Multivariate Pareto distributions: properties and examples
|
|
- Erika Wiggins
- 5 years ago
- Views:
Transcription
1 Multivariate Pareto distributions: properties and examples Ana Ferreira 1, Laurens de Haan 2 1 ISA UTL and CEAUL, Portugal 2 Erasmus Univ Rotterdam and CEAUL EVT2013 Vimeiro, September 8 11
2 Univariate Pareto distribution: P(X > x) = ( 1+γ x µ ) 1/γ, σ 0 < x µ σ < (0 ( γ)) 1, σ > 0, µ,γ R; read e (x µ)/σ if γ = 0. MDA (Balkema and de Haan (1974), Pickands (1975)) F (d.f.) is in the (maximum) domain of attraction of some Generalized extreme value distribution GEV γ if and only if s(t) > 0 such that lim t x P (X > t +x s(t) X > t) = (1+γx) 1/γ, 1+γx > 0, (1) where x = sup{x : F(x) < 1}; that is, above the high threshold t we get a Pareto tail in the limit.
3 Multivariate Pareto distributions Definition 1. (Falk et al. (2004); Michel (2008)) A multivariate generalized Pareto distribution is any multivariate distribution function that can be represented by 1+logGEV in a neighborhood of the right endpoint of GEV. Definition 2. (Rootzén and Tajvidi (2006)) A distribution function H is a multivariate generalized Pareto distribution if H(x) = 1 loggev(0) log GEV(x) GEV(x 0) for some GEV with non-degenerate margins and with 0 < G(0) < 1. (...)
4 Simple Multivariate Pareto Let, W = (W 1,...,W d ) T random vector (r.vect.) in R d + = [0, ) d, W = max 1 i d W i, ω 0 > 0, threshold parameter. Theorem 1. The following three statements are equivalent: 1.(a) E (W i / W ) > 0 for all i = 1,...,d, 1.(b) P ( W /ω 0 > x) = x 1, for x > 1 (stand. Pareto distr.), 1.(c) ( ) ( ) ω0 W P W B ω0 W W > rω 0 = P W B, (2) for all r > 1 and B B ( D + ω 0 ) with D + ω 0 := {w R d + : w = ω 0 }.
5 Simple Multivariate Pareto Theorem 1. (cont.) 2.(a) P ( W > ω 0 ) = 1, 2.(b) E (W i / W ) > 0 for all i = 1,...,d, 2.(c) P(W ra) = r 1 P(W A), (3) for all r > 1 and A B ( D + ω 0 ) with D + ω 0 := {w R d + : w ω 0 }. Recall stand. Pareto r.var.: P(Y > ry) = r 1 P(Y > y), y,r>1.
6 Simple Multivariate Pareto Theorem 1. (cont.) W = (W 1,...,W d ) = Y(V 1,...,V d ) = YV verifying: 3.(a) V R d + r. vect., V = ω 0 a.s. and EV i > 0 for all i = 1,...,d, 3.(b) Y is a stand. Pareto r.var., P(Y y) = 1 1/y, y > 1, 3.(c) Y and V are independent. Definition 1. W R d + characterized in Theorem 1. is simple Pareto with threshold parameter ω 0.
7 Some distribution formulas: If w > 0, ( P(W w) = E max 1 i d V i w i ω 0 ) ( ) V i E max 1 i d w i (which corresponds to Def.1. from Rootzén and Tajvidi (2006)). If w > 0 and w > ω 0, ( P(W > w) = E If E (min 1 i d V i ) > 0, for x R, min 1 i d ) V i. w i P(W > x W > ω 0 ) = { 1, x ω0 ω 0 /x, x > ω 0. For x R and for each i = 1,...,d, { 1, x ω0 P(W i > x W i > ω 0 ) = ω 0 /x, x > ω 0.
8 Definition 2: Multivariate GP r.vect. W µ,σ,γ R d Let W simple Pareto r.vect. γ = (γ 1,...,γ d ) R d extreme value index vector, µ = (µ 1,...,µ d ) R d mean value vector, σ = (σ 1,...,σ d ) > 0 scale vector. Define, W µ,σ,γ = µ+σ Wγ 1. γ
9 Domain of attraction Let X = (X 1,...,X d ) R d with continuous d.f. F, γ = (γ 1,...,γ d ) R d, a t = (a t,1,...,a t,d ) > 0, b t = (b t,1,...,b t,d ) R d, ( ) 1/γ T t X = 1+γ X bt a. t + Theorem 2. F is in the maximum domain of attraction of some GEV with normalizing constants a t > 0 and b t R d iff, 1(a) tp ((X i b t,i )/a t,i > x) (1+γ i x) 1/γ i, i, 1(b) P ( T t X > x T t X > 1) x 1, x > 1, 1(c) ( ) Tt X P T t X B T t X > 1 ρ(b), (4) B B( D + 1 ), ρ( B) = 0, ρ some probability measure on D + 1.
10 Example Let W = (W 1,W 2 ) = (YB,Y(1 B)) with B Bernoulli (1/2) independent of Y stand. Pareto. Then, The spectral measure, i.e. the probability measure of (V 1,V 2 ) = (B,1 B) is concentrated on {(0,1),(1,0)} with prob. 1/2 at each atom. Its d.f. is in the max-domain of atraction of some GEV with independent components, hence verifies asymptotic independence. Note that E(V1 V 2 ) = 0. For W R d, let λ i,j = P(W i > u W j > u) = E(V i V j ) E(V j ) and bivariate asymptotic independence holds if λ i,j = 0. That is, whenever E(V i V j ) = 0 we get asymptotic independence.
11 The Peaks-over-threshold method Corollary 1. Under the max-domain of attraction conditions lim P ( T t X A T t X > 1 ) = P (W A), (5) t A B(D 1 + ), P(W A) = 0, W simple Pareto r.vect. Recall univariate POT: ( X t lim t x P s(t) > x X > t Example of application: ) = (1+γx) 1/γ, 0 < x < (0 ( γ)) 1. Rainfall data at L locations (rain gauge stations) s 1,...,s L, described by some r.vect. X R L verifying the domain of attraction condition. Estimate probabilities: given the relations with max-stable processes, we can use known statistical methods e.g. estimation of the exponent measure. (6)
12 The POT method - Application Estimation of P ( T n/k X A Tn/k X > 1 ) on the basis of n i.i.d. observations of X. 1. Obtain ˆγ n/k,â n/k,ˆb n/k. 2. Obtain the normalized processes ( X T n/k X j = 1+ ˆγ j ˆb n/k n/k â n/k 3. ) 1/ˆγ n/k P ( T n/k X A T n/k X > 1 ) = +, j = 1,...,n. n j=1 I ( T n/k X j A & X j ˆb n/k > 0) ( ). n j=1 I X j ˆb n/k > 0
13 References Balkema, A.A. and de Haan, L. (1974) Residual life time at great age. Ann. Probab. 2, Ferreira, A. and de Haan, L. (2013) The generalized Pareto process; with a view towards application and simulation. Bernoulli: to appear. Falk, M., Hüsler, J. and Reiss, R.-D. (2010) Laws of Small Numbers: Extremes and Rare Events, Birkhäuser, Springer. Michel, R. (2008) Some notes on multivariate generalized Pareto distributions. J. Multiv. Analysis 99, Pickands, J. III (1975) Statistical inference using extreme order statistics. Ann. Statist. 3, Rootzén, H. and Tajvidi, N. (2006) Multivariate generalized Pareto distributions. Bernoulli 12,
The Generalized Pareto process; with application
The Generalized Pareto process; with application Ana Ferreira ISA, Universidade Técnica de Lisboa and CEAUL Tapada da Ajuda 1349-017 Lisboa, Portugal Laurens de Haan Erasmus University Rotterdam and CEAUL
More informationOn estimating extreme tail. probabilities of the integral of. a stochastic process
On estimating extreme tail probabilities of the integral of a stochastic process Ana Ferreira Instituto uperior de Agronomia, UTL and CEAUL Laurens de Haan University of Tilburg, Erasmus University Rotterdam
More informationModelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions
Modelling Multivariate Peaks-over-Thresholds using Generalized Pareto Distributions Anna Kiriliouk 1 Holger Rootzén 2 Johan Segers 1 Jennifer L. Wadsworth 3 1 Université catholique de Louvain (BE) 2 Chalmers
More informationarxiv: v2 [math.pr] 28 Jul 2011
ON EXTREME VALUE PROCESSES AND THE FUNCTIONAL D-NORM arxiv:1107.5136v2 [math.pr] 28 Jul 2011 STEFAN AULBACH, MICHAEL FALK AND MARTIN HOFMANN Abstract. We introduce some mathematical framework for functional
More informationA NOTE ON SECOND ORDER CONDITIONS IN EXTREME VALUE THEORY: LINKING GENERAL AND HEAVY TAIL CONDITIONS
REVSTAT Statistical Journal Volume 5, Number 3, November 2007, 285 304 A NOTE ON SECOND ORDER CONDITIONS IN EXTREME VALUE THEORY: LINKING GENERAL AND HEAVY TAIL CONDITIONS Authors: M. Isabel Fraga Alves
More informationMultivariate generalized Pareto distributions
Bernoulli 12(5), 2006, 917 930 Multivariate generalized Pareto distributions HOLGER ROOTZÉN 1 and NADER TAJVIDI 2 1 Chalmers University of Technology, S-412 96 Göteborg, Sweden. E-mail rootzen@math.chalmers.se
More informationMultivariate generalized Pareto distributions
Multivariate generalized Pareto distributions Holger Rootzén and Nader Tajvidi Abstract Statistical inference for extremes has been a subject of intensive research during the past couple of decades. One
More informationModels and estimation.
Bivariate generalized Pareto distribution practice: Models and estimation. Eötvös Loránd University, Budapest, Hungary 7 June 2011, ASMDA Conference, Rome, Italy Problem How can we properly estimate the
More informationThe extremal elliptical model: Theoretical properties and statistical inference
1/25 The extremal elliptical model: Theoretical properties and statistical inference Thomas OPITZ Supervisors: Jean-Noel Bacro, Pierre Ribereau Institute of Mathematics and Modeling in Montpellier (I3M)
More informationPREPRINT 2005:38. Multivariate Generalized Pareto Distributions HOLGER ROOTZÉN NADER TAJVIDI
PREPRINT 2005:38 Multivariate Generalized Pareto Distributions HOLGER ROOTZÉN NADER TAJVIDI Department of Mathematical Sciences Division of Mathematical Statistics CHALMERS UNIVERSITY OF TECHNOLOGY GÖTEBORG
More informationON EXTREME VALUE ANALYSIS OF A SPATIAL PROCESS
REVSTAT Statistical Journal Volume 6, Number 1, March 008, 71 81 ON EXTREME VALUE ANALYSIS OF A SPATIAL PROCESS Authors: Laurens de Haan Erasmus University Rotterdam and University Lisbon, The Netherlands
More informationA Note on Tail Behaviour of Distributions. the max domain of attraction of the Frechét / Weibull law under power normalization
ProbStat Forum, Volume 03, January 2010, Pages 01-10 ISSN 0974-3235 A Note on Tail Behaviour of Distributions in the Max Domain of Attraction of the Frechét/ Weibull Law under Power Normalization S.Ravi
More informationarxiv: v4 [math.pr] 7 Feb 2012
THE MULTIVARIATE PIECING-TOGETHER APPROACH REVISITED STEFAN AULBACH, MICHAEL FALK, AND MARTIN HOFMANN arxiv:1108.0920v4 math.pr] 7 Feb 2012 Abstract. The univariate Piecing-Together approach (PT) fits
More informationExtreme Value Analysis and Spatial Extremes
Extreme Value Analysis and Department of Statistics Purdue University 11/07/2013 Outline Motivation 1 Motivation 2 Extreme Value Theorem and 3 Bayesian Hierarchical Models Copula Models Max-stable Models
More informationBivariate generalized Pareto distribution
Bivariate generalized Pareto distribution in practice Eötvös Loránd University, Budapest, Hungary Minisymposium on Uncertainty Modelling 27 September 2011, CSASC 2011, Krems, Austria Outline Short summary
More informationExtreme Value Theory An Introduction
Laurens de Haan Ana Ferreira Extreme Value Theory An Introduction fi Springer Contents Preface List of Abbreviations and Symbols vii xv Part I One-Dimensional Observations 1 Limit Distributions and Domains
More informationExceedance probability of the integral of a stochastic process
Exceedance probability of the integral of a stochastic process Ana Ferreira IA, Universidade Técnica de Lisboa and CEAUL Laurens de Haan University of Tilburg, Erasmus University Rotterdam and CEAUL Chen
More informationESTIMATING BIVARIATE TAIL
Elena DI BERNARDINO b joint work with Clémentine PRIEUR a and Véronique MAUME-DESCHAMPS b a LJK, Université Joseph Fourier, Grenoble 1 b Laboratoire SAF, ISFA, Université Lyon 1 Framework Goal: estimating
More informationMath 576: Quantitative Risk Management
Math 576: Quantitative Risk Management Haijun Li lih@math.wsu.edu Department of Mathematics Washington State University Week 11 Haijun Li Math 576: Quantitative Risk Management Week 11 1 / 21 Outline 1
More informationarxiv:math/ v1 [math.st] 16 May 2006
The Annals of Statistics 006 Vol 34 No 46 68 DOI: 04/009053605000000886 c Institute of Mathematical Statistics 006 arxiv:math/0605436v [mathst] 6 May 006 SPATIAL EXTREMES: MODELS FOR THE STATIONARY CASE
More informationExtreme value statistics: from one dimension to many. Lecture 1: one dimension Lecture 2: many dimensions
Extreme value statistics: from one dimension to many Lecture 1: one dimension Lecture 2: many dimensions The challenge for extreme value statistics right now: to go from 1 or 2 dimensions to 50 or more
More informationOverview of Extreme Value Theory. Dr. Sawsan Hilal space
Overview of Extreme Value Theory Dr. Sawsan Hilal space Maths Department - University of Bahrain space November 2010 Outline Part-1: Univariate Extremes Motivation Threshold Exceedances Part-2: Bivariate
More informationNonlinear Time Series Modeling
Nonlinear Time Series Modeling Part II: Time Series Models in Finance Richard A. Davis Colorado State University (http://www.stat.colostate.edu/~rdavis/lectures) MaPhySto Workshop Copenhagen September
More informationEstimating Bivariate Tail: a copula based approach
Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Université Lyon 1 - ISFA, Institut de Science Financiere et d'assurances - AST&Risk (ANR Project) Joint work with Véronique Maume-Deschamps
More informationChapter 2 Asymptotics
Chapter Asymptotics.1 Asymptotic Behavior of Student s Pdf Proposition.1 For each x R d,asν, f ν,,a x g a, x..1 Proof Let a = 0. Using the well-known formula that Ɣz = π z e z z z 1 + O 1, as z,. z we
More informationExtreme Value Theory and Applications
Extreme Value Theory and Deauville - 04/10/2013 Extreme Value Theory and Introduction Asymptotic behavior of the Sum Extreme (from Latin exter, exterus, being on the outside) : Exceeding the ordinary,
More informationContributions for the study of high levels that persist over a xed period of time
Contributions for the study of high levels that persist over a xed period of time Marta Ferreira Universidade do Minho, Dep. Matemática, CMAT Luísa Canto e Castro Faculdade de Ciências da Universidade
More informationMFM Practitioner Module: Quantitiative Risk Management. John Dodson. October 14, 2015
MFM Practitioner Module: Quantitiative Risk Management October 14, 2015 The n-block maxima 1 is a random variable defined as M n max (X 1,..., X n ) for i.i.d. random variables X i with distribution function
More informationStatistics of Extremes
Statistics of Extremes Anthony Davison c 211 http://stat.epfl.ch Multivariate Extremes 19 Componentwise maxima.................................................. 194 Standardization........................................................
More informationMax-stable Processes for Threshold Exceedances in Spatial Extremes
Max-stable Processes for Threshold Exceedances in Spatial Extremes Soyoung Jeon A dissertation submitted to the faculty of the University of North Carolina at Chapel Hill in partial fulfillment of the
More informationAN ASYMPTOTICALLY UNBIASED MOMENT ESTIMATOR OF A NEGATIVE EXTREME VALUE INDEX. Departamento de Matemática. Abstract
AN ASYMPTOTICALLY UNBIASED ENT ESTIMATOR OF A NEGATIVE EXTREME VALUE INDEX Frederico Caeiro Departamento de Matemática Faculdade de Ciências e Tecnologia Universidade Nova de Lisboa 2829 516 Caparica,
More informationA Conditional Approach to Modeling Multivariate Extremes
A Approach to ing Multivariate Extremes By Heffernan & Tawn Department of Statistics Purdue University s April 30, 2014 Outline s s Multivariate Extremes s A central aim of multivariate extremes is trying
More informationAnalysis methods of heavy-tailed data
Institute of Control Sciences Russian Academy of Sciences, Moscow, Russia February, 13-18, 2006, Bamberg, Germany June, 19-23, 2006, Brest, France May, 14-19, 2007, Trondheim, Norway PhD course Chapter
More informationNONPARAMETRIC ESTIMATION OF THE CONDITIONAL TAIL INDEX
NONPARAMETRIC ESTIMATION OF THE CONDITIONAL TAIL INDE Laurent Gardes and Stéphane Girard INRIA Rhône-Alpes, Team Mistis, 655 avenue de l Europe, Montbonnot, 38334 Saint-Ismier Cedex, France. Stephane.Girard@inrialpes.fr
More informationDiscussion on Human life is unlimited but short by Holger Rootzén and Dmitrii Zholud
Extremes (2018) 21:405 410 https://doi.org/10.1007/s10687-018-0322-z Discussion on Human life is unlimited but short by Holger Rootzén and Dmitrii Zholud Chen Zhou 1 Received: 17 April 2018 / Accepted:
More informationExtreme Value Theory as a Theoretical Background for Power Law Behavior
Extreme Value Theory as a Theoretical Background for Power Law Behavior Simone Alfarano 1 and Thomas Lux 2 1 Department of Economics, University of Kiel, alfarano@bwl.uni-kiel.de 2 Department of Economics,
More informationON THE TAIL INDEX ESTIMATION OF AN AUTOREGRESSIVE PARETO PROCESS
Discussiones Mathematicae Probability and Statistics 33 (2013) 65 77 doi:10.7151/dmps.1149 ON THE TAIL INDEX ESTIMATION OF AN AUTOREGRESSIVE PARETO PROCESS Marta Ferreira Center of Mathematics of Minho
More informationJournal of Statistical Planning and Inference
Journal of Statistical Planning Inference 39 9 336 -- 3376 Contents lists available at ScienceDirect Journal of Statistical Planning Inference journal homepage: www.elsevier.com/locate/jspi Maximum lielihood
More informationMultivariate Heavy Tails, Asymptotic Independence and Beyond
Multivariate Heavy Tails, endence and Beyond Sidney Resnick School of Operations Research and Industrial Engineering Rhodes Hall Cornell University Ithaca NY 14853 USA http://www.orie.cornell.edu/ sid
More informationIs there evidence of log-periodicities in the tail of the distribution of seismic moments?
Is there evidence of log-periodicities in the tail of the distribution of seismic moments? Luísa Canto e Castro Sandra Dias CEAUL, Department of Statistics University of Lisbon, Portugal CEAUL, Department
More informationOn the Estimation and Application of Max-Stable Processes
On the Estimation and Application of Max-Stable Processes Zhengjun Zhang Department of Statistics University of Wisconsin Madison, WI 53706, USA Co-author: Richard Smith EVA 2009, Fort Collins, CO Z. Zhang
More informationClassical Extreme Value Theory - An Introduction
Chapter 1 Classical Extreme Value Theory - An Introduction 1.1 Introduction Asymptotic theory of functions of random variables plays a very important role in modern statistics. The objective of the asymptotic
More informationMultivariate extremes. Anne-Laure Fougeres. Laboratoire de Statistique et Probabilites. INSA de Toulouse - Universite Paul Sabatier 1
Multivariate extremes Anne-Laure Fougeres Laboratoire de Statistique et Probabilites INSA de Toulouse - Universite Paul Sabatier 1 1. Introduction. A wide variety of situations concerned with extreme events
More informationNonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution
Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution p. /2 Nonparametric Estimation of the Dependence Function for a Multivariate Extreme Value Distribution
More informationBayesian Inference for Clustered Extremes
Newcastle University, Newcastle-upon-Tyne, U.K. lee.fawcett@ncl.ac.uk 20th TIES Conference: Bologna, Italy, July 2009 Structure of this talk 1. Motivation and background 2. Review of existing methods Limitations/difficulties
More informationSpatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012
Spatial extreme value theory and properties of max-stable processes Poitiers, November 8-10, 2012 November 8, 2012 15:00 Clement Dombry Habilitation thesis defense (in french) 17:00 Snack buet November
More informationOn the Application of the Generalized Pareto Distribution for Statistical Extrapolation in the Assessment of Dynamic Stability in Irregular Waves
On the Application of the Generalized Pareto Distribution for Statistical Extrapolation in the Assessment of Dynamic Stability in Irregular Waves Bradley Campbell 1, Vadim Belenky 1, Vladas Pipiras 2 1.
More informationTwo practical tools for rainfall weather generators
Two practical tools for rainfall weather generators Philippe Naveau naveau@lsce.ipsl.fr Laboratoire des Sciences du Climat et l Environnement (LSCE) Gif-sur-Yvette, France FP7-ACQWA, GIS-PEPER, MIRACLE
More informationPractical conditions on Markov chains for weak convergence of tail empirical processes
Practical conditions on Markov chains for weak convergence of tail empirical processes Olivier Wintenberger University of Copenhagen and Paris VI Joint work with Rafa l Kulik and Philippe Soulier Toronto,
More informationThe Convergence Rate for the Normal Approximation of Extreme Sums
The Convergence Rate for the Normal Approximation of Extreme Sums Yongcheng Qi University of Minnesota Duluth WCNA 2008, Orlando, July 2-9, 2008 This talk is based on a joint work with Professor Shihong
More informationQuantitative Modeling of Operational Risk: Between g-and-h and EVT
: Between g-and-h and EVT Paul Embrechts Matthias Degen Dominik Lambrigger ETH Zurich (www.math.ethz.ch/ embrechts) Outline Basel II LDA g-and-h Aggregation Conclusion and References What is Basel II?
More informationAccounting for extreme-value dependence in multivariate data
Accounting for extreme-value dependence in multivariate data 38th ASTIN Colloquium Manchester, July 15, 2008 Outline 1. Dependence modeling through copulas 2. Rank-based inference 3. Extreme-value dependence
More informationBivariate generalized Pareto distribution in practice: models and estimation
Bivariate generalized Pareto distribution in practice: models and estimation November 9, 2011 Pál Rakonczai Department of Probability Theory and Statistics, Eötvös University, Hungary e-mail: paulo@cs.elte.hu
More informationIntroduction to Algorithmic Trading Strategies Lecture 10
Introduction to Algorithmic Trading Strategies Lecture 10 Risk Management Haksun Li haksun.li@numericalmethod.com www.numericalmethod.com Outline Value at Risk (VaR) Extreme Value Theory (EVT) References
More informationSome conditional extremes of a Markov chain
Some conditional extremes of a Markov chain Seminar at Edinburgh University, November 2005 Adam Butler, Biomathematics & Statistics Scotland Jonathan Tawn, Lancaster University Acknowledgements: Janet
More informationNew Classes of Multivariate Survival Functions
Xiao Qin 2 Richard L. Smith 2 Ruoen Ren School of Economics and Management Beihang University Beijing, China 2 Department of Statistics and Operations Research University of North Carolina Chapel Hill,
More informationEXPLICIT MULTIVARIATE BOUNDS OF CHEBYSHEV TYPE
Annales Univ. Sci. Budapest., Sect. Comp. 42 2014) 109 125 EXPLICIT MULTIVARIATE BOUNDS OF CHEBYSHEV TYPE Villő Csiszár Budapest, Hungary) Tamás Fegyverneki Budapest, Hungary) Tamás F. Móri Budapest, Hungary)
More informationA Bayesian Spatial Model for Exceedances Over a Threshold
A Bayesian Spatial odel for Exceedances Over a Threshold Fernando Ferraz do Nascimento and Bruno Sansó June 2, 2017 Abstract Extreme value theory focuses on the study of rare events and uses asymptotic
More informationPreface to the Third Edition
Preface to the Third Edition The main focus of extreme value theory has been undergoing a dramatic change. Instead of concentrating on maxima of observations, large observations are now in the focus, defined
More informationModelling extreme-value dependence in high dimensions using threshold exceedances
Modelling extreme-value dependence in high dimensions using threshold exceedances Anna Kiriliouk A thesis submitted to the Université catholique de Louvain in partial fulfillment of the requirements for
More informationAN EMPIRICAL TAIL INDEX and VaR ANALYSIS 9
AN EMPIRICAL TAIL INDEX and VaR ANALYSIS 9 M. Ivette Gomes Clara Viseu CEAUL and DEIO, FCUL ISCA, Universidade de Coimbra Universidade de Lisboa, Portugal CEAUL, Universidade de Lisboa, Portugal Abstract:
More informationOperational Risk and Pareto Lévy Copulas
Operational Risk and Pareto Lévy Copulas Claudia Klüppelberg Technische Universität München email: cklu@ma.tum.de http://www-m4.ma.tum.de References: - Böcker, K. and Klüppelberg, C. (25) Operational VaR
More informationExtremogram and Ex-Periodogram for heavy-tailed time series
Extremogram and Ex-Periodogram for heavy-tailed time series 1 Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia) and Yuwei Zhao (Ulm) 1 Jussieu, April 9, 2014 1 2 Extremal
More informationThe Behavior of Multivariate Maxima of Moving Maxima Processes
The Behavior of Multivariate Maxima of Moving Maxima Processes Zhengjun Zhang Department of Mathematics Washington University Saint Louis, MO 6313-4899 USA Richard L. Smith Department of Statistics University
More informationPeaks over thresholds modelling with multivariate generalized Pareto distributions
Peaks over thresholds modelling with multivariate generalized Pareto distributions Anna Kiriliouk Johan Segers Université catholique de Louvain Institut de Statistique, Biostatistique et Sciences Actuarielles
More informationSparse Representation of Multivariate Extremes with Applications to Anomaly Ranking
Sparse Representation of Multivariate Extremes with Applications to Anomaly Ranking Nicolas Goix Anne Sabourin Stéphan Clémençon Abstract Capturing the dependence structure of multivariate extreme events
More informationAssessing Dependence in Extreme Values
02/09/2016 1 Motivation Motivation 2 Comparison 3 Asymptotic Independence Component-wise Maxima Measures Estimation Limitations 4 Idea Results Motivation Given historical flood levels, how high should
More informationHeavy Tailed Time Series with Extremal Independence
Heavy Tailed Time Series with Extremal Independence Rafa l Kulik and Philippe Soulier Conference in honour of Prof. Herold Dehling Bochum January 16, 2015 Rafa l Kulik and Philippe Soulier Regular variation
More informationOn Extreme Value Statistics. maximum likelihood portfolio optimization extremal rainfall Internet auctions
On Extreme Value Statistics maximum likelihood portfolio optimization extremal rainfall Internet auctions ISBN: 978 90 5170 912 4 Cover design: Crasborn Graphics Designers bno, Valkenburg a.d. Geul This
More informationRefining the Central Limit Theorem Approximation via Extreme Value Theory
Refining the Central Limit Theorem Approximation via Extreme Value Theory Ulrich K. Müller Economics Department Princeton University February 2018 Abstract We suggest approximating the distribution of
More informationChapter 5 continued. Chapter 5 sections
Chapter 5 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationFall 2017 STAT 532 Homework Peter Hoff. 1. Let P be a probability measure on a collection of sets A.
1. Let P be a probability measure on a collection of sets A. (a) For each n N, let H n be a set in A such that H n H n+1. Show that P (H n ) monotonically converges to P ( k=1 H k) as n. (b) For each n
More informationIntroduction to Probability and Statistics (Continued)
Introduction to Probability and Statistics (Continued) Prof. icholas Zabaras Center for Informatics and Computational Science https://cics.nd.edu/ University of otre Dame otre Dame, Indiana, USA Email:
More informationAbstract: In this short note, I comment on the research of Pisarenko et al. (2014) regarding the
Comment on Pisarenko et al. Characterization of the Tail of the Distribution of Earthquake Magnitudes by Combining the GEV and GPD Descriptions of Extreme Value Theory Mathias Raschke Institution: freelancer
More informationRichard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC
EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research University of North Carolina Chapel Hill, NC 27599-3260 rls@email.unc.edu AMS Committee on Probability and Statistics
More informationEstimation of the extreme value index and high quantiles under random censoring
Estimation of the extreme value index and high quantiles under random censoring Jan Beirlant () & Emmanuel Delafosse (2) & Armelle Guillou (2) () Katholiee Universiteit Leuven, Department of Mathematics,
More informationVerification of extremes using proper scoring rules and extreme value theory
Verification of extremes using proper scoring rules and extreme value theory Maxime Taillardat 1,2,3 A-L. Fougères 3, P. Naveau 2 and O. Mestre 1 1 CNRM/Météo-France 2 LSCE 3 ICJ May 8, 2017 Plan 1 Extremes
More informationSemi-parametric tail inference through Probability-Weighted Moments
Semi-parametric tail inference through Probability-Weighted Moments Frederico Caeiro New University of Lisbon and CMA fac@fct.unl.pt and M. Ivette Gomes University of Lisbon, DEIO, CEAUL and FCUL ivette.gomes@fc.ul.pt
More informationA MODIFICATION OF HILL S TAIL INDEX ESTIMATOR
L. GLAVAŠ 1 J. JOCKOVIĆ 2 A MODIFICATION OF HILL S TAIL INDEX ESTIMATOR P. MLADENOVIĆ 3 1, 2, 3 University of Belgrade, Faculty of Mathematics, Belgrade, Serbia Abstract: In this paper, we study a class
More informationTail empirical process for long memory stochastic volatility models
Tail empirical process for long memory stochastic volatility models Rafa l Kulik and Philippe Soulier Carleton University, 5 May 2010 Rafa l Kulik and Philippe Soulier Quick review of limit theorems and
More informationA PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS
Statistica Sinica 20 2010, 365-378 A PRACTICAL WAY FOR ESTIMATING TAIL DEPENDENCE FUNCTIONS Liang Peng Georgia Institute of Technology Abstract: Estimating tail dependence functions is important for applications
More informationDependence Comparison of Multivariate Extremes via Stochastic Tail Orders
Dependence Comparison of Multivariate Extremes via Stochastic Tail Orders Haijun Li Department of Mathematics Washington State University Pullman, WA 99164, U.S.A. July 2012 Abstract A stochastic tail
More informationA New Estimator for a Tail Index
Acta Applicandae Mathematicae 00: 3, 2003. 2003 Kluwer Academic Publishers. Printed in the Netherlands. A New Estimator for a Tail Index V. PAULAUSKAS Department of Mathematics and Informatics, Vilnius
More informationTail Approximation of Value-at-Risk under Multivariate Regular Variation
Tail Approximation of Value-at-Risk under Multivariate Regular Variation Yannan Sun Haijun Li July 00 Abstract This paper presents a general tail approximation method for evaluating the Valueat-Risk of
More informationThe Delta Method and Applications
Chapter 5 The Delta Method and Applications 5.1 Local linear approximations Suppose that a particular random sequence converges in distribution to a particular constant. The idea of using a first-order
More informationHigh-Dimensional Extremes and Copulas
High-Dimensional Extremes and Copulas Haijun Li lih@math.wsu.edu Washington State University CIAS-CUFE, Beijing, January 3, 2014 Haijun Li High-Dimensional Extremes and Copulas CIAS-CUFE, Beijing, January
More informationExtremogram and ex-periodogram for heavy-tailed time series
Extremogram and ex-periodogram for heavy-tailed time series 1 Thomas Mikosch University of Copenhagen Joint work with Richard A. Davis (Columbia) and Yuwei Zhao (Ulm) 1 Zagreb, June 6, 2014 1 2 Extremal
More informationEstimation of the Angular Density in Multivariate Generalized Pareto Models
in Multivariate Generalized Pareto Models René Michel michel@mathematik.uni-wuerzburg.de Institute of Applied Mathematics and Statistics University of Würzburg, Germany 18.08.2005 / EVA 2005 The Multivariate
More informationAlgorithms for Uncertainty Quantification
Algorithms for Uncertainty Quantification Tobias Neckel, Ionuț-Gabriel Farcaș Lehrstuhl Informatik V Summer Semester 2017 Lecture 2: Repetition of probability theory and statistics Example: coin flip Example
More informationHIERARCHICAL MODELS IN EXTREME VALUE THEORY
HIERARCHICAL MODELS IN EXTREME VALUE THEORY Richard L. Smith Department of Statistics and Operations Research, University of North Carolina, Chapel Hill and Statistical and Applied Mathematical Sciences
More informationBias-corrected goodness-of-fit tests for Pareto-type behavior
Bias-corrected goodness-of-fit tests for Pareto-type behavior Yuri Goegebeur University of Southern Denmark, Department of Statistics JB Winsløws Ve 9B DK5000 Odense C, Denmark E-mail: yurigoegebeur@statsdudk
More informationBias Reduction in the Estimation of a Shape Second-order Parameter of a Heavy Right Tail Model
Bias Reduction in the Estimation of a Shape Second-order Parameter of a Heavy Right Tail Model Frederico Caeiro Universidade Nova de Lisboa, FCT and CMA M. Ivette Gomes Universidade de Lisboa, DEIO, CEAUL
More informationGARCH processes probabilistic properties (Part 1)
GARCH processes probabilistic properties (Part 1) Alexander Lindner Centre of Mathematical Sciences Technical University of Munich D 85747 Garching Germany lindner@ma.tum.de http://www-m1.ma.tum.de/m4/pers/lindner/
More informationMA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems
MA/ST 810 Mathematical-Statistical Modeling and Analysis of Complex Systems Review of Basic Probability The fundamentals, random variables, probability distributions Probability mass/density functions
More informationSimulation of Max Stable Processes
Simulation of Max Stable Processes Whitney Huang Department of Statistics Purdue University November 6, 2013 1 / 30 Outline 1 Max-Stable Processes 2 Unconditional Simulations 3 Conditional Simulations
More informationChapter 5. Chapter 5 sections
1 / 43 sections Discrete univariate distributions: 5.2 Bernoulli and Binomial distributions Just skim 5.3 Hypergeometric distributions 5.4 Poisson distributions Just skim 5.5 Negative Binomial distributions
More informationarxiv: v2 [math.st] 20 Nov 2016
A Lynden-Bell integral estimator for the tail inde of right-truncated data with a random threshold Nawel Haouas Abdelhakim Necir Djamel Meraghni Brahim Brahimi Laboratory of Applied Mathematics Mohamed
More informationBayesian Modelling of Extreme Rainfall Data
Bayesian Modelling of Extreme Rainfall Data Elizabeth Smith A thesis submitted for the degree of Doctor of Philosophy at the University of Newcastle upon Tyne September 2005 UNIVERSITY OF NEWCASTLE Bayesian
More informationLecture 2: Repetition of probability theory and statistics
Algorithms for Uncertainty Quantification SS8, IN2345 Tobias Neckel Scientific Computing in Computer Science TUM Lecture 2: Repetition of probability theory and statistics Concept of Building Block: Prerequisites:
More informationEmpirical Tail Index and VaR Analysis
International Conference on Mathematical and Statistical Modeling in Honor of Enrique Castillo. June 28-30, 2006 Empirical Tail Index and VaR Analysis M. Ivette Gomes Universidade de Lisboa, DEIO, CEAUL
More information