fit the GEV distribution using an aggregated marked POT process.

PeakOverThreshold by GPD approach:

The package
evd
includes univariate estimation for GPD approach by MLE.

The Bayesian analysis of univariate extreme value models using MCMC methods in the package
evdbayes
includes the likelihood to estimate GP distributions.

The package
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
evir
performs modelling of univariate GPD by maximum likelihood fitting.

The package
evmix
provides kernel density estimation and extreme value modelling. It also implements extreme value models and includes help on the choice of the threshold within those models using MLE.

The package
extremefit
provides modelization of exceedances over a threshold in the Pareto type tail. It computes an adaptive choice of the threshold.

The package
extRemes
provides EVDs univariate estimation for GPD approach by MLE. A nonstationarity through the parameters of the EVDs and Lmoments estimation for the stationnary case for the GPD distributions is also included.

The package
extremeStat
includes functions to fit multiple GPD distributions types available in the package
lmomco
using linear moments to estimate the parameters.

The package
fExtremes
includes the estimation of stationary models for the GPD by maximum likelihood and probability weighted moments.

The package
lmom
includes functions to fit probability distributions from GPD to data using the loworder Lmoments.

The package
lmomRFA
extends package
lmom
and implements all the major components for regional frequency analysis using Lmoments.

The package
texmex
provides a univariate extreme value modeling approach for GPD distributions by bootstrap, MCMC simulations and maximum likelihood for parameter estimation.

The package
POT
provides multiple estimators of the GPD parameters (MLE, LMoments, method of median, minimum density power divergence). Lmoments diagrams and from the properties of a nonhomogeneous Poisson process techniques are provided for the selection of the threshold.

The package
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
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
ismev
package).

The package
QRM
provides functions to fit and graphically assess the fit of the GPD.

The package
Renext
provides various functions to fit and assess the GPD distribution using an aggregated marked POT process.

The package
threshr
deals with the selection of thresholds using
a Bayesian leaveoneout crossvalidation approach in order to compare the
predictive performance resulting from a set of thresholds.

Extremal index estimation approach:

The package
evd
implements univariate estimation for extremal index estimation approach.

The package
evdbayes
includes point process characterisation

the package
evir
includes extremal index estimation.

The package
extRemes
also provides EVDs univariate estimation for the block maxima and poisson point process approache by MLE. It also incorporates a nonstationarity through the parameters.

The package
fExtremes
provides univariate data processing and modelling. It includes extremal index estimation.

The package
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
ReIns
provides functions for extremal index and splicing approaches in a reinsurance perspective.

The package
ptsuite
implements various estimation methods
for the shape parameter of Pareto distributed data.

Regression models

The package
VGAM
offers additive modelling for extreme value analysis. The estimation for vector generalised additive models 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
ismev
provides a collection of functions to fit a point process with explanatory variables (diagnostic plot, MLE) and follows the book of Coles (2001).

Copula approach:

The package
copula
provides utilities for exploring and modelling a wide range of commonly used copulas, see also the
Distributions
task view (copula section).

Mixture distribution or composite distribution approach:

The package
evmix
provides several functions to
fit mixture distributions: either parametric / GPD,
semiparametric / GPD or nonparametric / GPD.

Block Maxima approach:

The package
evd
provides functions for multivariate distributions. Modelling function allow estimation of parameters for class of bivariate extreme value distributions. Both parametric and nonparametric estimation of bivariate EVD can be performed.

PeakOverThreshold by GPD approach:

The package
evd
implements bivariate threshold modelling using censored likelihood methodology.

The single multivariate implementation in the package
evir
is a bivariate threshold method.

The package
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
POT
provides estimators of the GPD parameters in the bivariate case.

Tail dependence coefficient approach:

The package
RTDE
implements bivariate estimation for the tail dependence coefficient.

Block Maxima approach:

The package
lmomco
is similar to the
lmom
but also implements recent advances in Lmoments estimation, including Lmoments for censored data, trimmed Lmoments and Lmoment for multivariate analysis for GEV distributions.

The package
SpatialExtremes
provides maxstable processes and uses weighted pairwise likelihood estimator to fit the processes.

PeakOverThreshold by GPD approach:

The package
lmomco
also implements Lmoments multivariate analysis for GPD distributions.

The package
SpatialExtremes
includes GPD method to modelize spatial extremes using Bayesian hierarchical models with a conditionnal independence assumption.

The package
texmex
provides a conditional multivariate extreme value modeling approach which is useful for multivariate processes where interest is in events occuring such that only a subset of the margins are extreme.

Copula approach:

The package
copula
provides utilities for exploring and modelling a wide range of commonly used copulas. Extreme value copulas and nonparametric estimates of extreme value copulas are implemented. See also the
Distributions
task view (copula section).

The package
SpatialExtremes
includes copula distributions.
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

Lmoment 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

texmex

mrl

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

texmex

rl

Return level plot

POT

retlev

Return level plot

POT

Return

Return level plot

Renext

plot,lines

Graphics for multivariate extreme value analysis
Bivariate threshold choice plot

evd

bvtcplot

Dependence measure (chi) plot

POT

chimeas

Dependence measure (chi) plot

evd

chiplot

Dependence measure (chi) plot

texmex

chi

Dependence diagnostic plot within time series

POT

tsdep.plot

Extremal index plot

POT

exiplot

Extremal index plot

evd

exiplot

Madogram

SpatialExtremes

madogram

FMadogram

SpatialExtremes

fmadogram

LMadogram

SpatialExtremes

lmadogram

Variogram

SpatialExtremes

variogram

Pickands' dependence function plot

POT

pickdep

Spectral density plot

POT

specdens


E. Gilleland, M. Ribatet, A. Stephenson (2013).
A Software Review for Extreme Value Analysis,
Extremes
,
16
, 103119.

R.D. Reiss, M. Thomas (2007).
Statistical Analysis of Extreme Values with Applications to Insurance, Finance, Hydrology and Other Fields
, SpringerVerlag.

L. de Haan, A. Ferreira (2006).
Extreme Value Theory: An Introduction
, SpringerVerlag.

J. Beirlant, Y. Goegebeur, J. Teugels, J. Segers (2004).
Statistics of Extremes: Theory and Applications
, John Wiley & Sons.

B. Finkenstaedt, H. Rootzen (2004).
Extreme Values in Finance, Telecommunications, and the Environment
, Chapman & Hall/CRC.

S. Coles (2001).
An Introduction to Statistical Modeling of Extreme Values
, SpringerVerlag.

P. Embrechts, C. Klueppelberg, T. Mikosch (1997).
Modelling Extremal Events for Insurance and Finance
, SpringerVerlag.

S.I. Resnick (1987).
Extreme Values, Regular Variation and Point Processes
, SpringerVerlag.

Smith, R.L. (1987). Approximations in extreme value theory. Technical report 205, Center for
Stochastic Process, University of North Carolina, 1–34.

Suveges (2007) Likelihood estimation of the extremal index. Extremes, 10(1), 4155.

Suveges and Davison (2010), Model misspecification in peaks over threshold analysis. Annals of
Applied Statistics, 4(1), 203221.