Journées d'études

Cimpa-Sénégal

Méthodes statistiques pour l’évaluation des risques extrêmes
5 - 15 avril 2016
 

Résumé des cours

Cours 1: Introduction à la Statistique des valeurs extrêmes

Stéphane GIRARD (INRIA / Grenoble - France)

La prise en compte des évènements extrêmes (fortes précipitations, crues, cours exceptionnels d’actions, charges anormales ...) est souvent très importante dans la démarche statistique de modélisation du risque. C’est le comportement en queue de distribution qui est alors primordial et non le comportement en partie centrale comme en statistique usuelle.
La statistique des valeurs extrêmes offre une panoplie d’outils permettant la modélisation et l'estimation de la probabilité d’occurence des évènements extrêmes. En particulier, les points suivants seront abordés dans le cours :

1) Comportement asymptotique de la plus grande valeur d’un échantillon. Domaine d’attraction des lois de Fréchet, Weibull et Gumbel. Loi de Pareto Généralisée. Fonctions à variations régulières.

2) Estimation des paramètres de la loi de Pareto Généralisée. Estimateur de Hill. Application à l’estimation des quantiles extrêmes. Illustration sur simulations et données réelles.

3) Pour aller plus loin: le cas des lois à queues de type Weibull, estimateurs non-paramétriques pour les extrêmes en présence de covariable.

 

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Course 2: Ruin models for food safety assessment

Patrice BERTAIL, MODAL’X, Université Paris-Ouest, Jessica TRESSOU, INRA MIA Paris, UMR MIA 518

Food safety is now receiving increasing attention both in the public health community and in the scientific literature. Certain foods may indeed contain varying amounts of chemicals such as methylmercury (present in sea food), diox- ins (in poultry, meat) or mycotoxins (in cereals, dried fruits, etc.), which may cause major health problems when accumulating inside the body in excessive doses. This topic naturally interfaces with various disciplines, such as biology, economics, nutritional medicine, toxicology and of course applied mathematics with the aim to develop rigorous methods for quantitative risk assessment.

The study of dietary exposure to food contaminants has raised many stim- ulating questions and motivated the use and/or the development of appropri- ate statistical methods for analyzing the data . Most recent works are mainly centred on static approaches for modelling the quantity X of a specific food contaminant ingested on a short period of time and computing the probability that X exceeds a maximum tolerable dose, eventually causing adverse effects on human health. This can be done either by using Monte-carlo type methods (for large risks) or extreme value theory.

However, it is essential for successful modelling to take appropriate account of the kinetics in man of the chemical of interest, when considering contaminants such as methylmercury our running example in this course, with a half-life measured in weeks rather than days

In this course, we will also present some variations over the traditional ”ruin models” used in insurance which may be helpfull for modelling the dynamic of the exposure. The amount of contaminant present in the body evolves through its accumulation after repeated intakes (food consumption) and according to the pharmacokinetics governing its elimination/excretion, so that its temporal evo- lution is described by a piecewise-deterministic Markov process (PDM process in abbreviated form): the accumulation process is modeled by a marked point process in a standard fashion, while the elimination phenomenon is described by a differential equation with random coefficients, randomness accounting for the variability of the rate at which the total contaminant body burden decreases in between intakes due to metabolic factors. Such a process slightly extends ruin models and storage models with general release rules widely used in operations research and engineering for dealing with problems such as water storage in dams, in that one allows here the (content dependent) release rate to be ran- dom, as strongly advocated by biological modeling, and inter-intake times are not required to be exponentially distributed. In some particular case (linear rate in the phamacocinetic elimination) these models may be seen as shot noise models.

The purpose of the course is to introduce the basic tools of ruin models in the framework of food risk assessments, as well to emphasize the importance of developing new models and tools for analysing multivariate risks.

1. Food risk assessment : the static approach (3 hours)

Introduction to food risk assessment (available data for assessing ex- posure to a contaminant, tolerable doses, common approaches, em- pirical approach: principle and limit)

Extreme value theory: estimation of the Pareto index in the iid case (Hill estimator, moment estimator, Pickands estimator)

Bias correction and choice of k. Illustration on Mercury data

Characterizing at risk population by the introduction of covariates

in a generalized Pareto model

Limits of the static approach

2. The basic dynamic models : Cramer-Lundberg (without and with barrier).

(3 hours)

Cramer-Lundberg model (ruin probability)

Renewal theorem and integro-differential equation

Some bounds on the Sparre-Andersen model

Some generalisations : Cramer-Lundberg model with a barrier, mod- ulation, dependence, multidimensional models.

3. KDEM Kinedic Dynamic Exposure Models (3 hours)

Model definition and similarities to the insurance problematic

Probabilistic study of the model (Markov property, existence of a stationary distribution, tail behavior)

Statistical analysis (estimation by simulation, multilevel splitting, importance sampling for rare events estimation)

4. Tools and links with other models (shot noise, autoregressive models with random coefficcients) (3 hours)

Link with autoregressive models with random coefficients

Shot noise

Regenerative properties of the models

Applications : estimation of the tail index and extremal index of KDEM. Others examples

References:

S. Asmussen et H. Albrecher. Ruin probabilities, Advanced Series on Statistical Science & Applied Probability. World Scientific, New Jersey, 2010.

P. Bertail et S. Clémençon et J. Tressou (2009). Extreme values statistics for Markov chains via the (pseudo-) regenerative method, Extremes,12(4), 327-360.

Bertail P., Clémençon S. et Tressou J. (2013). Regenerative Block-Bootstrap Confidence Intervals for Tails and Extremal Indexes. Electronic Journal of Statistics 7, 1224-1248.

P. Bertail et S. Clémençon et J. Tressou (2008). A storage model with random release rate for modelling exposure to food contaminants. Math- ematical Bioscience and Engineering, 5(1), 35-60.

Bertail, P. et S. Clémençon et J. Tressou (2010). Statistical analysis of a dynamic model for food contaminant exposure with applications to dietary methylmercury contamination. Journal of Biological Dynamics, 4(2), 212- 234.

Verger P., Tressou J. et Clémençon S. (2007). Integration of time as a description parameter in risk characterization: application to methyl mercury. Regulatory Toxicology and Pharmacology. 49, 25-30.

Clémençon S. et Tressou J. (2009). Exposition aux risques alimentaires et processus stochastiques : le cas des contaminants chimiques. Journal de la SFDS. 150(1), 3-29.

Bertail P. et Tressou, J. (2006). Evaluation empirique des risques alimen- taires: une approche de type Monte-Carlo. Chapitre 7 dans Analyse des risques alimentaires, Ed.,Feinberg, Bertail,Tressou, Verger, Tech&Doc- Lavoisier.

Bertail P. et Tressou, J. (2006). Evaluation des petits risques - la th ́eorie des valeurs extrˆemes. Chapitre 9 dans Analyse des risques alimentaires, Ed., Feinberg, Bertail,Tressou, Verger, Tech&Doc-Lavoisier.

Bertail P., Boisson A., Lebon S. et Tressou J. (2006). Les m ́ethodes de cor- rection de biais des estimateurs de type Hill. Chapitre 10 dans Analyse des risques alimentaires, Ed., Feinberg, Bertail,Tressou, Verger, Tech&Doc- Lavoisier.

Bertail, P., Tressou, J. (2006). Incomplete U-Statistics for food risk as- sessment. Biometrics, 62, 66-74.

P. Embrechts, C. Klu ̈ppelberg et T. Mikosch (1997). Modelling Extremal Events for Insurance and Finance, Applications of Mathematics, Springer- Verlag.

 M. Gibaldi et D. Perrier(1982). Pharmacokinetics. Drugs and the Phar- maceutical Sciences : a Series of Textbooks and Monographs, Marcel Dekker, New York, Second Edition.

 

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Cours 3: Spatial extremes and applications

Anthony DAVIDSON (EPFL / Lausanne - Suisse)

Catastrophic events such as heavy rainfall, floods, windstorms, heatwaves and doughts take place in space and time, and accurate estimation of the associated risks must take this into account. Modelling of spatial and space-time extremes has therefore become a very active research area in the past decade. In these lectures I shall describe statistical models and methods that have been recently developed for the modelling of spatial extremes. The lectures will cover related notions of statistics of extremes, including max-stable and Pareto processes, asymptotic dependence and independence, extremal copulas, topics from classical geostatistics, methods for model estimation and assessment, and simulation for risk estimation. The ideas will be illustrated using applications from various areas, and computing will be discussed.

1. Multivariate extremes
- general framework
- models, measures of dependence
- point process formulation
- asymptotic dependence and independence
- inference

2. Spatial extremes
- basic ideas/goals
- Gaussian processes
- Latent variable models
- Copula models

3. Max-stable models
- basic ideas, measures of extremal dependence
- models
- inference
- applications

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Cours 4: Heavy-tail phenomena and serial dependence

Thomas MIKOSCH (University of Copenhagen / Danemark)

 

The modeling and estimation of extremes is particularly relevant when the data have heavy-tailed distributions. For example, in non-life insurance, in particular re-insurance, heavy-tailed claim sizes are standard. They are typically modeled by subexponential distributions, including the Pareto, Burr, log-gamma distribu- tions with power law tails, the log-normal and Weibull distributions with semi- exponential tails. Heavy tails are also relevant in finance: return data of speculative prices (such as foreign exchange rates, asset prices, stock indices,...) exhibit very large/small values in times of financial insecurity. Heavy-tailed distributions are also suitable models in the area of telecommunications, e.g. for time series of file sizes, transmission durations, throughput data.

The course aims at giving an introduction to probabilistic models commonly used for describing heavy-tail phenomena. In particular, we are interested in modeling serial dependence and heavy tails. Our focus will be on models whose distributions exhibit power law tails. In this context, univariate and multivariate regular vari- ation will be convenient tools and we will consider time series models which have power law tails for their marginal and finite-dimensional distributions.

The plan of the course is as follows:

1. Heavy tails and extremal dependence - some empirical facts
1.1 Heavy tails in real-life data
1.2 Extremal dependence/independence in real-life data
2. Modeling heavy tails and serial dependence via regularly varying
time series
2.1 Univariate regularly varying distributions
2.2 Multivariate regular variation
2.3 Operations on regularly varying vectors
2.4 Regularly varying stationary sequences
Resnick (1987,2007)
3. The extremogram - an autocorrelation function for serial
extremal dependence
3.1 A motivating example: the tail dependence coefficient
3.2 Definition
3.3 The sample extremogram
3.4 Variations on the theme: cross-extreomogram, return time extremogram
Davis, Mikosch (2009), Davis, Mikosch, Cribben (2012),
Mikosch and Zhao (2014,2015)
4. Special topics
4.1 The eigenvalues of heavy-tailed sample covariance matrices
4.2 Heavy-tailed large deviations
Davis, Heiny, Mikosch (2015), Nagaev (1979)

References

  •  Davis, R.A., Heiny, J. and Mikosch, T. (2015) Extreme value analysis for the sample autocovariance matrices of heavy-tailed multivariate time series. Report.
  • Davis, R.A. and Mikosch, T. (2009) The extremogram: a correlogram for extreme events. Bernoulli 15, 977–1009
  •  Davis, R.A., Mikosch, T. and Cribben, I. (2012) Towards estimating extremal serial de- pendence via the bootstrapped extremogram. J. Econometrics 170, 142–152.
  • Mikosch, T. and Zhao, Y. (2014) A Fourier analysis of extreme events. Bernoulli 20, 803– 845.
  • Mikosch, T. and Zhao, Y. (2015) The integrated periodogram of a dependent extremal event sequence. Stoch. Proc. Appl. (2015) 125, 3126–3169.
  •   Nagaev, S.V. (1979) Large deviations of sums of independent random variables. Ann. Probab. 7, 745–789.
  •  Resnick, S.I. (1987) Extreme Values, Regular Variation, and Point Processes. Springer, New York.
  • Resnick, S.I. (2007) Heavy Tail Phenomena: Probabilistic and Statistical Modeling. Springer, New York.

 

 

 

Cours 5: Introduction à la statistique spatiale

Liliane BEL (AGROPARISTECH / Paris - France)

Les processus entrant en jeu dans l'étude des risques environnementaux sont par nature spatialisés et leur modélisation requiert une compréhension
fine des dépendances induites par les proximités géographiques. La statistique spatiale a pour objet la prise en compte de ces dépendances pour établir des modèles, réaliser leur inférence et des prédictions associées. On s'intéressera dans ce cours à des champs aléatoires dans un domaine continu, à la modélisation des structures de dépendance et à la prédiction spatiale qui permet d'établir des cartes pour une région à partir d'un processus observé en un nombre limité de points du domaine. Des applications à l'environnement (pollution, hauteur de vagues) ou au climat (précipitations, vents) seront traitées.

  1. Champs aléatoires, processus stationnaires, à accroissements stationnaires
  2. Mesures de dépendance spatiale, fonction de covariance, variogramme
  3. Prédiction spatiale, krigeage (simple, ordinaire, universel), cokrigeage, simulations conditionnelles

 

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