A proposal of a bivariate Conditional Tail Expectation
|
|
- Constance Dawson
- 5 years ago
- Views:
Transcription
1 A proposal of a bivariate Conditional Tail Expectation Elena Di Bernardino a joint works with Areski Cousin b, Thomas Laloë c, Véronique Maume-Deschamps d and Clémentine Prieur e a, b, d Université Lyon 1, ISFA, Laboratoire SAF c Université de Nice Sophia-Antipolis, Laboratoire J-A Dieudonné e Université Joseph Fourier, Grenoble
2 Given an univariate continuous and strictly monotonic loss distribution function F X, VaR α (X ) = Q X (α) = F 1 X (α), α (0, 1). Shortcoming of VaR measure: Var is not a coherent risk measure (in the "Artzner sense", see Artzner, 1999) Var does not give any information about thickness of the upper tail of the loss To overcome problems of VaR Conditional Tail Expectation (CTE): CTE α (X ) = E[ X X VaR α (X ) ] = E[ X X Q X (α) ], (e.g. for properties of the CTE α (X ) see Denuit et al., 2005).
3 Our Bivariate Conditional Tail Expectation In the literature A new bivariate Conditional Tail Expectation Denition (Generalization of the CTE in dimension 2) Consider a random vector (X, Y ) satisfying regularity properties, with associate distribution function F. Let L(α) = {F (x, y) α}. For α (0, 1), we dene ( ) E[ X (X, Y ) L(α) ] CTE α (X, Y) =. E[ Y (X, Y ) L(α) ] Distributional approach to risk model: Our CTE α (X, Y ) does not use an aggregate variable (sum, min, max...) in order to analyze the bivariate risk's issue. Conversely CTE α (X, Y ) deals with the simultaneous joint damages considering the bivariate dependence structure of data in a specic risk's area (α-level set : L(α)).
4 Our Bivariate Conditional Tail Expectation In the literature Several bivariate generalizations of the classical univariate CTE have been proposed; mainly using as conditioning events the total risk or some univariate extreme risk in the portfolio. We recall for instance: CTE α (X, Y ) sum = E[ (X, Y ) X + Y > Q X +Y (α) ], CTE α (X, Y ) min = E[ (X, Y ) min{x, Y } > Q min{x,y } (α) ], CTE α (X, Y ) max = E[ (X, Y ) max{x, Y } > Q max{x,y } (α) ]. Remark: Conditioning events are the total risk or some univariate extreme risk in the portfolio (aggregate variable).
5 Our Bivariate Conditional Tail Expectation In the literature Two dierent developments of research : 1 Problem of estimating the level sets L(c) = {F (x) c}, with c (0, 1), of an unknown distribution function F on R 2 + with a plug-in approach; in order to provide a consistent estimator for CTE α (X, Y ) [Article in a non-compact setting with applications in multivariate risk theory, with Laloë L., Maume-Deschamps V. and Prieur C., submitted]. 2 Analysis of the CTE α (X, Y ) as risk measure: study of the classical properties (monotonicity, translation invariance, positive homogeneity,...), behavior with respect to dierent risk scenarios and stochastic ordering of marginals risks [Article Some proposals about bivariate risk measures, with Cousin A.].
6 Estimation of the level sets of a density function: Polonik (1995), Tsybakov (1997), Baíllo et al. (2001), Biau et al. (2007). Estimation of the level sets of a regression function in a compact setting: Cavalier (1997), Laloë (2009). Estimation of general compact level sets: Cuevas et al. (2006). An alternative approach, based on the geometric properties of the compact support sets: Cuevas and Fraiman (1997), Cuevas and Rodríguez-Casal (2004).
7 Goal: We consider the problem of estimating the level sets of a bivariate distribution function F. More precisely our goal is to build a consistent estimator of L(c) := {F (x) c}, for c (0, 1). Idea: We consider a plug-in approach that is L(c) is estimated by L n (c) := {F n (x) c}, for c (0, 1), where F n is a consistent estimator of F.
8 From a parametric formulation of L(c) (see Belzunce et al. 2007). Case: Survival Clayton Copula with marginals (Burr(1), Burr(2))
9 We state consistency results with respect to two proximity criteria between sets: the Hausdor distance and the volume of the symmetric dierence (physical proximity between sets). Problem of compactness property for the level sets we estimate. Figure: (left) Hausdor distance between sets X and Y ; (right) λ(x Y ), where λ stands for the Lebesgue measure on R 2 and for the symmetric dierence.
10 Let R 2 + := R 2 + \ (0, 0), F the set of continuous distribution functions f : R 2 + [0, 1] and F F. Given an i.i.d sample {X i } n i=1 in R2 + with distribution function F, we denote F n (.) = F n (X 1, X 2,..., X n,.) an estimator of F. Dene, for c (0, 1), the upper c-level set of F F and its plug-in estimator : L(c) := {x R 2 + : F (x) c}, In addition, given T > 0, we set L n (c) := {x R 2 + : F n (x) c}, {F = c} = {x R 2 + : F (x) = c}. L(c) T = {x [0, T ] 2 : F (x) c}, L n (c) T = {x [0, T ] 2 : F n (x) c}, {F = c} T = {x [0, T ] 2 : F (x) = c}. For any A R 2 + we note A its boundary.
11 For r > 0 and λ > 0, dene E := B({x R 2 + : F c r}, λ), m := inf x E ( F ) x, M H := sup x E (HF ) x, B(x, ξ) is the closed ball centered on x and with positive radius ξ, ( F ) x is the gradient of F evaluated at x, (HF ) x is the matrix norm induced by Euclidean distance of the Hessian matrix in x. Hausdor distance between A 1 and A 2 is dened by: d H (A 1, A 2 ) = inf{ε > 0 : A 1 B(A 2, ε), A 2 B(A 1, ε)}, where B(S, ε) = x S B(x, ε); A 1 and A 2 are compact sets in (R 2 +, d).
12 H: There exist γ > 0 and A > 0 such that, if t c γ then T > 0 such that {F = c} T and {F = t} T, d H ({F = c} T, {F = t} T ) A t c. Assumption H is satised under mild conditions. Proposition Let c (0, 1). Let F F be twice dierentiable on R 2 +. Assume there exist r > 0, ζ > 0 such that m > 0 and M H <. Then F satises Assumption H, with A = 2. m
13 From now on we note, for n N, and for T > 0, F F n = sup F (x) F n(x), F F n T = sup F (x) F n(x). x R 2 + x [0,T ] 2 Theorem (Consistency Hausdor distance) Let c (0, 1). Let F F be twice dierentiable on R 2 +. Assume that there exist r > 0, ζ > 0 such that m > 0 and M H <. Let T 1 > 0 such that for all t : t c r, L(t) T1. Let (T n) n N be an increasing sequence of positive values. Assume that, for each n, F n is continuous with probability one and that Then for n large enough, F F n 0, a.s. d H ( L(c) T n, L n(c) T n ) 6 A F F n T n, a.s., where A = 2 m.
14 d H ( L(c) T n, Ln(c)T n ) converges to zero and the quality of our plug-in estimator is obviously related to the quality of the estimator F n. Note that in the case c is close to one the constant A = 2 could be m large. In this case, we will need a large number of data to get a reasonable estimation. In order to overcome the problem of F n is continuous with probability one it can be considered a smooth version of estimator.
15 Let us introduce the following assumption: A1 There exist positive increasing sequences (v n) n N and (T n) n N such that v n [0,T n] 2 F F n p λ(dx) P n 0, for some 1 p <. Theorem (Consistency volume) Let c (0, 1). Let F F be twice dierentiable on R 2 +. Assume that there exist r > 0, ζ > 0 such that m > 0 and M H <. Let (v n) n N and (T n) n N positive increasing sequences such that Assumption A1 is satised and that for all t : t c r, L(t) T1. Then it holds that p n d λ (L(c) T n, L n(c) T n ) P 0, n with p n an increasing positive sequence such that p n = o ( 1 p+1 vn /T p p+1 n ).
16 Theorem Consistency volume provides a convergence rate, which is closely related to the choice of the sequence T n. A sequence T n whose divergence rate is large implies a convergence rate p n quite slow. Theorem Consistency volume does not require continuity assumption on F n. Analysis in the case: F n the bivariate empirical distribution function. Problem: optimal criterion for the choice of T n related with the p in Assumption A1.
17 Estimation of CTE α (X, Y ) Denition Consider a random vector (X, Y ) with associate distribution function F F. For α (0, 1), we dene the estimated bivariate α-conditional Tail Expectation ĈTE α (X, Y ) = n i=1 X i 1 {(Xi,Y i ) L n(α)} n i=1 1 {(X i,y i ) L n(α)} n i=1 Y i 1 {(Xi,Y i ) L n(α)} n i=1 1 {(X i,y i ) L n(α)}. (1) We introduce truncated versions of the theoretical and estimated CTE α : CTE T α (X, Y ) = E[(X, Y ) (X, Y ) L(α) T ], T ĈTEα (X, Y ), using L(α) T and L n (α) T i.e. the truncated versions L(α) and L n (α).
18 Estimation of CTE α (X, Y ) Theorem (Consistency of ĈTE α (X, Y )) Under regularity properties of (X, Y ), Assumptions of Theorem Consistency volume and with the same notation, it holds that T β n T n CTE n α (X, Y ) ĈTEα (X, Y ) P 0, (2) n r 2(1+r) where β n = min{ pn, a n }, with r > 0 such that the density f (X, Y ) L 1+r (λ) and a n = ( ) o n. The convergence in (2) must be interpreted as a componentwise convergence. In the case of a bounded density function f (X, Y ) we obtain β n = min{ p n, a n }.
19 On the choice of T n ( ) In the case of the bivariate empirical d.f. F n : β n = o n r 2 r 6 (1+r) 3 (1+r) /T n. ( In the case of a bounded density function f (X,Y ), β n = o n 1 6 /T 2 3 n ). Obviously it could be interesting to consider the convergence: T n CTEα (X, Y ) ĈTEα (X, Y ). the speed of convergence will also depend on the convergence rate to zero of CTEα (X, Y ) CTE T n α (X, Y ), then of P[(X, Y ) L(α) \ L(α) T n ]. We remark that (P[X T n or Y T n ]) 1 is increasing in T n, whereas the speed of convergence is decreasing in T n. This kind of compromise provides an illustration on how to choose T n, apart from satisfying the assumptions of consistency results above.
20 Study of CTE α (X, Y ) as risk measure (stochastic order for random variables and random vectors) CTE α (X, Y ) and Kendall distribution function or bivariate probability integral transformation (BIPIT) F (X, Y ). In this work we provide asymptotic results for a xed level c... Problem of constant A for c 1: Plug in method with Ln (c) := {(x, y) R 2 + : F (x, y) c}, for c 1, with F a tail estimator of F (Bivariate extreme value theory; rst part of my PhD thesis).
21 Artzner, P. and Delbaen, F. and Eber, J. M. and Heath, D., Coherent measures of risk. Math. Finance 9(3), (1999), A. Baíllo, J.A. Cuesta-Albertos, A. Cuevas, Convergence rates in nonparametric estimation of level sets. Statistics and Probability Letters 53, (2001), F. Belzunce, A. Castaño, A. Olvera-Cervantes, A. Suárez-Llorens, Quantile curves and dependence structure for bivariate distributions. Computational Statistics & Data Analysis 51, (2007), G. Biau, B. Cadre, B. Pelletier, A graph-based estimator of the number of clusters. ESAIM, Probability and Statistics 11, (2007), J. Cai, H. Li, Conditional tail expectations for multivariate phase-type distributions. Journal of Applied Probability 42(3), (2005), L. Cavalier, Nonparametric estimation of regression level sets. Statistics 29, (1997), A. Cuevas, R. Fraiman, A plug-in approach to support estimation. Annals of Statistics 25, (1997),
22 A. Cuevas, W. González-Manteiga, A. Rodríguez-Casal, Plug-in estimation of general level sets. Australian and New Zealand Journal of Statistics 48, (2006), A. Cuevas, A. Rodríguez-Casal, On boundary estimation. Advances in Applied Probability 36, (2004), M. Denuit, J. Dhaene, M. Goovaerts, R. Kaas. Actuarial Theory for Dependent Risks. Wiley, (2005). W. Polonik, Measuring mass concentration and estimating density contour clusters - an excess mass approach. Annals of Statistics 23, (1995), A. Rodríguez-Casal, Estimacíon de conjuntos y sus fronteras. Un enfoque geometrico. Ph.D. Thesis, University of Santiago de Compostela, (2003). A.B. Tsybakov, On nonparametric estimation of density level sets. Annals of Statistics 25, (1997),
Estimating 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 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 informationBregman superquantiles. Estimation methods and applications
Bregman superquantiles. Estimation methods and applications Institut de mathématiques de Toulouse 2 juin 2014 Joint work with F. Gamboa, A. Garivier (IMT) and B. Iooss (EDF R&D). 1 Coherent measure of
More informationVaR vs. Expected Shortfall
VaR vs. Expected Shortfall Risk Measures under Solvency II Dietmar Pfeifer (2004) Risk measures and premium principles a comparison VaR vs. Expected Shortfall Dependence and its implications for risk measures
More informationNonparametric estimation of tail risk measures from heavy-tailed distributions
Nonparametric estimation of tail risk measures from heavy-tailed distributions Jonthan El Methni, Laurent Gardes & Stéphane Girard 1 Tail risk measures Let Y R be a real random loss variable. The Value-at-Risk
More informationEstimation of multivariate critical layers: Applications to rainfall data
Elena Di Bernardino, ICRA 6 / RISK 2015 () Estimation of Multivariate critical layers Barcelona, May 26-29, 2015 Estimation of multivariate critical layers: Applications to rainfall data Elena Di Bernardino,
More informationSome multivariate risk indicators; minimization by using stochastic algorithms
Some multivariate risk indicators; minimization by using stochastic algorithms Véronique Maume-Deschamps, université Lyon 1 - ISFA, Joint work with P. Cénac and C. Prieur. AST&Risk (ANR Project) 1 / 51
More informationLu Cao. A Thesis in The Department of Mathematics and Statistics
Multivariate Robust Vector-Valued Range Value-at-Risk Lu Cao A Thesis in The Department of Mathematics and Statistics Presented in Partial Fulllment of the Requirements for the Degree of Master of Science
More informationA Measure of Monotonicity of Two Random Variables
Journal of Mathematics and Statistics 8 (): -8, 0 ISSN 549-3644 0 Science Publications A Measure of Monotonicity of Two Random Variables Farida Kachapova and Ilias Kachapov School of Computing and Mathematical
More informationMultivariate extensions of expectiles risk measures
Depend. Model. 2017; 5:20 44 Research Article Special Issue: Recent Developments in Quantitative Risk Management Open Access Véronique Maume-Deschamps*, Didier Rullière, and Khalil Said Multivariate extensions
More informationRegularly Varying Asymptotics for Tail Risk
Regularly Varying Asymptotics for Tail Risk Haijun Li Department of Mathematics Washington State University Humboldt Univ-Berlin Haijun Li Regularly Varying Asymptotics for Tail Risk Humboldt Univ-Berlin
More informationBregman superquantiles. Estimation methods and applications
Bregman superquantiles Estimation methods and applications Institut de mathématiques de Toulouse 17 novembre 2014 Joint work with F Gamboa, A Garivier (IMT) and B Iooss (EDF R&D) Bregman superquantiles
More informationQuantile prediction of a random eld extending the gaussian setting
Quantile prediction of a random eld extending the gaussian setting 1 Joint work with : Véronique Maume-Deschamps 1 and Didier Rullière 2 1 Institut Camille Jordan Université Lyon 1 2 Laboratoire des Sciences
More informationFrontier estimation based on extreme risk measures
Frontier estimation based on extreme risk measures by Jonathan EL METHNI in collaboration with Ste phane GIRARD & Laurent GARDES CMStatistics 2016 University of Seville December 2016 1 Risk measures 2
More informationA new approach for stochastic ordering of risks
A new approach for stochastic ordering of risks Liang Hong, PhD, FSA Department of Mathematics Robert Morris University Presented at 2014 Actuarial Research Conference UC Santa Barbara July 16, 2014 Liang
More informationEstimation of density level sets with a given probability content
Estimation of density level sets with a given probability content Benoît CADRE a, Bruno PELLETIER b, and Pierre PUDLO c a IRMAR, ENS Cachan Bretagne, CNRS, UEB Campus de Ker Lann Avenue Robert Schuman,
More informationLecture Quantitative Finance Spring Term 2015
on bivariate Lecture Quantitative Finance Spring Term 2015 Prof. Dr. Erich Walter Farkas Lecture 07: April 2, 2015 1 / 54 Outline on bivariate 1 2 bivariate 3 Distribution 4 5 6 7 8 Comments and conclusions
More informationMULTIVARIATE EXTREME VALUE ANALYSIS UNDER A
MULTIVARIATE EXTREME VALUE ANALYSIS UNDER A DIRECTIONAL APPROACH Raúl A. TORRES CNAM Paris Department of Statistics Universidad Carlos III de Madrid December 2015 Torres Díaz, Raúl A. Directional Extreme
More informationConditional Tail Expectations for Multivariate Phase Type Distributions
Conditional Tail Expectations for Multivariate Phase Type Distributions Jun Cai Department of Statistics and Actuarial Science University of Waterloo Waterloo, ON N2L 3G1, Canada Telphone: 1-519-8884567,
More informationFinancial Econometrics and Volatility Models Copulas
Financial Econometrics and Volatility Models Copulas Eric Zivot Updated: May 10, 2010 Reading MFTS, chapter 19 FMUND, chapters 6 and 7 Introduction Capturing co-movement between financial asset returns
More informationDistortion Risk Measures: Coherence and Stochastic Dominance
Distortion Risk Measures: Coherence and Stochastic Dominance Dr. Julia L. Wirch Dr. Mary R. Hardy Dept. of Actuarial Maths Dept. of Statistics and and Statistics Actuarial Science Heriot-Watt University
More informationMultivariate Stress Testing for Solvency
Multivariate Stress Testing for Solvency Alexander J. McNeil 1 1 Heriot-Watt University Edinburgh Vienna April 2012 a.j.mcneil@hw.ac.uk AJM Stress Testing 1 / 50 Regulation General Definition of Stress
More informationarxiv: v1 [stat.me] 27 Apr 2017
A note on Quantile curves based bivariate reliability concepts Sreelakshmi N Sudheesh K Kattumannil arxiv:74.8444v [stat.me] 27 Apr 27 Indian Statistical Institute, Chennai, India Abstract Several attempts
More informationExtreme geometric quantiles
1/ 30 Extreme geometric quantiles Stéphane GIRARD (Inria Grenoble Rhône-Alpes) Joint work with Gilles STUPFLER (Aix Marseille Université) ERCIM, Pisa, Italy, December 2014 2/ 30 Outline Extreme multivariate
More informationOn Kusuoka Representation of Law Invariant Risk Measures
MATHEMATICS OF OPERATIONS RESEARCH Vol. 38, No. 1, February 213, pp. 142 152 ISSN 364-765X (print) ISSN 1526-5471 (online) http://dx.doi.org/1.1287/moor.112.563 213 INFORMS On Kusuoka Representation of
More informationTail Mutual Exclusivity and Tail- Var Lower Bounds
Tail Mutual Exclusivity and Tail- Var Lower Bounds Ka Chun Cheung, Michel Denuit, Jan Dhaene AFI_15100 TAIL MUTUAL EXCLUSIVITY AND TAIL-VAR LOWER BOUNDS KA CHUN CHEUNG Department of Statistics and Actuarial
More informationNonparametric estimation of extreme risks from heavy-tailed distributions
Nonparametric estimation of extreme risks from heavy-tailed distributions Laurent GARDES joint work with Jonathan EL METHNI & Stéphane GIRARD December 2013 1 Introduction to risk measures 2 Introduction
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 informationAsymptotic distribution of the sample average value-at-risk
Asymptotic distribution of the sample average value-at-risk Stoyan V. Stoyanov Svetlozar T. Rachev September 3, 7 Abstract In this paper, we prove a result for the asymptotic distribution of the sample
More informationX
Correlation: Pitfalls and Alternatives Paul Embrechts, Alexander McNeil & Daniel Straumann Departement Mathematik, ETH Zentrum, CH-8092 Zürich Tel: +41 1 632 61 62, Fax: +41 1 632 15 23 embrechts/mcneil/strauman@math.ethz.ch
More informationEstimation of risk measures for extreme pluviometrical measurements
Estimation of risk measures for extreme pluviometrical measurements by Jonathan EL METHNI in collaboration with Laurent GARDES & Stéphane GIRARD 26th Annual Conference of The International Environmetrics
More informationAn axiomatic characterization of capital allocations of coherent risk measures
An axiomatic characterization of capital allocations of coherent risk measures Michael Kalkbrener Deutsche Bank AG Abstract An axiomatic definition of coherent capital allocations is given. It is shown
More informationRandom sets. Distributions, capacities and their applications. Ilya Molchanov. University of Bern, Switzerland
Random sets Distributions, capacities and their applications Ilya Molchanov University of Bern, Switzerland Molchanov Random sets - Lecture 1. Winter School Sandbjerg, Jan 2007 1 E = R d ) Definitions
More informationAsymptotic Bounds for the Distribution of the Sum of Dependent Random Variables
Asymptotic Bounds for the Distribution of the Sum of Dependent Random Variables Ruodu Wang November 26, 2013 Abstract Suppose X 1,, X n are random variables with the same known marginal distribution F
More informationA Brief Introduction to Copulas
A Brief Introduction to Copulas Speaker: Hua, Lei February 24, 2009 Department of Statistics University of British Columbia Outline Introduction Definition Properties Archimedean Copulas Constructing Copulas
More informationA GRAPH-BASED ESTIMATOR OF THE NUMBER OF CLUSTERS. Introduction
ESAIM: PS June 007, ol. 11, p. 7 80 DOI: 10.1051/ps:007019 ESAIM: Probability and Statistics www.edpsciences.org/ps A GRAPH-BASED ESTIMATOR OF THE NUMBER OF CLUSTERS Gérard Biau 1,Benoît Cadre 1 and Bruno
More informationAsymptotics of random sums of heavy-tailed negatively dependent random variables with applications
Asymptotics of random sums of heavy-tailed negatively dependent random variables with applications Remigijus Leipus (with Yang Yang, Yuebao Wang, Jonas Šiaulys) CIRM, Luminy, April 26-30, 2010 1. Preliminaries
More information1/37. Convexity theory. Victor Kitov
1/37 Convexity theory Victor Kitov 2/37 Table of Contents 1 2 Strictly convex functions 3 Concave & strictly concave functions 4 Kullback-Leibler divergence 3/37 Convex sets Denition 1 Set X is convex
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 informationEstimation of the functional Weibull-tail coefficient
1/ 29 Estimation of the functional Weibull-tail coefficient Stéphane Girard Inria Grenoble Rhône-Alpes & LJK, France http://mistis.inrialpes.fr/people/girard/ June 2016 joint work with Laurent Gardes,
More informationISSN Distortion risk measures for sums of dependent losses
Journal Afrika Statistika Journal Afrika Statistika Vol. 5, N 9, 21, page 26 267. ISSN 852-35 Distortion risk measures for sums of dependent losses Brahim Brahimi, Djamel Meraghni, and Abdelhakim Necir
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 informationMULTIVARIATE EXTENSIONS OF RISK MEASURES
MULTIVARIATE EXTENSIONS OF EXPECTILES RISK MEASURES Véronique Maume-Deschamps, Didier Rullière, Khalil Said To cite this version: Véronique Maume-Deschamps, Didier Rullière, Khalil Said. EXPECTILES RISK
More informationExtreme Negative Dependence and Risk Aggregation
Extreme Negative Dependence and Risk Aggregation Bin Wang and Ruodu Wang January, 015 Abstract We introduce the concept of an extremely negatively dependent (END) sequence of random variables with a given
More informationMultivariate comonotonicity, stochastic orders and risk measures
Multivariate comonotonicity, stochastic orders and risk measures Alfred Galichon (Ecole polytechnique) Brussels, May 25, 2012 Based on collaborations with: A. Charpentier (Rennes) G. Carlier (Dauphine)
More informationSome results on Denault s capital allocation rule
Faculty of Economics and Applied Economics Some results on Denault s capital allocation rule Steven Vanduffel and Jan Dhaene DEPARTMENT OF ACCOUNTANCY, FINANCE AND INSURANCE (AFI) AFI 0601 Some Results
More informationRecovering Copulae from Conditional Quantiles
Wolfgang K. Härdle Chen Huang Alexander Ristig Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics HumboldtUniversität zu Berlin http://lvb.wiwi.hu-berlin.de
More informationModeling of Dependence Structures in Risk Management and Solvency
Moeling of Depenence Structures in Risk Management an Solvency University of California, Santa Barbara 0. August 007 Doreen Straßburger Structure. Risk Measurement uner Solvency II. Copulas 3. Depenent
More informationOn tail dependence coecients of transformed multivariate Archimedean copulas
Tails and for Archim Copula () February 2015, University of Lille 3 On tail dependence coecients of transformed multivariate Archimedean copulas Elena Di Bernardino, CNAM, Paris, Département IMATH Séminaire
More informationDependence. Practitioner Course: Portfolio Optimization. John Dodson. September 10, Dependence. John Dodson. Outline.
Practitioner Course: Portfolio Optimization September 10, 2008 Before we define dependence, it is useful to define Random variables X and Y are independent iff For all x, y. In particular, F (X,Y ) (x,
More informationExpected Shortfall is not elicitable so what?
Expected Shortfall is not elicitable so what? Dirk Tasche Bank of England Prudential Regulation Authority 1 dirk.tasche@gmx.net Finance & Stochastics seminar Imperial College, November 20, 2013 1 The opinions
More informationRisk Aggregation with Dependence Uncertainty
Introduction Extreme Scenarios Asymptotic Behavior Challenges Risk Aggregation with Dependence Uncertainty Department of Statistics and Actuarial Science University of Waterloo, Canada Seminar at ETH Zurich
More informationA multivariate dependence measure for aggregating risks
A multivariate dependence measure for aggregating risks Jan Dhaene 1 Daniël Linders 2 Wim Schoutens 3 David Vyncke 4 December 1, 2013 1 KU Leuven, Leuven, Belgium. Email: jan.dhaene@econ.kuleuven.be 2
More informationarxiv: v1 [math.pr] 24 Aug 2018
On a class of norms generated by nonnegative integrable distributions arxiv:180808200v1 [mathpr] 24 Aug 2018 Michael Falk a and Gilles Stupfler b a Institute of Mathematics, University of Würzburg, Würzburg,
More informationDecomposability and time consistency of risk averse multistage programs
Decomposability and time consistency of risk averse multistage programs arxiv:1806.01497v1 [math.oc] 5 Jun 2018 A. Shapiro School of Industrial and Systems Engineering Georgia Institute of Technology Atlanta,
More informationLifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution
Lifetime Dependence Modelling using a Generalized Multivariate Pareto Distribution Daniel Alai Zinoviy Landsman Centre of Excellence in Population Ageing Research (CEPAR) School of Mathematics, Statistics
More informationRisk Aggregation. Paul Embrechts. Department of Mathematics, ETH Zurich Senior SFI Professor.
Risk Aggregation Paul Embrechts Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/~embrechts/ Joint work with P. Arbenz and G. Puccetti 1 / 33 The background Query by practitioner
More informationASYMPTOTIC MULTIVARIATE EXPECTILES
ASYMPTOTIC MULTIVARIATE EXPECTILES Véronique Maume-Deschamps Didier Rullière Khalil Said To cite this version: Véronique Maume-Deschamps Didier Rullière Khalil Said ASYMPTOTIC MULTIVARIATE EX- PECTILES
More informationMarshall-Olkin Bivariate Exponential Distribution: Generalisations and Applications
CHAPTER 6 Marshall-Olkin Bivariate Exponential Distribution: Generalisations and Applications 6.1 Introduction Exponential distributions have been introduced as a simple model for statistical analysis
More informationIntroduction Wavelet shrinage methods have been very successful in nonparametric regression. But so far most of the wavelet regression methods have be
Wavelet Estimation For Samples With Random Uniform Design T. Tony Cai Department of Statistics, Purdue University Lawrence D. Brown Department of Statistics, University of Pennsylvania Abstract We show
More informationExplicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables
Explicit Bounds for the Distribution Function of the Sum of Dependent Normally Distributed Random Variables Walter Schneider July 26, 20 Abstract In this paper an analytic expression is given for the bounds
More informationDistributionally Robust Discrete Optimization with Entropic Value-at-Risk
Distributionally Robust Discrete Optimization with Entropic Value-at-Risk Daniel Zhuoyu Long Department of SEEM, The Chinese University of Hong Kong, zylong@se.cuhk.edu.hk Jin Qi NUS Business School, National
More informationCharacterization of Upper Comonotonicity via Tail Convex Order
Characterization of Upper Comonotonicity via Tail Convex Order Hee Seok Nam a,, Qihe Tang a, Fan Yang b a Department of Statistics and Actuarial Science, University of Iowa, 241 Schaeffer Hall, Iowa City,
More informationEstimating Bivariate Tail: a copula based approach
Estimating Bivariate Tail: a copula based approach Elena Di Bernardino, Véronique Maume-Deschamps, Clémentine rieur To cite this version: Elena Di Bernardino, Véronique Maume-Deschamps, Clémentine rieur.
More informationThe newsvendor problem with convex risk
UNIVERSIDAD CARLOS III DE MADRID WORKING PAPERS Working Paper Business Economic Series WP. 16-06. December, 12 nd, 2016. ISSN 1989-8843 Instituto para el Desarrollo Empresarial Universidad Carlos III de
More informationBuered Probability of Exceedance: Mathematical Properties and Optimization
Buered Probability of Exceedance: Mathematical Properties and Optimization Alexander Mafusalov, Stan Uryasev RESEARCH REPORT 2014-1 Risk Management and Financial Engineering Lab Department of Industrial
More informationIntroduction to Dependence Modelling
Introduction to Dependence Modelling Carole Bernard Berlin, May 2015. 1 Outline Modeling Dependence Part 1: Introduction 1 General concepts on dependence. 2 in 2 or N 3 dimensions. 3 Minimizing the expectation
More informationOn Kusuoka representation of law invariant risk measures
On Kusuoka representation of law invariant risk measures Alexander Shapiro School of Industrial & Systems Engineering, Georgia Institute of Technology, 765 Ferst Drive, Atlanta, GA 3332. Abstract. In this
More information7) Important properties of functions: homogeneity, homotheticity, convexity and quasi-convexity
30C00300 Mathematical Methods for Economists (6 cr) 7) Important properties of functions: homogeneity, homotheticity, convexity and quasi-convexity Abolfazl Keshvari Ph.D. Aalto University School of Business
More informationModelling Dependence with Copulas and Applications to Risk Management. Filip Lindskog, RiskLab, ETH Zürich
Modelling Dependence with Copulas and Applications to Risk Management Filip Lindskog, RiskLab, ETH Zürich 02-07-2000 Home page: http://www.math.ethz.ch/ lindskog E-mail: lindskog@math.ethz.ch RiskLab:
More informationOPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION
OPTIMAL TRANSPORTATION PLANS AND CONVERGENCE IN DISTRIBUTION J.A. Cuesta-Albertos 1, C. Matrán 2 and A. Tuero-Díaz 1 1 Departamento de Matemáticas, Estadística y Computación. Universidad de Cantabria.
More informationA Note on the Swiss Solvency Test Risk Measure
A Note on the Swiss Solvency Test Risk Measure Damir Filipović and Nicolas Vogelpoth Vienna Institute of Finance Nordbergstrasse 15 A-1090 Vienna, Austria first version: 16 August 2006, this version: 16
More informationLinear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) 1.1 The Formal Denition of a Vector Space
Linear Algebra (part 1) : Vector Spaces (by Evan Dummit, 2017, v. 1.07) Contents 1 Vector Spaces 1 1.1 The Formal Denition of a Vector Space.................................. 1 1.2 Subspaces...................................................
More informationAN IMPORTANCE SAMPLING METHOD FOR PORTFOLIO CVaR ESTIMATION WITH GAUSSIAN COPULA MODELS
Proceedings of the 2010 Winter Simulation Conference B. Johansson, S. Jain, J. Montoya-Torres, J. Hugan, and E. Yücesan, eds. AN IMPORTANCE SAMPLING METHOD FOR PORTFOLIO CVaR ESTIMATION WITH GAUSSIAN COPULA
More informationPolitecnico di Torino. Porto Institutional Repository
Politecnico di Torino Porto Institutional Repository [Article] On preservation of ageing under minimum for dependent random lifetimes Original Citation: Pellerey F.; Zalzadeh S. (204). On preservation
More informationExpected Shortfall is not elicitable so what?
Expected Shortfall is not elicitable so what? Dirk Tasche Bank of England Prudential Regulation Authority 1 dirk.tasche@gmx.net Modern Risk Management of Insurance Firms Hannover, January 23, 2014 1 The
More informationMultivariate Operational Risk: Dependence Modelling with Lévy Copulas
Multivariate Operational Risk: Dependence Modelling with Lévy Copulas Klaus Böcker Claudia Klüppelberg Abstract Simultaneous modelling of operational risks occurring in different event type/business line
More informationApplications of axiomatic capital allocation and generalized weighted allocation
6 2010 11 ( ) Journal of East China Normal University (Natural Science) No. 6 Nov. 2010 Article ID: 1000-5641(2010)06-0146-10 Applications of axiomatic capital allocation and generalized weighted allocation
More informationOperations Research Letters. On a time consistency concept in risk averse multistage stochastic programming
Operations Research Letters 37 2009 143 147 Contents lists available at ScienceDirect Operations Research Letters journal homepage: www.elsevier.com/locate/orl On a time consistency concept in risk averse
More informationStochastic orders: a brief introduction and Bruno s contributions. Franco Pellerey
Stochastic orders: a brief introduction and Bruno s contributions. Franco Pellerey Stochastic orders (comparisons) Among his main interests in research activity A field where his contributions are still
More informationRisk Aggregation and Model Uncertainty
Risk Aggregation and Model Uncertainty Paul Embrechts RiskLab, Department of Mathematics, ETH Zurich Senior SFI Professor www.math.ethz.ch/ embrechts/ Joint work with A. Beleraj, G. Puccetti and L. Rüschendorf
More informationNon parametric estimation of Archimedean copulas and tail dependence. Paris, february 19, 2015.
Non parametric estimation of Archimedean copulas and tail dependence Elena Di Bernardino a and Didier Rullière b Paris, february 19, 2015. a CNAM, Paris, Département IMATH, b ISFA, Université Lyon 1, Laboratoire
More informationRisk Aggregation with Dependence Uncertainty
Risk Aggregation with Dependence Uncertainty Carole Bernard, Xiao Jiang and Ruodu Wang November 213 Abstract Risk aggregation with dependence uncertainty refers to the sum of individual risks with known
More informationTime Varying Hierarchical Archimedean Copulae (HALOC)
Time Varying Hierarchical Archimedean Copulae () Wolfgang Härdle Ostap Okhrin Yarema Okhrin Ladislaus von Bortkiewicz Chair of Statistics C.A.S.E. Center for Applied Statistics and Economics Humboldt-Universität
More informationHomework 1 Due: Wednesday, September 28, 2016
0-704 Information Processing and Learning Fall 06 Homework Due: Wednesday, September 8, 06 Notes: For positive integers k, [k] := {,..., k} denotes te set of te first k positive integers. Wen p and Y q
More informationModelling Dependent Credit Risks
Modelling Dependent Credit Risks Filip Lindskog, RiskLab, ETH Zürich 30 November 2000 Home page:http://www.math.ethz.ch/ lindskog E-mail:lindskog@math.ethz.ch RiskLab:http://www.risklab.ch Modelling Dependent
More informationOptimal Ridge Detection using Coverage Risk
Optimal Ridge Detection using Coverage Risk Yen-Chi Chen Department of Statistics Carnegie Mellon University yenchic@andrew.cmu.edu Shirley Ho Department of Physics Carnegie Mellon University shirleyh@andrew.cmu.edu
More informationEfficient estimation of a semiparametric dynamic copula model
Efficient estimation of a semiparametric dynamic copula model Christian Hafner Olga Reznikova Institute of Statistics Université catholique de Louvain Louvain-la-Neuve, Blgium 30 January 2009 Young Researchers
More informationCopulas and Measures of Dependence
1 Copulas and Measures of Dependence Uttara Naik-Nimbalkar December 28, 2014 Measures for determining the relationship between two variables: the Pearson s correlation coefficient, Kendalls tau and Spearmans
More informationStatistical Learning Theory
Statistical Learning Theory Fundamentals Miguel A. Veganzones Grupo Inteligencia Computacional Universidad del País Vasco (Grupo Inteligencia Vapnik Computacional Universidad del País Vasco) UPV/EHU 1
More informationExperience Rating in General Insurance by Credibility Estimation
Experience Rating in General Insurance by Credibility Estimation Xian Zhou Department of Applied Finance and Actuarial Studies Macquarie University, Sydney, Australia Abstract This work presents a new
More informationConsensus formation in the Deuant model
Consensus formation in the Deuant model Timo Hirscher Chalmers University of Technology Seminarium i matematisk statistik, KTH October 29, 2014 Outline The seminar is structured as follows: Introduction:
More informationRuin, Operational Risk and How Fast Stochastic Processes Mix
Ruin, Operational Risk and How Fast Stochastic Processes Mix Paul Embrechts ETH Zürich Based on joint work with: - Roger Kaufmann (ETH Zürich) - Gennady Samorodnitsky (Cornell University) Basel Committee
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 informationCopula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011
Copula Regression RAHUL A. PARSA DRAKE UNIVERSITY & STUART A. KLUGMAN SOCIETY OF ACTUARIES CASUALTY ACTUARIAL SOCIETY MAY 18,2011 Outline Ordinary Least Squares (OLS) Regression Generalized Linear Models
More informationTranslation invariant and positive homogeneous risk measures and portfolio management. Zinoviy Landsman Department of Statistics, University of Haifa
ranslation invariant and positive homogeneous risk measures and portfolio management 1 Zinoviy Landsman Department of Statistics, University of Haifa -It is wide-spread opinion the use of any positive
More informationNotes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed
18.466 Notes, March 4, 2013, R. Dudley Maximum likelihood estimation: actual or supposed 1. MLEs in exponential families Let f(x,θ) for x X and θ Θ be a likelihood function, that is, for present purposes,
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 informationDISCUSSION PAPER 2016/43
I N S T I T U T D E S T A T I S T I Q U E B I O S T A T I S T I Q U E E T S C I E N C E S A C T U A R I E L L E S ( I S B A ) DISCUSSION PAPER 2016/43 Bounds on Kendall s Tau for Zero-Inflated Continuous
More informationWhen is a copula constant? A test for changing relationships
When is a copula constant? A test for changing relationships Fabio Busetti and Andrew Harvey Bank of Italy and University of Cambridge November 2007 usetti and Harvey (Bank of Italy and University of Cambridge)
More information