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CRAN Task View: Extreme Value Analysis

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 theFinance task view, for environmental analysis in theEnvironmetrics task view. General implementation of probability distributions is studied in theDistributions 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

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 aslog andlower tail, see the viewDistributions 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, thenieve package provides symbolic differentiation for two EVT probability distribution (GPD and GEV) in order to compute the log-likelihood.

Bayesian approach

• The packageextRemes provides bayesian estimation.
  
• The packageMCMC4Extremes proposes some functions to perform posterior estimation for some distribution, with an emphasis to extreme value distributions.
• The packagerevdbayes provides the Bayesian analysis of univariate extreme value models using direct random sampling from the posterior distribution, that is, without using MCMC methods.
  
• The packagetexmex 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.

[^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

• The packageclimextRemes 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 packageevd provides functions for a wide range of univariate distributions. Modelling function allow estimation of parameters for standard univariate extreme value methods.
• The packageevir performs modelling of univariate GEV distributions by maximum likelihood fitting.
• The packageextRemes 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 packageextremeStat includes functions to fit multiple GEV distributions types available in the packagelmomco using linear moments to estimate the parameters.
• The packagefExtremes 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 packageismev provides a collection of three functions to fit the GEV (diagnostic plot, MLE, likelihood profile) and follows the book of Coles (2001).
  
• The packagelmom has functions to fit probability distributions from GEV distributions to data using the low-order L-moments.
• The packagelmomRFA extends packagelmom and implements all the major components for regional frequency analysis using L-moments.
• The packageQRM provides a function to fit GEV in Quantitative Risk Management perspective.
• The packageRenext provides various functions to fit the GEV distribution using an aggregated marked POT process.

Summary of GEV density functions and GEV fitting functions

Extremal index estimation approach

• The packageevd implements univariate estimation for extremal index estimation approach.
• The packageevir includes extremal index estimation.
• The packageextRemes 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 packageextremefit provides modelization of exceedances over a threshold in the Pareto type tail. It computes an adaptive choice of the threshold.
  
• The packageExtremeRisks 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 packagefExtremes provides univariate data processing and modelling. It includes extremal index estimation.
• The packagemev 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 packageReIns provides functions for extremal index and splicing approaches in a reinsurance perspective.
• The packageevgam implements a moment-based estimator of extremal index based on Ferro and Segers (2003).

Mixture distribution or composite distribution approach

• The packageevmix 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

• The packageercv provides a methodology to fit a generalized Pareto distribution, together with an automatic threshold selection algorithm.
• The packageeva provides Goodness-of-fit tests for selection of r in the r-largest order statistics and threshold selection.
• The packageevd includes univariate estimation for GPD approach by MLE.
• The packageevir performs modelling of univariate GPD by maximum likelihood fitting.
• The packageextRemes 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 packageextremeStat includes functions to fit multiple GPD distributions types available in the packagelmomco using linear moments to estimate the parameters.
• The packagefExtremes includes the estimation of stationary models for the GPD by maximum likelihood and probability weighted moments.
• The packageismev 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 packagelmom includes functions to fit probability distributions from GPD to data using the low-order L-moments.
• The packagelmomRFA extends packagelmom and implements all the major components for regional frequency analysis using L-moments.
• The packagemev provides functions to simulate data from GPD and multiple method to estimate the parameters (optimization, MLE, Bayesian methods and the method used in theismev package).
  
• The packagePOT 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 packageQRM provides functions to fit and graphically assess the fit of the GPD.
• The packageReIns provides a function to fit the GPD distribution as well as the extended Pareto distribution.
  
• The packageRenext provides various functions to fit and assess the GPD distribution using an aggregated marked POT process.
• The packageSpatialExtremes 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 packagetexmex 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 packageNHPoisson provides a function to fit non-homogeneous Poisson processes for peak over threshold analysis.
• The packageheavytails provides peak over threshold analysis as well as other types of estimators.

Summary of GPD density functions and GPD fitting functions

Record models:

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

Regression models:

• The packageVGAM 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 packageismev provides a collection of functions to fit a point process with explanatory variables (diagnostic plot, MLE) and follows the book of Coles (2001).
• The packagetexmex 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 packageevgam provides methods for fitting various extreme value distributions with parameters of generalised additive model (GAM) form.
• The packageGJRM 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

• The packagethreshr 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 packageercv provides a methodology to fit a generalized Pareto distribution, together with an automatic threshold selection algorithm.
• The packagePOT 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

Copula approach

• The packagecopula provides utilities for exploring and modelling a wide range of commonly used copulas, see also theDistributions task view (copula section).
  
• The packagefCopulae provides utilities to fit bivariate extreme copulas.

Maxima approach

• The packageevd 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 packageextremis in the bivariate setting.

Peak-Over-Threshold by GPD approach

• The packageevd implements bivariate threshold modelling using censored likelihood methodology.
• The single multivariate implementation in the packageevir is a bivariate threshold method.
• The packageextremefit 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 packagePOT provides estimators of the GPD parameters in the bivariate case.

Tail dependence coefficient approach

• The packageRTDE implements bivariate estimation for the tail dependence coefficient.

Multivariate Extreme Value Theory

Bayesian approach

• The packageSpatialExtremes provides tools for the statistical modelling of spatial extremes using Bayesian hierarchical models (fitting, checking, selection).
• The packageExtremalDep also provides function to fit a multivariate extreme value using Bayesian inference.

Copula approach

• The packageSpatialExtremes provides functions to estimate a copula-based model to spatial extremes as well as model checking and selection.
• The packagecopula 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 theDistributions task view (copula section).
• The packageSimCop has functionalities for simulation of some bivariate extreme value distributions and the multivariate logistic model, or Gumbel copula.

Multivariate Maxima

• The packagelmomco is similar to thelmom 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 packageSpatialExtremes 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 inExtremalDep.
• TheBMAmevt 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

• The packagelmomco also implements L-moments multivariate analysis for GPD distributions.
• The packagegraphicalExtremes 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

• The packageSpatialExtremes provides functions to estimate non parametrically the extremal coefficient function as well as model checking and selection.
• The packageExtremeRisks provides risk measures such as Expectile, Value-at-Risk, for multivariate independent marginals.
• The packagetailDepFun provides functions implementing minimal distance estimation methods for parametric tail dependence models.

Statistical tests

• Thecopula package includes three tests of max-stability assumption.

Classical graphics

Graphics for univariate extreme value analysis

Graphics for multivariate extreme value analysis

Bibliography

Review papers

• L. Belzile, C. Dutang, P. Northrop, T. Opitz (2023),A modeler’s guide to extreme value software, Extremes,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,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,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,doi:10.1007/978-3-7643-7399-3.
• L. de Haan, A. Ferreira (2006).Extreme Value Theory: An Introduction, Springer-Verlag,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,doi:10.1002/0470012382.
• B. Finkenstaedt, H. Rootzen (2004).Extreme Values in Finance, Telecommunications, and the Environment , Chapman & Hall/CRC,doi:10.1201/9780203483350.
• S. Coles (2001).An Introduction to Statistical Modeling of Extreme Values, Springer-Verlag,doi:10.1007/978-1-4471-3675-0.
• P. Embrechts, C. Klueppelberg, T. Mikosch (1997).Modelling Extremal Events for Insurance and Finance, Springer-Verlag,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,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.

CRAN packages

Other resources

• CRAN Task View:Distributions
• CRAN Task View:Environmetrics
• CRAN Task View:Finance