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106 results on '"Laurent Gardes"'

1. An Integrated Functional Weissman Estimator for Conditional Extreme Quantiles

Author

Laurent Gardes and Gilles Stupfler

Subjects
heavy-tailed distribution, functional random covariate, extreme quantile, tail index, asymptotic normality, Statistics, HA1-4737, Probabilities. Mathematical statistics, QA273-280
Abstract

It is well-known that estimating extreme quantiles, namely, quantiles lying beyond the range of the available data, is a nontrivial problem that involves the analysis of tail behavior through the estimation of the extreme-value index. For heavy-tailed distributions, on which this paper focuses, the extreme-value index is often called the tail index and extreme quantile estimation typically involves an extrapolation procedure. Besides, in various applications, the random variable of interest can be linked to a random covariate. In such a situation, extreme quantiles and the tail index are functions of the covariate and are referred to as conditional extreme quantiles and the conditional tail index, respectively. The goal of this paper is to provide classes of estimators of these quantities when there is a functional (i.e. possibly infinite-dimensional) covariate. Our estimators are obtained by combining regression techniques with a generalization of a classical extrapolation formula. We analyze the asymptotic properties of these estimators, and we illustrate the finite-sample performance of our conditional extreme quantile estimator on a simulation study and on a real chemometric data set.

Published
2019
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2. On the Estimation of the Second Order Parameter for Heavy-Tailed Distributions

Author

El hadji Deme, Laurent Gardes, and Stéphane Girard

Subjects
extreme-value theory, heavy-tailed distribution, extreme-value index, second order parameter, asymptotic properties, Statistics, HA1-4737, Probabilities. Mathematical statistics, QA273-280
Abstract

The extreme-value index γ is an important parameter in extreme-value theory since it controls the first order behavior of the distribution tail. In the literature, numerous estimators of this parameter have been proposed especially in the case of heavy-tailed distributions, which is the situation considered here. Most of these estimators depend on the k largest observations of the underlying sample. Their bias is controlled by the second order parameter ρ. In order to reduce the bias of γ’s estimators or to select the best number k of observations to use, the knowledge of ρ is essential. In this paper, we propose a simple approach to estimate the second order parameter ρ leading to both existing and new estimators. We establish a general result that can be used to easily prove the asymptotic normality of a large number of estimators proposed in the literature or to compare different estimators within a given family. Some illustrations on simulations are also provided.

Published
2013
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3. Comparison of Weibull Tail-Coefficient Estimators

Author

Laurent Gardes and Stéphane Girard

Subjects
Weibull tail-coefficient, extreme quantile, extreme value theory, asymptotic normality, Statistics, HA1-4737, Probabilities. Mathematical statistics, QA273-280
Abstract

We address the problem of estimating the Weibull tail-coefficient which is the regular variation exponent of the inverse failure rate function. We propose a family of estimators of this coefficient and an associate extreme quantile estimator. Their asymptotic normality are established and their asymptotic mean-square errors are compared. The results are illustrated on some finite sample situations.

Published
2006
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8. Xplique: A Deep Learning Explainability Toolbox.

Author

Thomas Fel, Lucas Hervier, David Vigouroux, Antonin Poché, Justin Plakoo, Rémi Cadène, Mathieu Chalvidal, Julien Colin, Thibaut Boissin, Louis Béthune, Agustin Martin Picard, Claire Nicodeme, Laurent Gardes, Grégory Flandin, and Thomas Serre

Published
2022
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9. Dataset Definition Standard (DDS).

Author

Cyril Cappi, Camille Chapdelaine, Laurent Gardes, Eric Jenn, Baptiste Lefèvre, Sylvaine Picard, and Thomas Soumarmon

Published
2021

10. Nonparametric asymptotic confidence intervals for extreme quantiles

Author

Laurent Gardes, Samuel Maistre, and Gardes, Laurent

Subjects
Statistics and Probability, confidence interval, Statistics, Probability and Uncertainty, heavy-tailed distribution, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Extreme quantiles
Abstract

In this paper, we propose new asymptotic confidence intervals for extreme quantiles i.e., for quantiles located outside the range of the available data. We restrict ourselves to the situation where the underlying distribution is heavy-tailed. While asymptotic confidence intervals are mostly constructed around a pivotal quantity, we consider here an alternative approach based on the distribution of order statistics sampled from a uniform distribution. The convergence of the coverage probability to the nominal one is established under a classical second-order condition. The finite sample behavior is also examined and our methodology is applied to a real dataset.

Published
2022
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15. On tail-risk measures for non-integrable heavy-tailed random variables

Author

Laurent Gardes and Gardes, Laurent

Subjects
Statistics and Probability, Economics and Econometrics, extreme value theory, Risk measure, Statistics, Probability and Uncertainty, heavy-tailed distribution, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], finite sample
Abstract

The assessment of risk for heavy-tailed distributions is a crucial question in various fields of application. An important family of risk measures is provided by the class of distortion risk (DR) measures which encompasses the Value-at-Risk and the Tail-Value-at-Risk measures. The Tail-Value-at-Risk is a coherent risk measure (which is not the case for the Value-at-Risk) but it is defined only for integrable quantile functions that is to say for heavy-tailed distributions with a tail index smaller than 1. Moreover, it is a matter of fact that the performance of the empirical estimator is strongly deteriorated when the tail index becomes close to 1. The main contribution of this work is the introduction and the estimation of a new risk measure which is defined for all heavy-tailed distributions and which is tail-equivalent to a coherent DR measure when the tail of the underlying distribution is not too heavy. Its finite sample performance is discussed on a fire claims dataset.

Published
2022
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17. On the estimation of the variability in the distribution tail

Author

Laurent Gardes, Stéphane Girard, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension (STATIFY), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Chaire Stress Test - BNP Paribas/Ecole polytechnique/Fondation de l'X., ANR-15-IDEX-0002,UGA,IDEX UGA(2015), ANR-19-CE40-0013,ExtremReg,Régression extrême avec applications à l'économétrie, l'environnement et à la finance(2019), Modelling and Inference of Complex and Structured Stochastic Systems [2016-2019] (MISTIS [2016-2019]), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019] (Grenoble INP [2007-2019])-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Institut polytechnique de Grenoble - Grenoble Institute of Technology [2007-2019] (Grenoble INP [2007-2019])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), ANR-15-IDEX-02,UGA,IDEX UGA(2016), ANR-19-CE40-0013,ExtremReg,Extremal Regression with Applications to Econometrics, Environment and Finance(2019), and Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann [2016-2019] (LJK [2016-2019])

Subjects
Statistics and Probability, distribution tail, extreme-value theory, Risk measure, Estimator, Sample (statistics), 01 natural sciences, Measure (mathematics), risk measure, 010104 statistics & probability, 03 medical and health sciences, 0302 clinical medicine, Transformation (function), Distribution (mathematics), [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Range (statistics), Gini functional, 030211 gastroenterology & hepatology, Statistical physics, 0101 mathematics, Statistics, Probability and Uncertainty, Extreme value theory, variability measure, Mathematics
Abstract

We propose a new measure of variability in the tail of a distribution by applying a Box–Cox transformation of parameter $$p \ge 0$$ to the tail-Gini functional. It is shown that the so-called Box–Cox Tail Gini Variability measure is a valid variability measure whose condition of existence may be as weak as necessary thanks to the tuning parameter p. The tail behaviour of the measure is investigated under a general extreme-value condition on the distribution tail. We then show how to estimate the Box–Cox Tail Gini Variability measure within the range of the data. These methods provide us with basic estimators that are then extrapolated using the extreme-value assumption to estimate the variability in the very far tails. The finite sample behaviour of the estimators is illustrated both on simulated and real data.

Published
2021
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20. Ensuring Dataset Quality for Machine Learning Certification

Author

Laurent Gardes, Cyril Cappi, Sylvaine Picard, Camille Chapdelaine, Thomas Soumarmon, Eric Jenn, Baptiste Lefèvre, SAFRAN Group, SNCF, ANR-19-P3IA-0004,ANITI,Artificial and Natural Intelligence Toulouse Institute(2019), MAMALET, Franck, and Artificial and Natural Intelligence Toulouse Institute - - ANITI2019 - ANR-19-P3IA-0004 - P3IA - VALID

Subjects
FOS: Computer and information sciences, [INFO.INFO-AI] Computer Science [cs]/Artificial Intelligence [cs.AI], Computer Science - Machine Learning, 020205 medical informatics, Computer science, Process (engineering), media_common.quotation_subject, Machine Learning (stat.ML), Context (language use), 02 engineering and technology, Certification, Machine learning, computer.software_genre, Statistics - Computation, Machine Learning (cs.LG), Domain (software engineering), [INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI], 03 medical and health sciences, 0302 clinical medicine, Statistics - Machine Learning, Data integrity, 0202 electrical engineering, electronic engineering, information engineering, Quality (business), 030212 general & internal medicine, Computation (stat.CO), media_common, business.industry, datasets, certification process, machine learning, Life-critical system, Artificial intelligence, business, Engineering design process, computer
Abstract

In this paper, we address the problem of dataset quality in the context of Machine Learning (ML)-based critical systems. We briefly analyse the applicability of some existing standards dealing with data and show that the specificities of the ML context are neither properly captured nor taken into ac-count. As a first answer to this concerning situation, we propose a dataset specification and verification process, and apply it on a signal recognition system from the railway domain. In addi-tion, we also give a list of recommendations for the collection and management of datasets. This work is one step towards the dataset engineering process that will be required for ML to be used on safety critical systems.

Published
2021

21. An extreme quantile estimator for the log-generalized Weibull-tail model

Author

Stéphane Girard, Laurent Gardes, Anne Dutfoy, Clément Albert, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), PERformance et prévention des Risques Industriels du parC par la simuLation et les EtudeS (EDF R&D PERICLES), EDF R&D (EDF R&D), EDF (EDF)-EDF (EDF), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension (STATIFY), ANR-15-IDEX-0002,UGA,IDEX UGA(2015), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)

Subjects
Statistics and Probability, Extreme-value theory, Economics and Econometrics, 010504 meteorology & atmospheric sciences, Estimator, Asymptotic distribution, Sample (statistics), 01 natural sciences, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Applied mathematics, Extended regular variation, 0101 mathematics, Statistics, Probability and Uncertainty, Extreme value theory, Extreme quantiles, 0105 earth and related environmental sciences, Quantile, Weibull distribution, Mathematics
Abstract

International audience; A new estimator for extreme quantiles is proposed under the log-generalized Weibull-tail model, ntroduced by (de Valk, C., Extremes, pp. 661--686, vol. 19, 2016). This model relies on a new regular variation condition which, in some situations, permits to extrapolate further into the tails than the classical assumption in extreme-value theory. The asymptotic normality of the estimator is established and its finite sample properties are illustrated both on simulated and real datasets.

Published
2020
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22. Beyond tail median and conditional tail expectation: Extreme risk estimation using tail L p ‐optimization

Author

Gilles Stupfler, Laurent Gardes, Stéphane Girard, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), Modèles statistiques bayésiens et des valeurs extrêmes pour données structurées et de grande dimension (STATIFY), School of Mathematical Sciences [Nottingham], University of Nottingham, UK (UON), Chaire Stress Test - BNP Paribas/Ecole polytechnique/Fondation de l'X., ANR-15-IDEX-0002,UGA,IDEX UGA(2015), and ANR-19-CE40-0013,ExtremReg,Régression extrême avec applications à l'économétrie, l'environnement et à la finance(2019)

Subjects
Statistics and Probability, asymptotic normality, Asymptotic distribution, extreme value statistics, 01 natural sciences, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0502 economics and business, Statistics, Range (statistics), 0101 mathematics, heavy-tailed distribution, Extreme value theory, 050205 econometrics, Event (probability theory), Mathematics, median shortfall, 05 social sciences, Estimator, semiparametric extrapolation, Lp−optimisation, Tail value at risk, Heavy-tailed distribution, conditional tail expectation, Statistics, Probability and Uncertainty, Quantile
Abstract

The Conditional Tail Expectation is an indicator of tail behaviour that has recently gained traction in actuarial and financial applications. Contrary to the quantile or Value-at-Risk, it takes into account the frequency of a tail event together with the probabilistic behaviour of the variable of interest on this event. However, the asymptotic normality of the empirical Conditional Tail Expectation estimator requires that the underlying distribution possess a finite variance; this can be a strong restriction in heavy-tailed models which constitute the favoured class of models in actuarial and financial applications. One possible solution in very heavy-tailed models where this assumption fails could be to use the more robust Median Shortfall, but this quantity is actually just a quantile, which therefore only gives information about the frequency of a tail event and not about its typical magnitude. We construct a synthetic class of tail L p −medians, which encompasses the Median Shortfall (for p = 1) and Conditional Tail Expectation (for p = 2). We show that, for 1 < p < 2, a tail L p −median always takes into account both the frequency and magnitude of tail events, and its empirical estimator is, within the range of the data, asymptotically normal under a condition weaker than a finite variance. We extrapolate this estimator, along with another technique, to proper extreme levels using the heavy-tailed framework. The estimators are showcased on a simulation study and on a set of real fire insurance data showing evidence of a very heavy right tail.

Published
2019
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23. Nonparametric confidence interval for conditional quantiles with large-dimensional covariates

Author

Laurent Gardes, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and ANR-19-CE40-0013,ExtremReg,Régression extrême avec applications à l'économétrie, l'environnement et à la finance(2019)

Subjects
Statistics and Probability, dimension reduction, 05 social sciences, Order statistic, Coverage probability, Nonparametric statistics, Conditional probability distribution, 01 natural sciences, Confidence interval, 010104 statistics & probability, Conditional independence, confidence interval, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Extreme conditional quantiles, Dimension reduction DR, 0502 economics and business, Statistics, 0101 mathematics, Statistics, Probability and Uncertainty, CDF-based nonparametric confidence interval, 050205 econometrics, Mathematics, Quantile
Abstract

International audience; The first part of the paper is dedicated to the construction of a nonparametric confidence interval for a conditional quantile with a level depending on the sample size. When this level tends to 0 or 1 as the sample size increases, the conditional quantile is said to be extreme and is located in the tail of the conditional distribution. The proposed confidence interval is constructed by approximating the distribution of the order statistics selected with a nearest neighbor approach by a Beta distribution. We show that its coverage probability converges to the preselected probability and its accuracy is illustrated on a simulation study. When the dimension of the covariate increases, the coverage probability of the confidence interval can be very different from the nominal probability. This is a well known consequence of the data sparsity especially in the tail of the distribution. In a second part, a dimension reduction procedure is proposed in order to select more appropriate nearest neighbors in the right tail of the distribution and in turn to obtain a better coverage probability for extreme conditional quantiles. This procedure is based on the Tail Conditional Independence assumption introduced in (Gardes, Extremes, pp. 57--95, 18(3), 2018).

Published
2020
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24. Estimation of extreme conditional quantiles under a general tail first order condition

Author

Laurent Gardes, Armelle Guillou, Claire Roman, Institut de Recherche Mathématique Avancée (IRMA), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)

Subjects
Statistics and Probability, Estimation, Class (set theory), Extreme quantile, asymptotic normality, 05 social sciences, Asymptotic distribution, Estimator, 01 natural sciences, 010104 statistics & probability, 13. Climate action, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Kernel (statistics), 0502 economics and business, Order (group theory), First derivative test, Applied mathematics, local estimation, 0101 mathematics, 050205 econometrics, Quantile, Mathematics
Abstract

We consider the estimation of an extreme conditional quantile. In a first part, we propose a new tail condition in order to establish the asymptotic distribution of an extreme conditional quantile estimator. Next, a general class of estimators is introduced, which encompasses, among others, kernel or nearest neighbors types of estimators. A unified theorem of the asymptotic normality for this general class of estimators is provided under the new tail condition and illustrated on the different well-known examples. A comparison between different estimators belonging to this class is provided on a small simulation study and illustrated on a real dataset on earthquake magnitudes.

Published
2020
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25. A general estimator for the extreme value index: applications to conditional and heteroscedastic extremes

Author

Laurent Gardes, Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Heteroscedasticity, Economics, Econometrics and Finance (miscellaneous), Estimator, heteroscedastic extremes, Sample (statistics), Distribution (mathematics), Survival function, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Large set (Ramsey theory), Consistency (statistics), Extreme value index, AMS Subject Classifications: 62G05, 62G20, 62G30, 62G32, Consistent estimator, Econometrics, Consistency, Engineering (miscellaneous), conditional extremes, Mathematics
Abstract

International audience; The tail behavior of a survival function is controlled by the extreme value index. The aim of this paper is to propose a general procedure for the estimation of this parameter in the case where the observations are not necessarily distributed from the same distribution. The idea is to estimate in a consistent way the survival function and to apply a general functional to obtain a consistent estimator for the extreme value index. This procedure permits to deal with a large set of models such as conditional extremes and heteroscedastic extremes. The consistency of the obtained estimator is established under general conditions. A simulation study and a concrete application on financial data are proposed to illustrate the finite sample behavior of the proposed procedure.

Published
2015
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26. A mixture model for dimension reduction

Author

Laurent Gardes, Jean-Luc Dortet-Bernadet, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Finite mixture, Dimensionality reduction, 05 social sciences, Mixture of distributions, Regression analysis, Maximum likelihood estimates, Mixture model, 01 natural sciences, Sliced Inverse Regression, 010104 statistics & probability, Joint probability distribution, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0502 economics and business, Statistics, Sliced inverse regression, Dimension reduction, Applied mathematics, 0101 mathematics, Subspace topology, 050205 econometrics, Mathematics
Abstract

International audience; The existence of a Dimension Reduction (DR) subspace is a common assumption in regression analysis when dealing with high-dimensional predictors. The estimation of such a DR subspace has received considerable attention in the past few years, the most popular method being undoubtedly the Sliced Inverse Regression. We propose in this paper a new estimation procedure of the DR subspace by assuming that the joint distribution of the predictor and the response variables is a finite mixture of distributions. The new method is compared through a simulation study to some classical methods.

Published
2017
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27. Frontier estimation based on extreme risk measures

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Modelling and Inference of Complex and Structured Stochastic Systems ( MISTIS ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Institut National Polytechnique de Grenoble ( INPG ), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS ), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST]
Abstract

International audience; Value-at-risk, Conditional Tail Expectation, Conditional Tail Variance and Conditional Tail Moment are classical risk measures. In statistical terms,the Value-at-risk is the upper a-quantile of the loss distribution where alpha in (0;1) is the confidence level. Here, we focus on the properties of theserisk measures for extreme losses (where alpha tends to zero and is thus no longer fixed). To assign probabilities to extreme losses we assume that the distribution satisfies a von-Mises condition which allows us to work in the general setting, whether the extreme-value index is positive, negative or zero i.e. for all domains of attraction. We also consider these risk measures in the presence of a covariate. The main goal is to propose estimators of the above risk measures for all domains of attraction, for extreme losses, and to include a covariate in the estimation. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of our estimators is established and their finite sample behavior is illustrated on simulated data and also on a motivating application in the reliability of nuclear reactors.

Published
2016

28. Estimation of the functional Weibull-tail coefficient

Author

Laurent Gardes, Stéphane Girard, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS ), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), and Girard, Stephane

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Conditional Weibull tail-coefficient, Statistics::Methodology, Nonparametric estimation, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Extreme quantiles
Abstract

International audience; We present a nonparametric family of estimators for the tail index of a Weibull tail-distribution when functional covariate is available. Our estimators are based on a kernel estimator of extreme conditional quantiles, extending a previous work Daouia et al. (2013) to the infinite dimensional case. Asymptotic normality of the estimators is proved under mild regularity conditions. Their finite sample performances are illustrated both on simulated and real data. We refer to Gardes and Girard (2016) for further details.Daouia, A., Gardes, L., Girard, S. (2013). On kernel smoothing for extremal quantile regression. Bernoulli, 19, 2557–2589.Gardes, L., Girard, S. (2016). On the estimation of the functional Weibull tail-coefficient, Journal of Multivariate Analysis, to appear, http://dx.doi.org/10.1016/j.jmva.2015.05.007

Published
2016

29. Estimation of extreme quantiles from heavy and light tailed distributions

Author

Armelle Guillou, Stéphane Girard, Laurent Gardes, Jonathan El Methni, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Asymptotic distribution, 01 natural sciences, 010104 statistics & probability, Weibull tail-distributions, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0502 economics and business, Statistics, Pareto-type distributions, Asymptotic normality, 0101 mathematics, Maximum domain of attraction, Exponentiated Weibull distribution, 050205 econometrics, Mathematics, Weibull distribution, Estimation, Extreme quantile, Applied Mathematics, 05 social sciences, Estimator, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Trimmed estimator, Data set, Statistics, Probability and Uncertainty, Quantile
Abstract

International audience; In Gardes et al. (2011), a new family of distributions is introduced, depending on two parameters tau and theta which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators for theta and extreme quantiles are also proposed, but they both depend on the unknown parameter tau, making them useless in practical situations. In this paper, we propose an estimator of tau which is independent of theta. Plugging our estimator of tau in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unifi ed way. The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.

Published
2012
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30. Conditional extremes from heavy-tailed distributions: an application to the estimation of extreme rainfall return levels

Author

Stéphane Girard, Laurent Gardes, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), and ANR-07-VULN-0006,MEDUP,Forecast and projection in climate scenario of Mediterranean intense events: Uncertainties and Propagation on environment.(2007)

Subjects
Statistics and Probability, 62G32, 62G05, 62E20, Economics, Econometrics and Finance (miscellaneous), 0207 environmental engineering, Asymptotic distribution, Sample (statistics), 02 engineering and technology, 01 natural sciences, k-nearest neighbors algorithm, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], extreme rainfalls, Statistics, Covariate, Econometrics, 0101 mathematics, heavy-tailed distribution, 020701 environmental engineering, Conditional extreme quantiles, Engineering (miscellaneous), Mathematics, Pointwise, Estimator, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], nearest neighbor estimator, 13. Climate action, Heavy-tailed distribution, Quantile
Abstract

International audience; This paper is dedicated to the estimation of extreme quantiles and the tail index from heavy-tailed distributions when a covariate is recorded simultaneously with the quantity of interest. A nearest neighbor approach is used to construct our estimators. Their asymptotic normality is established under mild regularity conditions and their finite sample properties are illustrated on a simulation study. An application to the estimation of pointwise return levels of extreme rainfalls in the Cevennes-Vivarais region is provided.

Published
2010
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31. Estimation non-paramétrique de mesures de risque pour des lois conditionnelles à queues lourdes avec application à des extrêmes pluviométriques

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Girard, Stephane, Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Modelling and Inference of Complex and Structured Stochastic Systems ( MISTIS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jean Kuntzmann ( LJK ), and Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Institut National Polytechnique de Grenoble ( INPG )

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [ MATH.MATH-NA ] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-NA] Mathematics [math]/Numerical Analysis [math.NA], [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA]
Abstract

National audience; L'étude et la maîtrise des risques extrêmes est d'un grand intéret afin d'anticiper des catastrophes et ainsi permettre l'aménagement du territoire. Les hydrologues s'intéressent à l'estimation de la hauteur de pluie journalière pouvant être dépassée par exemple tous les 100 ans. On parle dans ce cas de pluie centennale. Cette mesure de risque est appelée Value-at-Risk et représente le quantile d'ordre p de la fonction de survie de la variable aléatoire d'intérêt. Bien que cette mesure de risque soit la plus utilisée, elle ne fournit qu'une information ponctuelle et donc sous-estime l'impactdu sinistre. Dans le but de lui trouver une alternative, d'autres mesures de risques ont été proposées afin de prendre en compte les incertitudes sur les évènements extrêmes.Nos travaux consistent à introduire et à estimer une nouvelle mesure de risque appellée Conditional Tail Moment. Elle est définie comme le moment d'ordre a > 0 de la loi de la variable aléatoire d'intérêt au-delà du quantile d'ordre p. Estimer le Conditional Tail Moment permet d'estimer toutes les mesures de risque basées sur les moments conditionnels telles que la Conditional Tail Expectation, la Conditional Value-at-Risk ou la Condi-tional Tail Variance. On s'intéresse à l'estimation de ces mesures de risque dans un cadre extrême, c'est-à-dire lorsque p tend vers 0 lorsque la taille de l'échantillon augmente. On suppose également que la loi de la variable aléatoire d'intérêt est à queue lourde et qu'elle dépend d'une covariable. Les estimateurs proposés combinent des méthodes d'estimation non-paramétrique à noyau avec des méthodes issues de la statistique des valeurs extrêmes. On établira le comportement asymptotique de nos estimateurs. On appliquera nos travaux à un jeu de données pluviométriques fourni par le Laboratoire d'étude des Transferts en Hydrologie et Environnement de Grenoble. On dispose des hauteurs de pluies journalières en millimètres entre les années 1958 et 2000 sur 523 stations situées dans la région des Cévennes-Vivarais. Dans ce contexte, les variables d'intérêt sont les précipitations journalières et les covariables sont les coordonnées géographiques des stations. En conclusion, on pourra estimer des mesures de risque extrêmes en des sites où l'on ne dispose pas de mesure.

Published
2015

32. Estimating extreme quantiles under random truncation

Author

Laurent Gardes, Gilles Stupfler, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Groupement de Recherche en Économie Quantitative d'Aix-Marseille (GREQAM), École des hautes études en sciences sociales (EHESS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), École Centrale de Marseille (ECM)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), and Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA)

Subjects
Statistics and Probability, Asymptotic distribution, 01 natural sciences, [SHS]Humanities and Social Sciences, 010104 statistics & probability, Consistency (statistics), [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0103 physical sciences, Statistics, Asymptotic normality, Truncation (statistics), 0101 mathematics, 010303 astronomy & astrophysics, Mathematics, Weak consistency, Extreme quantile, Estimator, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [SHS.ECO]Humanities and Social Sciences/Economics and Finance, Heavy-tailed distribution, Consistency, Statistics, Probability and Uncertainty, Random variable, Tail index, Quantile
Abstract

AD; International audience; The goal of this paper is to provide estimators of the tail index and extreme quantiles of a heavy-tailed random variable when it is right-truncated. The weak consistency and asymptotic normality of the estimators are established. The finite sample performance of our estimators is illustrated on a simulation study and we showcase our estimators on a real set of failure data..

Published
2015
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33. Asymptotic distribution of a Pickands-type estimator of the extreme-value index

Author

Laurent Gardes and Stéphane Girard

Subjects
Index (economics), Distribution (number theory), Rate of convergence, Statistics, Rare events, Applied mathematics, Estimator, Asymptotic distribution, Extreme value index, General Medicine, Type (model theory), Mathematics
Abstract

One of the main goals of extreme-value analysis is to estimate the probability of rare events given a sample from an unknown distribution. The upper tail behavior of this distribution is described by the extreme-value index ξ. The aim of this Note is to establish the asymptotic distribution of the estimator of ξ∈R introduced in Gardes and Girard [A Pickands-type estimator of the extreme-value index, Technical Report LMC-IMAG, RR-1063, 2004]. We also give its rate of convergence in some typical situations. To cite this article: L. Gardes, S. Girard, C. R. Acad. Sci. Paris, Ser. I 341 (2005).

Published
2005
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34. Estimating Extreme Quantiles of Weibull Tail Distributions

Author

Stéphane Girard and Laurent Gardes

Subjects
Statistics and Probability, Statistics::Theory, Statistics::Applications, Weibull modulus, Asymptotic distribution, Estimator, Trimmed estimator, Consistent estimator, Statistics, Statistics::Methodology, Applied mathematics, Exponentiated Weibull distribution, Mathematics, Quantile, Weibull distribution
Abstract

We present a new estimator of extreme quantiles dedicated to Weibull tail distributions. This estimate is based on a consistent estimator of the Weibull tail coefficient. This parameter is defined as the regular variation coefficient of the inverse cumulative hazard function. We give conditions in order to obtain the weak consistency and the asymptotic distribution of the extreme quantiles estimator. Its asymptotic as well as its finite sample performances are compared to classical ones.

Published
2005
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35. Double-thresholded estimator of extreme value index

Author

Laurent Gardes

Subjects
Delta method, Efficient estimator, Minimum-variance unbiased estimator, Bias of an estimator, Statistics, Stein's unbiased risk estimate, Consistent estimator, General Medicine, Trimmed estimator, Rao–Blackwell theorem, Mathematics
Abstract

The purpose of this Note is to propose an estimator of the extreme value index constructed by using only the number of points exceeding random thresholds. We prove the weak consistency and the asymptotic normality of this estimator. We deduce from this last result that the rate of convergence of our estimator is in a power of the sample size. To our knowledge, this rate of convergence is not reached by any other estimate of the extreme value index. Through a simulation, we compare our estimator to the moment estimator (Dekkers et al., Ann. Statist. 17 (1989) 1833–1855). To cite this article: L. Gardes, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

Published
2003
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36. Non-parametric estimation of extreme risk measures from conditional heavy-tailed distributions

Author

Stéphane Girard, Jonathan El Methni, Laurent Gardes, Université de Genève (UNIGE), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), and Université de Genève = University of Geneva (UNIGE)

Subjects
Statistics and Probability, Asymptotic distribution, Conditional probability distribution, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Tail value at risk, Moment (mathematics), Regular conditional probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Statistics, Econometrics, Kernel regression, Statistics, Probability and Uncertainty, Extreme value theory, Conditional variance, Mathematics
Abstract

International audience; In this paper, we introduce a new risk measure, the so-called Conditional Tail Moment. It is the moment of order a>0 of the loss distribution above the upper alpha-quantile. Estimating the Conditional Tail Moment permits to estimate all risk measures based on conditional moments such as Conditional Tail Expectation, Conditional Value-at-Risk or Conditional Tail Variance. Here, we focus on the estimation of these risk measures in case of extreme losses (where alpha converges to 0). It is moreover assumed that the loss distribution is heavy-tailed and depends on a covariate. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of the estimators is established and their finite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cévennes-Vivarais region (France).

Published
2014
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37. Estimating the conditional tail index with an integrated conditional log-quantile estimator in the random covariate case

Author

Laurent Gardes, Gilles Stupfler, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Groupement de Recherche en Économie Quantitative d'Aix-Marseille (GREQAM), École Centrale de Marseille (ECM)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), Stupfler, Gilles, and École des hautes études en sciences sociales (EHESS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[STAT]Statistics [stat], consistency, random covariate, Heavy-tailed distribution, tail index, [MATH] Mathematics [math], [MATH]Mathematics [math], asymptotic normality, [STAT] Statistics [stat]
Abstract

It is well known that the tail behavior of a heavy-tailed distribution is controlled by a parameter called the tail index. Such a parameter is therefore of primary interest in extreme value analysis, particularly to estimate extreme quantiles. In various applications, the random variable of interest can be linked to a finite-dimensional random covariate. In such a situation, the tail index is function of the covariate and is referred to as the conditional tail index. The goal of this paper is to provide a class of estimators of this quantity. The pointwise weak consistency and asymptotic normality of these estimators are established. We illustrate the finite sample performance of our technique on a simulation study and on a real hurricane data set.

Published
2014

38. Kernel estimation of extreme risk measures for all domains of attraction

Author

Jonathan El Methni, Stéphane Girard, Laurent Gardes, Université Paris Descartes - Paris 5 (UPD5), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS ), Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK ), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut National de Recherche en Informatique et en Automatique (Inria)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019])-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes [2016-2019] (UGA [2016-2019]), Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF), Mathématiques Appliquées à Paris 5 ( MAP5 - UMR 8145 ), Université Paris Descartes - Paris 5 ( UPD5 ) -Institut National des Sciences Mathématiques et de leurs Interactions-Centre National de la Recherche Scientifique ( CNRS ), Modelling and Inference of Complex and Structured Stochastic Systems ( MISTIS ), Institut National de Recherche en Informatique et en Automatique ( Inria ) -Institut National de Recherche en Informatique et en Automatique ( Inria ) -Laboratoire Jean Kuntzmann ( LJK ), Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Université Pierre Mendès France - Grenoble 2 ( UPMF ) -Université Joseph Fourier - Grenoble 1 ( UJF ) -Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique ( CNRS ) -Université Grenoble Alpes ( UGA ) -Institut National Polytechnique de Grenoble ( INPG ), Institut de Recherche Mathématique Avancée ( IRMA ), Université de Strasbourg ( UNISTRA ) -Centre National de la Recherche Scientifique ( CNRS ), Université Paris Descartes - Paris 5 ( UPD5 ), and El Methni, Jonathan

Subjects
[STAT.TH] Statistics [stat]/Statistics Theory [stat.TH], Conditional Tail Expectation, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Risk measures, Extreme-value statistics, Asymptotic normality, [ STAT.TH ] Statistics [stat]/Statistics Theory [stat.TH], Conditional Tail, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Value-at-risk, Kernel smoothing, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Conditional Value-at-risk, Von- Mises condition
Abstract

International audience; Value-at-risk, Conditional Tail Expectation, Conditional Value-at-risk and Conditional Tail Variance are classical risk measures. In statistical terms, the Value-at-risk is the upper α-quantile of the loss distribution where α ∈ (0, 1) is the confidence level. Here, we focus on the properties of these risk measures for extreme losses (where α → 0 is no longer fixed). To assign probabilities to extreme losses we assume that the distribution satisfies a von-Mises condition which allows us to work in the general setting, whether the extreme-value index is positive, negative or zero i.e. for all domains of attraction. We also consider these risk measures in the presence of a covariate. The main goal of this communication is to propose estimators of the above risk measures for all domains of attraction, for extreme losses, and to include a covariate in the estimation. The estimation method thus combines nonparametric kernel methods with extreme-value statistics. The asymptotic distribution of our estimators is established and their finite sam- ple behavior is illustrated on simulated data and on a real data set of daily rainfall.

Published
2014

39. Estimation of the conditional tail index using a smoothed local Hill estimator

Author

Gilles Stupfler, Laurent Gardes, Institut de Recherche Mathématique Avancée (IRMA), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Centre d'Études et de Recherche en Gestion d'Aix-Marseille (CERGAM), Aix Marseille Université (AMU)-Université de Toulon (UTLN), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, Estimation, Weak consistency, random covariate, Economics, Econometrics and Finance (miscellaneous), Estimator, Asymptotic distribution, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], uniform consistency, 62G05, 62G20, 62G30, 62G32, Distribution (mathematics), Heavy-tailed distribution, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], pointwise asymptotic normality, Covariate, Statistics, Engineering (miscellaneous), Mathematics, Quantile
Abstract

International audience; For heavy-tailed distributions, the so-called tail index is an important parameter that controls the behavior of the tail distribution and is thus of primary interest to estimate extreme quantiles. In this paper, the estimation of the tail index is considered in the presence of a finite-dimensional random covariate. Uniform weak consistency and asymptotic normality of the proposed estimator are established and some illustrations on simulations are provided.

Published
2014

40. Estimation de mesures de risque pour des pluies extrêmes dans la région Cévennes-Vivarais

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), and Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)

Subjects
Risk measure, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Extreme-value statistics, Heavy-tailed distributions, ddc:150, statistique des valeurs extrêmes, 13. Climate action, Mesure de risque, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], estimateurs à noyau, Kernel estimators, [MATH.MATH-NA]Mathematics [math]/Numerical Analysis [math.NA], Water Science and Technology, lois à queues lourdes
Abstract

International audience; Many risk measures can be found in the literature such as the Value-at-Risk and the Conditional Tail Expectation. In statistical terms, the Value-at-Risk is a upper quantile of the distribution of the variable of interest. In hydrology, the Value-at-Risk of the rainfall distribution is the return level. The Conditional Tail Expectation is the mean of the rainfalls larger than the Value-at-Risk. Here, we focus on the estimation of these risk measures in case of extreme rainfall modeled by heavy-tailed distributions. In order to take into account the geographical factors, we also assume that these risk measures depend on a covariate. We present the theoretical properties of our estimators and we illustrate their behaviour on a real data set of daily rainfalls in the Cévennes-Vivarais region.; On dénombre de nombreuses mesures de risque dans la littérature dont la Value-at-Risk et la Conditional Tail Expectation. En termes statistiques, la Value-at-Risk est un quantile de la distribution de la variable aléatoire d'intérêt. En termes hydrologiques, la Value-at-Risk de la distribution des pluies est le niveau de retour. La Conditional Tail Expectation est la moyenne des précipitations plus élevées que la Value-at-Risk. On s'intéresse à l'estimation de ces mesures de risque dans le cas de pluies extrêmes modélisées par des lois à queues lourdes. Afin de prendre en compte les facteurs géographiques dans notre estimation on considèrera aussi ces mesures de risque en présence d'une covariable. On donnera les propriétés théoriques de nos estimateurs et on illustrera leurs comportements sur un jeu de données pluviométriques provenant de la région Cévennes-Vivarais.

Published
2013
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41. On kernel smoothing for extremal quantile regression

Author

Stéphane Girard, Abdelaati Daouia, Laurent Gardes, Groupe de recherche en économie mathématique et quantitative (GREMAQ), Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), Université Toulouse 1 Capitole (UT1), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées, Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), UCL - SSH/IMMAQ/ISBA - Institut de Statistique, Biostatistique et Sciences Actuarielles, Université de Strasbourg & CNRS, IRMA, France, and INRIA Rhône-Alpes & LJK, team Mistis, France

Subjects
Extreme-value index, Statistics and Probability, Statistics::Theory, kernel smoothing, asymptotic normality, extreme quantile, Mathematics - Statistics Theory, Statistics Theory (math.ST), 01 natural sciences, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0502 economics and business, FOS: Mathematics, Kernel smoothing, Asymptotic normality, Applied mathematics, Statistics::Methodology, 0101 mathematics, extreme-value index, 050205 econometrics, Mathematics, Extreme quantile, Von-Mises condition, 05 social sciences, Estimator, Conditional probability distribution, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Quantile function, 16. Peace & justice, Asymptotic theory (statistics), Regression, Quantile regression, Nonparametric regression, Kernel (statistics), von-Mises condition, kerne l smoothing, Quantile
Abstract

Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory far enough into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Specifically, we elucidate their limit distributions when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their (intermediate or extreme) order with the oscillations of the quantile function and a von-Mises property of the conditional distribution. A simulation experiment and an illustration on real data were presented. The real data are the American electric data where the estimation of conditional extremes is found to be of genuine interest., Comment: Published in at http://dx.doi.org/10.3150/12-BEJ466 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm)

Published
2013
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42. Nonparametric estimation of extreme risks from heavy-tailed distributions

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
conditional quantiles, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], kernel estimator, Conditional tail expectation, heavy-tailed distributions, risk measures, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; Value-at-risk, conditional tail expectation [1], conditional value-at-risk [4] and conditional tail variance [5] are classical risk measures. For instance, the value-at-risk is de ned as the upper alpha- quantile of the loss distribution where alpha is the confi dence level. In this communication, we propose nonparametric estimators of these risk measures for extreme losses, i.e. when alpha converges to0 and in the case of heavy-tailed distributions depending on covariates. Let us note that the presence of covariates in extreme value theory has already been investigated for instance in [2, 3]. The asymptotic distribution of the estimators is established and their fi nite sample behavior is illustrated both on simulated data and on a real data set of daily rainfalls in the Cevennes-Vivarais region (France). [1] Artzner, P., Delbaen, F., Eber, J.M., and Heath, D. (1999) Coherent measures of risk, Mathematical Finance, 9, 203-228. [2] Daouia, A., Gardes, L., Girard, S. and Lekina, A. (2011) Kernel estimators of extreme level curves, Test, 20, 311-333. [3] Daouia, A., Gardes, L. and Girard, S. (2013) On kernel smoothing for extremal quantile regression, Bernoulli, to appear. [4] Rockafellar, R.T., Uryasev, S. (2000) Optimization of conditional value-at-risk, Journal of Risk, 2, 21-42.

Published
2013

43. Nonparametric extremal quantile regression

Author

Abdelaati Daouia, Laurent Gardes, Stéphane Girard, Groupe de recherche en économie mathématique et quantitative (GREMAQ), Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; Nonparametric regression quantiles obtained by inverting a kernel estimator of the conditional distribution of the response are long established in statistics [1,3,4]. Attention has been, however, restricted to ordinary quantiles staying away from the tails of the conditional distribution. The purpose of this paper is to extend their asymptotic theory into the tails. We focus on extremal quantile regression estimators of a response variable given a vector of covariates in the general setting, whether the conditional extreme-value index is positive, negative, or zero. Their limit distribution is established when they are located in the range of the data or near and even beyond the sample boundary, under technical conditions that link the speed of convergence of their order with the oscillations of the quantile function and a von-Mises property of the conditional distribution [2]. A simulation experiment and an illustration on American electric data are presented. [1] Berlinet, A., Gannoun, A. and Matzner-Lober, E. (2001) Asymptotic normality of convergent estimates of conditional quantiles, Statistics, 35, 139--169. [2] Daouia, A., Gardes, L. and Girard, S. (2013) On kernel smoothing for extremal quantile regression, Bernoulli, to appear. [3] Samanta, T. (1989) Non-parametric estimation of conditional quantiles, Statistics and Probability Letters, 7, 407--412. [4] Stute, W. (1986) Conditional empirical processes, The Annals of Statistics, 14, 638--647.

Published
2013

44. Estimation de quantiles extrêmes pour les lois à queue de type Weibull : une synthèse bibliographique

Author

Laurent Gardes, Stéphane Girard, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)

Subjects
Synthèse bibliographique, 62-00, 62G32, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Lois à queue de type Weibull, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

National audience; Cet article est une synthèse bibliographique des méthodes d'estimation de quantiles extrêmes pour les lois à queue de type Weibull. Ces lois ont une fonction de survie qui décroit vers zéro à une vitesse exponentielle. Nous montrons comment cette problématique s'inscrit plus largement dans la théorie des valeurs extrêmes.

Published
2013

45. Estimation de mesures de risque extrêmes

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Girard, Stephane, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH] Statistics [stat]/Statistics Theory [stat.TH], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST]
Abstract

organisées par la Société Française de Statistique; National audience; Des mesures de risques classiques sont la Value-at-Risk, la Conditional Tail Expectation, la Conditional Value-at-Risk et la Conditional Tail Variance. En termes statistique, la Value-at-Risk est le quantile de niveau de con fiance alpha de la distribution des pertes. On s'intéresse aux propriétés de ces mesures de risque dans le cas de pertes extrêmes (où alpha n'est plus fixé mais tend vers 0) qu'on supposera modélisées par des lois à queues lourdes. On considérera aussi ces mesures de risque avec la présence d'une covariable. On ajoute ainsi deux difficultés dans l'estimation de mesures de risque. Par conséquent, le but principal de cette communication est de proposer des estimateurs de toutes les mesures de risque énoncées ci-dessus pour des pertes extrêmes dans le cas de lois à queues lourdes en présence d'une covariable. On établira les propriétés asymptotiques de nos estimateurs et on illustrera leurs comportements sur des données simulées et sur un jeu de données pluviométriques.

Published
2013

46. Estimating the conditional tail index by integrating a kernel conditional quantile estimator

Author

Laurent Gardes, Antoine Schorgen, Armelle Guillou, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, [STAT.AP]Statistics [stat]/Applications [stat.AP], Applied Mathematics, 05 social sciences, Asymptotic distribution, Estimator, Context (language use), 01 natural sciences, 010104 statistics & probability, Heavy-tailed distribution, Kernel (statistics), 0502 economics and business, Covariate, Statistics, 0101 mathematics, Statistics, Probability and Uncertainty, Minimax estimator, 050205 econometrics, Mathematics, Quantile
Abstract

International audience; This paper deals with the estimation of an extreme value index of a heavy-tailed distribution in the presence of covariates. A class of estimators is proposed in this context and its asymptotic normality established under mild regularity conditions. These estimators are functions of a kernel conditional quantile estimator depending on some tuning parameters. The finite sample properties of our estimators are illustrated on a small simulation study.

Published
2012
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47. Estimation de l'espérance conditionnelle des pertes extrêmes dans le cas de lois à queues lourdes en présence d'une covariable

Author

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), and Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS)

Subjects
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], conditional tail expectation, estimateur à noyau, quantiles conditionnels, statistique des valeurs extrêmes, lois à queue lourde, normalité asymptotique, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

organisées par la Société Française de Statistique; National audience; L'espérance conditionnelle des pertes extrêmes (Conditional Tail Expectation, CTE) est une importante mesure de risque fréquemment utilisée en actuariat et en finance. Elle représente la perte attendue au delà d'un certain quantile. Dans la littérature, il existe plusieurs méthodes d'estimation de la CTE basées sur l'estimation du quantile par exemple dans le cas de lois à queues lourdes. L'objectif de cette communication est double. On propose un estimateur de la CTE dans le cas de lois à queues lourdes en présence d'une covariable et cela pour des quantiles conditionnels extrêmes.

Published
2012

48. Evaluation of classical spatial-analysis schemes of extreme rainfall

Author

Jean-Dominique Creutin, Julie Carreau, Sandrine Anquetin, Stéphane Girard, Gilles Molinié, Davide Ceresetti, Laurent Gardes, Eugen Ursu, Laboratoire d'étude des transferts en hydrologie et environnement (LTHE), Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Grenoble (INPG)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut national des sciences de l'Univers (INSU - CNRS)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA), Groupe de Recherche en Economie Théorique et Appliquée (GREThA), Université de Bordeaux (UB)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), Laboratoire des Sciences du Climat et de l'Environnement [Gif-sur-Yvette] (LSCE), Université Paris-Saclay-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Centre National de la Recherche Scientifique (CNRS), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), ANR: 24651,24651, Observatoire des Sciences de l'Univers de Grenoble (OSUG), Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF), Institut national des sciences de l'Univers (INSU - CNRS)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Université de Versailles Saint-Quentin-en-Yvelines (UVSQ), Centre National de la Recherche Scientifique (CNRS)-Université de Strasbourg (UNISTRA), ANR-07-VULN-0006,MEDUP,Forecast and projection in climate scenario of Mediterranean intense events: Uncertainties and Propagation on environment.(2007), Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS)-Observatoire des Sciences de l'Univers de Grenoble (OSUG), Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry]), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), Centre National de la Recherche Scientifique (CNRS)-Université de Bordeaux (UB), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Laboratoire Jean Kuntzmann (LJK), Centre National de la Recherche Scientifique (CNRS)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria), Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Institut national des sciences de l'Univers (INSU - CNRS)-Institut national de recherche en sciences et technologies pour l'environnement et l'agriculture (IRSTEA)-Université Savoie Mont Blanc (USMB [Université de Savoie] [Université de Chambéry])-Centre National de la Recherche Scientifique (CNRS)-Institut National Polytechnique de Grenoble (INPG)-Centre National de la Recherche Scientifique (CNRS), and Université de Versailles Saint-Quentin-en-Yvelines (UVSQ)-Commissariat à l'énergie atomique et aux énergies alternatives (CEA)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Paris-Saclay-Centre National de la Recherche Scientifique (CNRS)

Subjects
010504 meteorology & atmospheric sciences, 0207 environmental engineering, 02 engineering and technology, 01 natural sciences, lcsh:TD1-1066, Kriging, Statistics, lcsh:Environmental technology. Sanitary engineering, 020701 environmental engineering, lcsh:Environmental sciences, 0105 earth and related environmental sciences, Block (data storage), Mathematics, lcsh:GE1-350, Artificial neural network, Rain gauge, lcsh:QE1-996.5, lcsh:Geography. Anthropology. Recreation, Statistical model, [SHS.ECO]Humanities and Social Sciences/Economics and Finance, lcsh:Geology, lcsh:G, General Earth and Planetary Sciences, Maxima, Interpolation, Quantile
Abstract

Extreme rainfall is classically estimated using raingauge data at raingauge locations. An important related issue is to assess return levels of extreme rainfall at ungauged sites. Classical methods consist in interpolating extreme-value models. In this paper, such methods are referred to as regionalization schemes. Our goal is to evaluate three classical regionalization schemes. Each scheme consists of an extreme-value model (block maxima, peaks over threshold) taken from extreme-value theory plus a method to interpolate the parameters of the statistical model throughout the Cévennes-Vivarais region. From the interpolated parameters, the 100-yr quantile level can be estimated over this whole region. A reference regionalization scheme is made of the couple block maxima/kriging, where kriging is an optimal interpolation method. The two other schemes differ from the reference by replacing either the extreme-value model block maxima by peaks over threshold or kriging by a neural network interpolation procedure. Hyper-parameters are selected by cross-validation and the three regionalization schemes are compared by double cross-validation. Our evaluation criteria are based on the ability to interpolate the 100-yr return level both in terms of precision and spatial distribution. It turns out that the best results are obtained by the regionalization scheme combining the peaks-over-threshold method with kriging.

Published
2012
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49. Nadaraya's estimates for large quantiles and free disposal support curves

Author

Abdelaati Daouia, Laurent Gardes, Stéphane Girard, Groupe de recherche en économie mathématique et quantitative (GREMAQ), Université Toulouse Capitole (UT Capitole), Université de Toulouse (UT)-Université de Toulouse (UT)-Institut National de la Recherche Agronomique (INRA)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS), I. Van Keilegom and P. Wilson, Centre National de la Recherche Scientifique (CNRS)-École des hautes études en sciences sociales (EHESS)-Institut National de la Recherche Agronomique (INRA)-Université Toulouse 1 Capitole (UT1), Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut Polytechnique de Grenoble - Grenoble Institute of Technology-Centre National de la Recherche Scientifique (CNRS)-Université Grenoble Alpes (UGA)-Institut National Polytechnique de Grenoble (INPG), and Université Fédérale Toulouse Midi-Pyrénées-Université Fédérale Toulouse Midi-Pyrénées

Subjects
Kernel estimator, Extreme values, 05 social sciences, Order statistic, Asymptotic distribution, Estimator, Multivariate normal distribution, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Quantile function, 01 natural sciences, 010104 statistics & probability, Large quantile, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], 0502 economics and business, Statistics, Econometrics, Partial boundary, 0101 mathematics, 62G08, 62G32, 62E20, Extreme value theory, 050205 econometrics, Quantile, Weibull distribution, Mathematics
Abstract

International audience; A new characterization of partial boundaries of a free disposal multivariate support, lying near the true support curve, is introduced by making use of large quantiles of a simple transformation of the underlying multivariate distribution. Pointwise empirical and smoothed estimators of the full and partial support curves are built as extreme sample and smoothed quantiles. The extreme-value theory holds then automatically for the empirical frontiers and we show that some fundamental properties of extreme order statistics carry over to Nadaraya's estimates of upper quantile-based frontiers. The benefits of the new class of partial boundaries are illustrated through simulated examples and a real data set, and both empirical and smoothed estimates are compared via Monte Carlo experiments. When the transformed distribution is attracted to the Weibull extreme-value type distribution, the smoothed estimator of the full frontier outperforms frankly the sample estimator in terms of both bias and Mean-Squared Error, under optimal bandwidth. In this domain of attraction, Nadaraya's estimates of extreme quantiles might be superior to the sample versions in terms of MSE although they have a higher bias. However, smoothing seems to be useless in the heavy tailed case.

Published
2012
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50. Functional kernel estimators of large conditional quantiles

Author

Laurent Gardes, Stéphane Girard, Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg (UNISTRA)-Centre National de la Recherche Scientifique (CNRS), Modelling and Inference of Complex and Structured Stochastic Systems (MISTIS), Inria Grenoble - Rhône-Alpes, Institut National de Recherche en Informatique et en Automatique (Inria)-Institut National de Recherche en Informatique et en Automatique (Inria)-Laboratoire Jean Kuntzmann (LJK), and Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)-Université Pierre Mendès France - Grenoble 2 (UPMF)-Université Joseph Fourier - Grenoble 1 (UJF)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP )-Centre National de la Recherche Scientifique (CNRS)

Subjects
Statistics and Probability, 62G30, Gaussian, Mathematics - Statistics Theory, Statistics Theory (math.ST), heavy-tailed distributions, 01 natural sciences, 62G32, 62G30, 62E20, 010104 statistics & probability, symbols.namesake, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Conditional quantiles, 0502 economics and business, Covariate, FOS: Mathematics, Applied mathematics, Statistics::Methodology, 0101 mathematics, Extreme value theory, 050205 econometrics, Mathematics, 62E20, extreme-value theory, 05 social sciences, Estimator, [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH], Rate of convergence, Sample size determination, Kernel (statistics), symbols, Statistics, Probability and Uncertainty, 62G32, Quantile, functional kernel estimator
Abstract

International audience; We address the estimation of conditional quantiles when the covariate is functional and when the order of the quantiles converges to one as the sample size increases. In a first time, we investigate to what extent these large conditional quantiles can still be estimated through a functional kernel estimator of the conditional survival function. Sufficient conditions on the rate of convergence of their order to one are provided to obtain asymptotically Gaussian distributed estimators. In a second time, basing on these result, a functional Weissman estimator is derived, permitting to estimate large conditional quantiles of arbitrary large order. These results are illustrated on finite sample situations.

Published
2012
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