Extreme value theory is a branch of statistics dealing with the extreme deviations from the bulk of probability distributions. More specifically, it focuses on the limiting distributions for the minimum or the maximum of a large collection of random observations from the same arbitrary (unknown) distribution. Let x₁< ... < x_n denote n ordered observations from a random variable X representing some quantity of interest. A p_n-quantile of X is the value q_pn such that the probability that X is greater than q_pn is p_n, i.e. P(X > q_pn) =p_n. When p_n < 1/n, such a quantile is said to be extreme since it is usually greater than the maximum observation x_n (see Figure 1). To estimate such extreme quantiles requires therefore specific methods to extrapolate information beyond the observed values of X. Those methods are based on Extreme value theory. This kind of issues appeared in hydrology. One objective was to assess risk for highly unusual events, such as 100-year floods, starting from flows measured over 50 years.

   

PhD theseshere

T. Moins, "Bayesian computational methods for estimating extreme quantiles from environmental data", Université Grenoble Alpes, septembre 2023. [pdf].

M. Allouche, "Contributions to generative modeling and dictionary learning: Theory and application", Institut Polytechnique de Paris, décembre 2022 [pdf].

M. Bousebata, "Bayesian estimation of extreme risk measures: Implication for the insurance of natural disasters", Université Grenoble Alpes, mars 2022 [pdf].

A. A. Ahmad, "Modélisation semi-paramétrique des extrêmes conditionnels", Université Gaston Berger, Sénégal, septembre 2020 [pdf].

C. Albert, "Estimation des limites d'extrapolation par les lois de valeurs extrêmes. Application à des données environnementales", Université Grenoble Alpes, décembre 2018 [pdf].

E. Deme, "Quelques contributions à la théorie univariée des valeurs extrêmes. Estimation des mesures de risque actuariel pour des pertes à queues lourdes", Université Gaston Berger, Sénégal, juin 2013 [pdf].

J. El-methni, "Différentes contributions à l'estimation de quantiles extrêmes", Université Grenoble 1, octobre 2013 [pdf].

G. Stupfler "Un modèle de Markov caché en assurance et Estimation de frontière et de point terminal", Université de Strasbourg, novembre 2011 [pdf].

A. Lekina, " Estimation non-paramétrique des quantiles extrêmes conditionnels , Université Grenoble 1, octobre 2010 [pdf].

L. Gardes, "Estimation d'une fonction quantile extrême", Université Montpellier 2, octobre 2003 [pdf].

M. Garrido, "Modélisation des événements rares et estimation des quantiles extrêmes, méthodes de sélection de modèles pour les queues de distribution", Université Grenoble 1, juin 2002 [pdf].

Conditional extremes

J. Arbel, S.Girard & H. Lorenzo, "Shrinkage for Extreme Partial Least Squares", Statistics and Computing, 34, 181, 2024. [Associated technical report: pdf], [Code: github].

M. Bousebata, G. Enjolras & S. Girard, "Extreme Partial Least-Squares", Journal of Multivariate Analysis, 194, 105101, 2023. [Associated technical report: pdf], [Code: github].

S. Girard, G. Stupfler & A. Usseglio-Carleve, "Nonparametric extreme conditional expectile estimation", Scandinavian Journal of Statistics, 49, 78--115, 2022. [Associated technical report: pdf].

S. Girard, G. Stupfler & A. Usseglio-Carleve, "Functional estimation of extreme conditional expectiles", Econometrics and Statistics, 21, 131--158, 2022. [Associated technical report: pdf].

S. Girard, G. Stupfler & A. Usseglio-Carleve, "Extreme Lp-quantile kernel regression", In Advances in Contemporary Statistics and Econometrics, A. Daouia and A. Ruiz-Gazen (eds.), Springer, 197-219, 2021. [Associated technical report:pdf].

A. Ahmad, E. Deme, A. Diop, S. Girard & A. Usseglio-Carleve. "Estimation of extreme quantiles from heavy-tailed distributions in a location-dispersion regression model", Electronic Journal of Statistics, 14, 4421--4456, 2020. [Associated technical report: pdf].

A. Ag Ahmad, E. Deme, A. Diop & S. Girard. "Estimation of the tail-index in a conditional location-scale family of heavy-tailed distributions", Dependence Modeling, 7, 394--417, 2019. [associated technical report: pdf].

L. Gardes & S. Girard. "On the estimation of the functional Weibull tail-coefficient", Journal of Multivariate Analysis, 146, 24--45, 2016 [associated technical report: pdf].

S. Girard & S. Louhichi. "On the strong consistency of the kernel estimator of extreme conditional quantiles", In E. Ould-Said et al, editors, Functional Statistics and Applications, Springer, pages 59--77, 2015. [associated technical report:pdf].

L. Gardes & S. Girard. "Nonparametric estimation of the conditional tail copula", Journal of Multivariate Analysis, 137, 1--16, 2015. [associated technical report: pdf].

A. Daouia, L. Gardes & S. Girard. "On kernel smoothing for extremal quantile regression", Bernoulli, 19, 2557--2589, 2013. [associated technical report: pdf].

L. Gardes & S. Girard. "Functional kernel estimators of large conditional quantiles", Electronic Journal of Statistics, 6, 1715--1744, 2012. [associated technical report: pdf].

A. Daouia, L. Gardes & S. Girard "Nadaraya's estimates for large quantiles and free disposal support curves" In I. Van Keilegom and P. Wilson, editors, Exploring research frontiers in contemporary statistics and econometrics, pages 1--22, Springer, 2012. [Associated technical report: pdf].

J. Carreau & S. Girard. "Spatial extreme quantile estimation using a weighted log-likelihood approach", Journal de la Société Française de Statistique, 152(3), 66--83 , 2011. [pdf].

L. Gardes & S. Girard "Functional kernel estimators of conditional extreme quantiles" In F. Ferraty, editor, Recent advances in functional data analysis and related topics, pages 135-140, Springer, Physica-Verlag, 2011. [Associated technical report: pdf].

A. Daouia, L. Gardes, S. Girard & A. Lekina. "Kernel estimators of extreme level curves", Test, 20(2), 311--333, 2011. [Associated technical report: pdf].

L. Gardes & S. Girard. "Conditional extremes from heavy-tailed distributions: An application to the estimation of extreme rainfall return levels", Extremes, 13(2), 177--204, 2010. [Associated technical report: pdf].

L. Gardes, S. Girard & A. Lekina. "Functional nonparametric estimation of conditional extreme quantiles", Journal of Multivariate Analysis, 101, 419--433, 2010. [Associated technical report: pdf].

L. Gardes & S. Girard. "A moving window approach for nonparametric estimation of the conditional tail index", Journal of Multivariate Analysis, 99, 2368--2388, 2008. [Associated technical report: pdf].

Estimation of risk measures

M. Allouche, S. Girard & E. Gobet, "Learning extreme Expected Shortfall and Conditional Tail Moments with neural networks. Application to cryptocurrency data", Neural Networks, 182, 106903, 2025. [associated technical report:pdf].

M. Allouche, J. El-methni & S. Girard, "Reduced-bias estimation of the extreme conditional tail expectation for Box-Cox transforms of heavy-tailed distributions", Journal of Statistical Planning and Inference, 233, 106189, 2024. [associated technical report:pdf], [code].

J. Arbel, S. Girard, H. Nguyen & A. Usseglio-Carleve, "Multivariate expectile-based distribution: properties, Bayesian inference, and applications", Journal of Statistical Planning and Inference, 225, 146--170, 2023. [associated technical report: pdf], [code].

S. Girard, G. Stupfler & A. Usseglio-Carleve, "On automatic bias reduction for extreme expectile estimation", Statistics and Computing, 32, 64, 2022. [associated technical report: pdf], [code].

S. Girard, G. Stupfler & A. Usseglio-Carleve, "Extreme conditional expectile estimation in heavy-tailed heteroscedastic regression models", Annals of Statistics, 49(6), 3358--3382, 2021. [associated technical report: pdf].

L. Gardes & S. Girard, "On the estimation of the variability in the distribution tail", Test, 30, 884--907, 2021. [associated technical report: pdf].

A. Daouia, S. Girard & G. Stupfler, "ExpectHill estimation, extreme risk and heavy tails", Journal of Econometrics, 221(1), 97--117, 2021. [associated technical report: pdf].

A. Daouia, S. Girard & G. Stupfler, "Tail expectile process and risk assessment", Bernoulli, 26, 531--556, 2020. [associated technical report: pdf].

L. Gardes, S. Girard & G. Stupfler, "Beyond tail median and conditional tail expectation: extreme risk estimation using tail Lp-optimisation", Scandinavian Journal of Statistics, 47, 922--949, 2020. [associated technical report: pdf].

A. Daouia, S. Girard & G. Stupfler. "Extreme M-quantiles as risk measures: From L1 to Lp optimization", Bernoulli, 25, 264--309, 2019. [associated technical report: pdf].

J. El Methni, L. Gardes & S. Girard. "Kernel estimation of extreme regression risk measures", Electronic Journal of Statistics, 12, 359--398, 2018. [associated technical report: pdf].

A. Daouia, S. Girard & G. Stupfler. "Estimation of Tail Risk based on Extreme Expectiles", Journal of the Royal Statistical Society series B, 80, 262--292, 2018. [associated technical report: pdf].

S. Girard & G. Stupfler. "Intriguing properties of extreme geometric quantiles", REVSTAT - Statistical Journal, 15, 107--139, 2017. [associated technical report: pdf].

S. Girard & G. Stupfler. "Extreme geometric quantiles in a multivariate regular variation framework", Extremes, 18, 629--663, 2015. [associated technical report: pdf].

E. Deme, S. Girard & A. Guillou. "Reduced-bias estimators of the Conditional Tail Expectation for heavy-tailed distributions" In M. Hallin et al, editors, Mathematical Statistics and Limit Theorems, pages 105--123, Springer, 2015. [Associated technical report: pdf].

J. El Methni, L. Gardes & S. Girard. "Nonparametric estimation of extreme risk measures from conditional heavy-tailed distributions", Scandinavian Journal of Statistics, 41, 988--1012, 2014. [associated technical report: pdf].

E. Deme, S. Girard & A. Guillou. "Reduced-bias estimator of the Proportional Hazard Premium for heavy-tailed distributions", Insurance: Mathematics and Economics, 52, 550--559, 2013. [Associated technical report: pdf].

Weibull-tail distributions

J. El Methni & S. Girard. "A refined extreme quantile estimator for Weibull tail-distributions", REVSTAT - Statistical Journal, to appear, 2024. [Associated technical report: pdf].

M. Vladimirova, S. Girard, N. Hien & J. Arbel. "Sub-Weibull distributions: generalizing sub-Gaussian and sub-Exponential properties to heavier-tailed distributions", Stat, 9, e318, 2020. [Associated technical report: pdf].

L. Gardes & S. Girard. "Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique", Journal de la Société Française de Statistique, 154, 98--118, 2013. [Journal paper: pdf] [Associated technical report: pdf].

J. El Methni, L. Gardes, S. Girard & A. Guillou. "Estimation of extreme quantiles from heavy and light tailed distributions", Journal of Statistical Planning and Inference, 142(10), 2735--2747, 2012. [Associated technical report: pdf].

L. Gardes, S. Girard & A. Guillou. "Weibull tail-distributions revisited: a new look at some tail estimators", Journal of Statistical Planning and Inference, 141(1), 429--444, 2011. [Associated technical report: pdf].

J. Diebolt, L. Gardes, S. Girard & A. Guillou. "Bias-reduced extreme quantiles estimators of Weibull-tail distributions", Journal of Statistical Planning and Inference, 138, 1389--1401, 2008. [Associated technical report: ps/pdf].

L. Gardes & S. Girard. "Estimation of the Weibull tail-coefficient with linear combination of upper order statistics", Journal of Statistical Planning and Inference, 138, 1416--1427, 2008. [Associated technical report: ps/pdf].

J. Diebolt, L. Gardes, S. Girard & A. Guillou. "Bias-reduced estimators of the Weibull tail-coefficient", Test, 17, 311--331, 2008. [Associated technical report: pdf].

L. Gardes & S. Girard. "Comparison of Weibull tail-coefficient estimators", REVSTAT - Statistical Journal, 4(2):163-188, 2006. [Associated technical report: ps/pdf], [Journal paper: pdf].

L. Gardes & S. Girard. "Estimating extreme quantiles of Weibull tail-distributions", Communication in Statistics - Theory and Methods, 34, 1065-1080, 2005. [Associated technical report: pdf].

S. Girard. "A Hill type estimate of the Weibull tail-coefficient", Communication in Statistics - Theory and Methods , 33(2), 205-234, 2004. [Associated technical report: pdf ps].

Bayesian approach

T. Moins, J. Arbel, S. Girard & A. Dutfoy. "Reparameterization of extreme value framework for improved Bayesian workflow", Computational Statistics and Data Analysis, 187, 107807, 2023. [Associated technical report: pdf], [code: github].

J. Diebolt, M. El-Aroui, M. Garrido & S. Girard. "Quasi-conjugate bayes estimates for GPD parameters and application to heavy tails modelling", Extremes, 8:57--78, 2005. [Associated technical report: ps/pdf].

Estimation of the extreme value index

M. Stehlik, J. Kiselak, M. Vaciulis, P. Jordanova, L.N. Soza, Z. Fabian, P. Hermann, L. Strelec, A. Rivera, S. Girard & S. Torres, "Priority statement and some properties of t-lgHill estimator", Extremes, 23, 493--499, 2020. [Associated technical report: pdf].

P. Jordanova, Z. Fabian, P. Hermann, L. Strelec, A. Rivera, S. Girard, S. Torres & M. Stehlik. "Weak properties and robustness of t-Hill estimators", Extremes, 19, 591--626, 2016. [Associated technical report: pdf].

E. Deme, L. Gardes & S. Girard. "On the estimation of the second order parameter for heavy-tailed distributions", REVSTAT - Statistical Journal, 11, 277--299, 2013. [Associated technical report: pdf].

L. Gardes & S. Girard. "Asymptotic properties of a Pickands type estimator of the extreme value index", In Louis R. Velle, editor, Focus on probability theory, Nova Science, New-York, 133-149, 2006. [Associated technical report: ps/pdf or ps].

L. Gardes & S. Girard. "Asymptotic distribution of a Pickands-type estimator of the extreme value index", Comptes-Rendus de l'Académie des Sciences, t. 341, Série I:53-58, 2005.

Extreme quantiles estimation

M. Allouche, S. Girard & E. Gobet, "Estimation of extreme quantiles from heavy-tailed distributions with neural networks", Statistics and Computing, 34, 12, 2024. [Associated technical report: pdf, Software: code.]

M. Allouche, J. El-methni & S. Girard. "A refined Weissman estimator for extreme quantiles", Extremes, 26, 545--572, 2023. [Associated technical report: pdf, Software: code].

C. Albert, A. Dutfoy & S. Girard. "Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles", Extremes, 23, 349--380, 2020. [Associated technical report: pdf].

C. Albert, A. Dutfoy, L.Gardes & S. Girard. "An extreme quantile estimator for the log-generalized Weibull-tail model", Econometrics and Statistics, 13, 137--174, 2020. [Associated technical report: pdf].

J. Diebolt, M. Garrido & S. Girard "Asymptotic normality of the ET method for extreme quantile estimation. Application to the ET test", Comptes-Rendus de l'Académie des Sciences, t. 337, Série I:213-218, 2003. [Associated technical report: ps/pdf].

J. Diebolt & S. Girard. "A Note on the asymptotic normality of the ET method for extreme quantile estimation", Statistics and Probability Letters, 62(4), 397-406, 2003. [Associated technical report: ps/pdf].

S. Girard & J. Diebolt. "Consistency of the ET method and smooth variations", Comptes-Rendus de l'Académie des Sciences, t. 329, Série I, 821-826, 1999.

Endpoint estimation

S. Girard, A. Guillou & G. Stupfler. "Estimating an endpoint with high order moments in the Weibull domain of attraction", Statistics and Probability Letters, 82, 2136-2144, 2012. [Associated technical report: pdf].

S. Girard, A. Guillou & G. Stupfler. "Estimating an endpoint with high order moments", Test, 21, 697--729, 2012. [Associated technical report: pdf].

Miscellaneous

S. Girard, T. Opitz & A. Usseglio-Carleve. "ANOVEX: ANalysis of Variability for heavy-tailed EXtremes", Electronic Journal of Statistics, 18(2), 5258--5303, 2024. [Associated technical report: pdf].

M. Allouche, S. Girard & E. Gobet, "EV-GAN: Simulation of extreme events with ReLU neural networks", Journal of Machine Learning Research, 23(150), 1--39, 2022. [Associated technical report: pdf].

J. Diebolt, M. Garrido & S. Girard "A goodness-of-fit test for the distribution tail" In M. Ahsanulah and S. Kirmani, editors, Extreme value distributions, p. 95-109, Nova Science, New-York, 2007. [Associated technical report: pdf].

Applications

O. Chelly, L. Amsaleg, T. Furon, S. Girard, M. Houle, K. Kawarabayashi & M. Nett. "Extreme-Value-Theoretic Estimation of Local Intrinsic Dimensionality", Journal of Data Mining and Knowledge Discovery, 32 (6), 1768--1805, 2018. [Associated technical report: pdf].

M. Stehlik, P. Aguirre, S. Girard, P. Jordanova, J. Kiselák, S. Torres-Leiva, Z. Sadovsky & A. Rivera. "On ecosystems dynamics", Ecological Complexity, 29, 10--29, 2017. [Associated technical report: pdf].

J. El Methni, L. Gardes & S. Girard. "Estimation de mesures de risque pour des pluies extrêmes dans la région Cévennes Vivarais", La Houille Blanche, 4, 26--31, 2015.

J. Carreau, D. Ceresetti, E. Ursu, S. Anquetin, J.D. Creutin, L. Gardes, S. Girard & G. Molinié. "Evaluation of classical spatial-analysis schemes of extreme rainfall", Natural Hazards and Earth System Sciences, 12, 3229--3240, 2012.

Software

J. Diebolt, J. Ecarnot, M. Garrido, S. Girard & D. Lagrange. "Le logiciel Extremes, un outil pour l'étude des queues de distribution", La revue de Modulad, 30, 53-60, 2003.