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1. A refined Weissman estimator for extreme quantiles

Author

Michaël Allouche, Jonathan El Methni, Stéphane Girard, Centre de Mathématiques Appliquées - Ecole Polytechnique (CMAP), École polytechnique (X)-Centre National de la Recherche Scientifique (CNRS), Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité), 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), Girard, Stephane, IDEX UGA - - UGA2015 - ANR-15-IDEX-0002 - IDEX - VALID, and El Methni, Jonathan

Subjects
Statistics and Probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Economics, Econometrics and Finance (miscellaneous), [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Engineering (miscellaneous), ComputingMilieux_MISCELLANEOUS
Abstract

International audience; Weissman extrapolation methodology for estimating extreme quantiles from heavy-tailed distributions is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail-index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting refined Weissman estimator. The asymptotic normality of the latter estimator is established and a data-driven method is introduced for the practical selection of the intermediate sequences. Our approach is compared to Weissman estimator and to six bias reduced estimators of extreme quantiles on a large scale simulation study. It appears that the refined Weissman estimator outperforms its competitors in a wide variety of situations, especially in the challenging high bias cases. Finally, an illustration on an actuarial real data set is provided.

Published
2022

2. I didn't feel like drinking, but I guess why: Evaluative conditioning changes on explicit attitudes toward alcohol and healthy foods depends on contingency awareness

Author

Jonathan El Methni, Nikki Rutte, Reinout W. Wiers, Oulmann Zerhouni, Edgar Werkman, Katrijn Houben, Section Eating Disorders and Obesity, RS: FPN CPS II, and Ontwikkelingspsychologie (Psychologie, FMG)

Subjects
IMPLICIT ATTITUDES, Health (social science), Contingency awareness, media_common.quotation_subject, Experimental and Cognitive Psychology, Alcohol, 050105 experimental psychology, Education, Developmental psychology, chemistry.chemical_compound, Healthy food, Developmental and Educational Psychology, Evaluative conditioning, CUES, 0501 psychology and cognitive sciences, 050102 behavioral science & comparative psychology, Valence (psychology), BRAIN, media_common, ASSOCIATIONS, Action tendencies, Addiction, 05 social sciences, Explicit attitudes, Neuropsychology and Physiological Psychology, chemistry, Food, Implicit attitude, Psychology
Abstract

Background and objectivesAddictive behaviors are influenced both by impulsive processes headed toward consumption, and by higher-order, reflective, control-oriented processes. Our objective here was to (i) change automatically-activated action tendencies and explicit attitudes toward alcohol and softdrinks, as well as fat and healthy food through evaluative conditioning (EC) and (ii) investigate the role of contingency awareness in producing an EC effect.MethodsIn the EC conditions, alcohol-related stimuli or fat foods (CS) were associated with negative pictures (US), while stimuli related to softdrinks or healthy foods (CS) were associated with positive pictures (US). In the control conditions, CSs were paired with neutral pictures (US). Measures of explicit attitudes and approach tendencies (AAT) were assessed before and after the EC procedure. Participants (n = 95) then indicated whether they remembered, for each CS, the valence of the associated US.ResultsSignificant EC effects were found on explicit attitudes toward alcohol and marginally significant change on attitudes toward fat foods. The effect on alcohol appeared to depend on awareness of CS-US contingencies. No effects of EC were found on approach tendencies.LimitationsDifficulties to find EC effects on action tendencies can be attributed to structural features of the AAT and strategies during EC.ConclusionsWe conclude that EC alone may not be efficient in reducing addictive behaviors in populations with reduced capacity to process CS-US pairs and for which impulsive behaviors are strong predictors of consumption.

Published
2019

3. A bias-reduced version of the Weissman extreme quantile estimator

Author

Jonathan El Methni, Stéphane Girard, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université de Paris (UP), 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), and Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)-Université Paris Cité (UPCité)

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

International audience; Weissman extrapolation device for estimating extreme quantiles is based on two estimators: an order statistic to estimate an intermediate quantile and an estimator of the tail index. The common practice is to select the same intermediate sequence for both estimators. In this work, we show how an adapted choice of two different intermediate sequences leads to a reduction of the asymptotic bias associated with the resulting Weissman estimator. Our approach is compared to other bias reduced estimators of extreme quantiles both on simulated and real data.

Published
2021

4. 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

5. Improved estimators of extreme Wang distortion risk measures for very heavy-tailed distributions

Author

Gilles Stupfler, Jonathan El Methni, MAP5 - Mathématiques Appliquées à Paris 5 (MAP5), Université Paris Descartes - Paris 5 (UPD5) - Institut National des Sciences Mathématiques et de leurs Interactions - Centre National de la Recherche Scientifique (CNRS), School of Mathematical Sciences [ Nottingham], University of Nottingham, UK (UON), Groupement de Recherche en Économie Quantitative d'Aix-Marseille (GREQAM), Centre National de la Recherche Scientifique (CNRS) - Université de la Méditerranée - Aix-Marseille 2 - Université Paul Cézanne - Aix-Marseille 3 - École des hautes études en sciences sociales (EHESS) - Aix Marseille Université (AMU) - Ecole Centrale de Marseille (ECM), 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), School of Mathematical Sciences [Nottingham], École Centrale de Marseille (ECM)-École des hautes études en sciences sociales (EHESS)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU), École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École des hautes études en sciences sociales (EHESS), École des hautes études en sciences sociales (EHESS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), 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 ), University of Nottingham, UK ( UON ), Groupement de Recherche en Économie Quantitative d'Aix-Marseille ( GREQAM ), Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS ) -Aix Marseille Université ( AMU ) -École des hautes études en sciences sociales ( EHESS ), and El Methni, Jonathan

Subjects
Statistics and Probability, Economics and Econometrics, Wang distortion risk measure, asymptotic normality, Asymptotic distribution, 01 natural sciences, extreme value statistics, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Distortion, 0502 economics and business, Statistics, Econometrics, winsorising, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Extreme value theory, heavy-tailed distribution, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], AMS Subject Classifications: 62G05, 62G30, 62G30, 62G32, 050205 econometrics, Mathematics, trimming, 05 social sciences, Asymptotic normality, Extreme value statistics, Heavy-tailed distribution, Trimming, Wang distortion risk measure, Winsorising, Estimator, Distribution (mathematics), Heavy-tailed distribution, Statistics, Probability and Uncertainty, Random variable, Quantile
Abstract

International audience; A general way to study the extremes of a random variable is to consider the family of its Wang distortion risk measures. This class of risk measures encompasses several popular indicators such as the classical quantile/Value-at-Risk, the Tail-Value-at-Risk and the recently introduced Conditional Tail Moments, among others. A couple of very recent studies have focused on the estimation of extreme analogues of such quantities. In this paper, we consider trimmed and winsorised versions of the empirical counterparts of extreme Wang distortion risk measures. We analyse their asymptotic properties, and we show that we can construct bias-corrected trimmed or winsorised estimators of extreme Wang distortion risk measures who appear to perform overall better than their standard empirical counterparts in practice when the underlying distribution has a very heavy right tail. We also showcase our technique on a set of real fire insurance data.

Published
2016

6. Extreme versions of Wang risk measures and their estimation for heavy-tailed distributions

Author

Gilles Stupfler, Jonathan El Methni, 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), 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), École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS)-Aix Marseille Université (AMU)-École des hautes études en sciences sociales (EHESS), École des hautes études en sciences sociales (EHESS)-Aix Marseille Université (AMU)-École Centrale de Marseille (ECM)-Centre National de la Recherche Scientifique (CNRS), 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 ), Groupement de Recherche en Économie Quantitative d'Aix-Marseille ( GREQAM ), Ecole Centrale de Marseille ( ECM ) -Centre National de la Recherche Scientifique ( CNRS ) -Aix Marseille Université ( AMU ) -École des hautes études en sciences sociales ( EHESS ), and El Methni, Jonathan

Subjects
[ MATH ] Mathematics [math], Statistics and Probability, Conditional Tail Moment, Asymptotic distribution, Sample (statistics), Asymptotic Normality, 01 natural sciences, 010104 statistics & probability, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Distortion, 0502 economics and business, Statistics, Distortion risk measure, Econometrics, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], [MATH]Mathematics [math], 0101 mathematics, Extreme value theory, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], 050205 econometrics, Mathematics, Distortion Risk Measure, 05 social sciences, Estimator, Heavy-Tailed Distribution, Heavy-tailed distribution, Statistics, Probability and Uncertainty, Extreme- Value Statistics, Random variable, extreme-value statistics
Abstract

International audience; Among the many possible ways to study the right tail of a real-valued random variable, a particularly general one is given by considering the family of its Wang distortion risk measures. This class of risk measures encompasses various interesting indicators, such as the widely used Value-at-Risk and Tail Value-at-Risk, which are especially popular in actuarial science, for instance. In this communication, we first build simple extreme analogues of Wang distortion risk measures and we show how this makes it possible to consider many standard measures of extreme risk, including the usual extreme Value-at-Risk or Tail-Value-at-Risk, as well as the recently introduced extreme Conditional Tail Moment, in a unified framework. We then introduce adapted estimators when the random variable of interest has a heavy-tailed distribution and we prove their asymptotic normality. The finite sample performance of our estimators is assessed on a simulation study and we showcase our techniques on an actuarial data set.

Published
2016
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7. 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

8. 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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9. 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

10. 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

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