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23 results on '"Jonathan El-Methni"'

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. Determination of risks of lower adherence to CPAP treatment before their first use by patients

Author

Caroline Poulet, Marc Gandit, Petrus J Pauwels, Chrystèle Deschaux, Jonathan El Methni, Marie Charavel, and Julie Bros

Subjects
Typology, medicine.medical_specialty, medicine.medical_treatment, Patient care, Illness perceptions, 03 medical and health sciences, 0302 clinical medicine, medicine, Humans, Cpap treatment, Longitudinal Studies, Prospective Studies, 030212 general & internal medicine, Continuous positive airway pressure, Intensive care medicine, Applied Psychology, Sleep Apnea, Obstructive, Continuous Positive Airway Pressure, business.industry, Theory of planned behavior, Sleep apnea, medicine.disease, respiratory tract diseases, Patient Compliance, Health behavior, business, 030217 neurology & neurosurgery
Abstract

A better understanding of Continuous Positive Airway Pressure (CPAP) adherence is a priority in improving patient care. To Identify adherence typology with a longitudinal approach, and explore the early determinants of lower adherence to CPAP. Obstructive sleep apnea patients ( N = 204). Prospective and longitudinal study.A classification into four profiles was observed: “ Regular Adherents,” “ Non-Regular Adherents,” “ Persistent Non-Adherents,” and “ Non-Persistent Non-Adherents.” Specific biopsychosocial factors make it possible to evaluate the risk of belonging to a lower adherence profile. We propose a novel approach of CPAP treatment adherence. Several pre-determinants have been identified.

Published
2020
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3. 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

4. Quality‐based pharmacokinetic model selection on DCE‐MRI for characterizing orbital lesions

Author

Mathieu Zmuda, Olivier Bergès, Jonathan El Methni, Jean-Claude Sadik, Louise Marais, Augustin Lecler, Julien Savatovsky, Daniel Balvay, O. Galatoire, Kevin Zuber, Laure Fournier, C.A. Cuenod, and Edgar Farah

Subjects
Adult, Male, Multivariate analysis, Quality Assurance, Health Care, Population, Contrast Media, Blood volume, Logistic regression, Permeability, 030218 nuclear medicine & medical imaging, 03 medical and health sciences, 0302 clinical medicine, Image Processing, Computer-Assisted, Medical imaging, Humans, Medicine, Radiology, Nuclear Medicine and imaging, Prospective Studies, Stage (cooking), education, education.field_of_study, Receiver operating characteristic, business.industry, Univariate, Reproducibility of Results, Middle Aged, Magnetic Resonance Imaging, Perfusion, ROC Curve, Multivariate Analysis, Orbital Neoplasms, Female, Nuclear medicine, business, Algorithms
Abstract

BACKGROUND Although several studies have evaluated dynamic contrast-enhanced (DCE) MRI in the orbit, showing its utility when detecting and diagnosing orbital lesions, none have evaluated the pharmacokinetic models. PURPOSE To provide a quality-based pharmacokinetic model selection for characterizing orbital lesions using DCE-MRI at 3.0T. STUDY TYPE Prospective. POPULATION From December 2015 to April 2017, 151 patients with an orbital lesion underwent MRI prior to surgery, including a high temporal resolution DCE sequence, divided into one training and one test dataset with 100 and 51 patients, respectively. FIELD STRENGTH/SEQUENCE 3T/DCE. ASSESSMENT Six different pharmacokinetic models were tested. STATISTICAL TESTS Univariate and multivariate analyses were performed using Wilcoxon-2-sample tests and a logistic regression to compare parameters between malignant and benign tumors for each pharmacokinetic model for the whole cohort. Receiver operating characteristic (ROC) curve analyses were performed on the training dataset to determine area under curve (AUC) and optimal cutoff values for each pharmacokinetic model, then validated on the test dataset to calculate sensitivity, specificity, and accuracy. RESULTS Regardless of the model, tissue blood flow and tissue blood volume values were significantly higher in malignant vs. benign lesions: 103.8-195.1 vs. 65-113.8, P [

Published
2019
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5. 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

6. Renewable energy policy effectiveness: A panel data analysis across Europe and Latin America

Author

Jonathan El-Methni, German Bersalli, Philippe Menanteau, Institute for Advanced Sustainability Studies [Potsdam] (IASS), Laboratoire d'Economie Appliquée de Grenoble (GAEL), Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)-Université Grenoble Alpes (UGA)-Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), Université Grenoble Alpes (UGA), 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), Institut polytechnique de Grenoble - Grenoble Institute of Technology (Grenoble INP ), and Université Grenoble Alpes (UGA)-Université Grenoble Alpes (UGA)-Centre National de la Recherche Scientifique (CNRS)-Institut National de Recherche pour l’Agriculture, l’Alimentation et l’Environnement (INRAE)

Subjects
Renewable energy, Latin Americans, Economic policy, 020209 energy, media_common.quotation_subject, Public policy, 02 engineering and technology, 7. Clean energy, Promotion (rank), Electricity, 0202 electrical engineering, electronic engineering, information engineering, Common value auction, Decarbonisation, Emerging markets, Panel data, media_common, Renewable Energy, Sustainability and the Environment, [SHS.ECO]Humanities and Social Sciences/Economics and Finance, Investment (macroeconomics), Auction, Europe, Latin America, Incentive, 8. Economic growth, Business
Abstract

International audience; Renewable energy (RE) technologies for electricity generation are a central pillar of energy sector decarbonisation strategies worldwide. Public policies to promote their diffusion have been in place in developed economies since 1980, and, since the 2000s, a growing number of emerging countries began implemented such policies. The Latin American countries have been proactive in RE promotion, but few attempts have been made to evaluate the results. This article proposes an econometric analysis of the effectiveness of RE policies, based on panel data for 20 Latin American and 30 European countries, over 20 years. The results converge for the influence of promotion policies in general: they have a positive and statistically significant effect on RE investment, being the principal determinant in both regions. Nevertheless, on their own, tax incentives are insufficient to assure the deployment of RE technologies. This article also highlights specificities in policy approaches and motivations across both regions and explains why auction became the main instrument in Latin American countries.

Published
2020
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7. Effects of circadian cortisol on the development of a health habit

Author

Carole Rovère, Fabienne d'Arripe-Longueville, Rémi Radel, Jonathan El Methni, Christopher S. Easthope, Lars Schwabe, Marion Fournier, University of Zurich, Fournier, Marion, Laboratoire Motricité Humaine Expertise Sport Santé (LAMHESS), Université Côte d'Azur (UCA)-Université de Toulon (UTLN)-Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA), Institut de pharmacologie moléculaire et cellulaire (IPMC), Université Nice Sophia Antipolis (... - 2019) (UNS), COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-COMUE Université Côte d'Azur (2015-2019) (COMUE UCA)-Centre National de la Recherche Scientifique (CNRS), Spinal Cord Injury Center, Balgrist University Hospital, Department of Cognitive Psychology, Institute for Psychology, University of Hamburg, Mathématiques Appliquées Paris 5 (MAP5 - UMR 8145), and Université Paris Descartes - Paris 5 (UPD5)-Institut National des Sciences Mathématiques et de leurs Interactions (INSMI)-Centre National de la Recherche Scientifique (CNRS)

Subjects
Adult, Male, Evening, Hydrocortisone, media_common.quotation_subject, education, Health Behavior, [SHS.PSY]Humanities and Social Sciences/Psychology, Automaticity, 610 Medicine & health, Bedtime, 3202 Applied Psychology, Developmental psychology, 03 medical and health sciences, 2738 Psychiatry and Mental Health, Habits, Young Adult, 0302 clinical medicine, Memory, Muscle Stretching Exercises, medicine, Humans, 030212 general & internal medicine, Circadian rhythm, Young adult, Saliva, Applied Psychology, Morning, media_common, [SCCO.NEUR]Cognitive science/Neuroscience, Circadian Rhythm, Exercise Therapy, Psychiatry and Mental health, 10046 Balgrist University Hospital, Swiss Spinal Cord Injury Center, Female, Habit, Psychology, hormones, hormone substitutes, and hormone antagonists, 030217 neurology & neurosurgery, medicine.drug
Abstract

Objective Given the impact of individuals' habits on health, it is important to study how behaviors can become habitual. Cortisol has been well documented to have a role in habit formation. This study aimed to elucidate the influence of the circadian rhythm of cortisol on habit formation in a real-life setting. Method Forty-eight students were followed for 90 days during which they attempted to adopt a health behavior (psoas iliac stretch). They were randomly assigned to perform the stretch either upon waking in the morning, when cortisol concentrations are high, or before evening bedtime, when cortisol levels approach the nadir. A smartphone application was used to assess the Self-Report Behavioural Automaticity Index every day and to provide reminders for salivary measurements every 30 days. The speed of the health habit formation process was calculated by modeling the learning curves. Results Extrapolation of the curves indicated that the morning group achieved automaticity at an earlier time point (105.95 days) than did the evening group (154.01 days). In addition, the cortisol level during the performance of the health behavior was identified as a significant mediator of the time point when the health behavior became habitual. Conclusion The present findings suggest that the time course of the development of healthy habits depends on the time of the day and that the effect is mediated through diurnal variation in cortisol levels. Future studies are now needed to determine to what extent cortisol rhythmicity can help individuals to adopt new health behaviors. (PsycINFO Database Record

Published
2017
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8. 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

9. 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

10. 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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11. 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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12. 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

13. 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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14. 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

15. 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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16. 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

17. 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

18. 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

19. Estimation d'un paramètre de queue commun aux lois de type Weibull et au domaine d'attraction de Fréchet

Author

Stéphane Girard, Laurent Gardes, Armelle Guillou, 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
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

organisées par la Société Française de Statistique; National audience

Published
2011

20. Kernel estimation of extreme regression risk measures

Author

Stéphane Girard, Jonathan El Methni, Laurent Gardes, MAP5 - Mathématiques Appliquées à Paris 5 (MAP5 - UMR 8145), Centre National de la Recherche Scientifique (CNRS) - Institut National des Sciences Mathématiques et de leurs Interactions - Université Paris Descartes - Paris 5 (UPD5), Institut de Recherche Mathématique Avancée (IRMA), Université de Strasbourg - 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 Grenoble Alpes (UGA) - Centre National de la Recherche Scientifique (CNRS) - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Université Joseph Fourier - Grenoble 1 (UJF) - Université Pierre Mendès France - Grenoble 2 (UPMF) - Université Grenoble Alpes (UGA) - Centre National de la Recherche Scientifique (CNRS) - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Université Joseph Fourier - Grenoble 1 (UJF) - Université Pierre Mendès France - Grenoble 2 (UPMF) - 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), 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]), ANR-15-IDEX-0002,UGA,IDEX UGA(2015), 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 ), 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 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), ANR IDEX UGA,CDP-RISK,Risk @ Univ. Grenoble Alpes(2018), 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), Institut National de Recherche en Informatique et en Automatique (Inria) - Institut National de Recherche en Informatique et en Automatique (Inria) - Institut National Polytechnique de Grenoble (INPG) - Laboratoire Jean Kuntzmann (LJK), and Université Joseph Fourier - Grenoble 1 (UJF) - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Centre National de la Recherche Scientifique (CNRS) - Université Pierre Mendès France - Grenoble 2 (UPMF) - Université Grenoble Alpes (UGA) - Université Joseph Fourier - Grenoble 1 (UJF) - Institut Polytechnique de Grenoble - Grenoble Institute of Technology - Centre National de la Recherche Scientifique (CNRS) - Université Pierre Mendès France - Grenoble 2 (UPMF) - Université Grenoble Alpes (UGA)

Subjects
Statistics and Probability, Extreme-value index, 0209 industrial biotechnology, 62G30, Kernel estimator, Kernel density estimation, AMS 2000 subject classification: 62G32 - 62G30 - 62E20, Asymptotic distribution, 02 engineering and technology, 01 natural sciences, Extreme-value analysis, Risk mea, 010104 statistics & probability, 020901 industrial engineering & automation, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], Statistics, Covariate, Asymptotic normality, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], 0101 mathematics, Extreme value theory, [MATH.MATH-ST] Mathematics [math]/Statistics [math.ST], Mathematics, 62E20, Estimator, Risk measures, AMS 2000 subject classification: 62G32, Moment (mathematics), Conditional tail moment, Skewness, Kernel (statistics), Statistics, Probability and Uncertainty, 62G32
Abstract

International audience; The Regression Conditional Tail Moment (RCTM) is the risk measure defined as the moment of order b ≥ 0 of a loss distribution above the upper α-quantile where α ∈ (0, 1) and when a covariate information is available. The purpose of this communication is first to establish the asymptotic properties of the RCTM in case of extreme losses, i.e when α → 0 is no longer fixed, under general extreme-value conditions on their distribution tail. In particular, no assumption is made on the sign of the associated extreme-value index. Second, the asymptotic normality of a kernel estimator of the RCTM is established, which allows to derive similar results for estimators of related risk measures such as the Regression Conditional Tail Expectation/Variance/Skewness. When the distribution tail is upper bounded, an application to frontier estimation is also proposed. The results are illustrated both on simulated data and on a real dataset in the field of nuclear reactors reliability.

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21. Estimation of risk measures for extreme pluviometrical measurements

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
risk measure, [MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], heavy-tailed distributions, [ MATH.MATH-ST ] Mathematics [math]/Statistics [math.ST], extreme-value statistics, kernel estimators
Abstract

International audience; Extreme rainfall statistics are often used when a flood has occurred to assess the rarity of such an event. A typical problem is to estimate the amount that will fall on a day of exceptionally heavy rainfall which is expected to occur every T years. Usually, hydrologists are interested in the value T = 100 corresponding to a centenary event. Statistically speaking, the problem is to estimate the T-year return level which is a upper quantile of the distribution of the variable of interest also called Value-at-Risk. This risk measure however suffers from several weaknesses. The estimation of a single extreme quantile only gives incomplete information on the extremes of a random variable. To put it differently, it may well be the case that a light-tailed distribution (e.g. a Gaussian distribution) and a heavy-tailed distribution share a quantile at some common level, although they clearly do not have the same behavior in their extremes. An alternative risk measure is the Conditional Tail Expectation which is the mean of the rainfalls larger than the Value-at-Risk. This risk measure thus takes into account the whole information contained in the upper tail of the distribution. It is frequently encountered in financial investment or in the insurance industry 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 esti-mators. The behaviour and the efficiency of our estimators are illustrated on rainfall observations in the Cévennes-Vivarais region (southern part of France). This data set is provided by the French meteorological service Météo-France and consists in daily rainfalls measured at 523 raingauge stations from 1958 to 2000. In this context, the variable of interest is the daily rainfall measured in millimeters. The covariate is the three dimensional geographical location (longitude, latitude and altitude).

22. Estimation of a new parameter discriminating between Weibull tail-distributions and heavy-tailed distributions

Author

Stéphane Girard, Jonathan El Methni, Laurent Gardes, 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
[MATH.MATH-ST]Mathematics [math]/Statistics [math.ST], [STAT.TH]Statistics [stat]/Statistics Theory [stat.TH]
Abstract

International audience; The Gnedenko theorem is a general result in extreme value theory establishing the asymptotic distribution of extreme order statistics. The maximum of a sample of independent and identically distributed random variables after proper renormalization converges in distribution to one of the three possible maximum domain of attraction: Fr échet, Weibull and Gumbel. In a lot of applications (hydrology, nance, etc...), the Fr échet maximum domain of attraction and the Gumbel maximum domain of attraction are used. The Gumbel maximum domain of attraction encompasses a large variety of distributions. Here, we focus on a subfamily of distributions called Weibull tail-distributions which depends on the Weibull tail-coeffi cient. Numerous works are dedicated to the estimation of this coeffi cient, and to the estimation of the tail index. In order to explain why the same methodology can be used to estimate the Weibull tail-coeffi cient and the tail index, the authors proposed a family of distributions which encompasses the whole Fr échet maximum domain of attraction as well as Weibull tail-distributions. These distributions depend on 2 parameters $\tau\in[0,1]$ and $\theta>0$. The fi rst one, $\tau$, allows us to represent a large panel of distribution tails ranging from Weibull-type tails ($\tau=0$) to distributions belonging to the maximum domain of attraction of Fr échet ($\tau=1$). The main goal of this communication is to propose an estimator for $\tau$ independent of $\theta$. Under some assumptions we establish the asymptotic distribution of this estimator.

23. Estimation of Extreme Risk Measures from Heavy-tailed distributions

Author

Jonathan El Methni, 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), 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, 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 it is assumed that we are in the case of heavy-tailed distributions. We also consider these risk measures in the presence of a covariate. Let us note that the presence of a covariate has already been investigated in extreme value theory. The main goal of this communication is to propose estimators of the above risk measures in the case of heavy-tailed distributions, for extreme losses, and to include a covariate in the estimation. The asymptotic distribution of our estimators is established and their finite sample behavior is illustrated on simulated data and on a real data set of pluviometrical measurements.

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