MULTIVARIATE EXTREME VALUE ANALYSIS UNDER A

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1 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 Value Analysis December / 57

2 MULTIVARIATE FRAMEWORK AND DIRECTIONS (A) Classical direction u (B) Auxiliary direction u Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

3 MULTIVARIATE FRAMEWORK AND DIRECTIONS Same view, different perspectives Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

4 DRAWBACKS IN THE MULTIVARIATE SETTING The lack of a total order in high dimensions. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

5 DRAWBACKS IN THE MULTIVARIATE SETTING The lack of a total order in high dimensions. The dependence among the variables belonging to a system. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

6 DRAWBACKS IN THE MULTIVARIATE SETTING The lack of a total order in high dimensions. The dependence among the variables belonging to a system. There are many interesting directions to analyze the data. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

7 DRAWBACKS IN THE MULTIVARIATE SETTING The lack of a total order in high dimensions. The dependence among the variables belonging to a system. There are many interesting directions to analyze the data. The costs of computing in high dimensions. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

8 OBJECTIVES Introduce a directional multivariate setting for extreme value analysis Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

9 OBJECTIVES Introduce a directional multivariate setting for extreme value analysis 1 Considering the dependence among the variables. 2 Giving the possibility of analyzing the variables considering external information, manager preferences or intrinsic system characteristics. 3 Improving the interpretation of the analysis of extremes. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

10 OBJECTIVES Introduce a directional multivariate setting for extreme value analysis 1 Considering the dependence among the variables. 2 Giving the possibility of analyzing the variables considering external information, manager preferences or intrinsic system characteristics. 3 Improving the interpretation of the analysis of extremes. 4 Providing a non-parametric procedure for estimation to compute the analysis in high dimensions. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

11 OUTLINE 1 DIRECTIONAL BASIC CONCEPTS 2 NON-PARAMETRIC ESTIMATION 3 APPLICATIONS IN EXTREME VALUE ANALYSIS Environmental Application in Financial Risk Measures Current Research about Estimation 4 CONCLUSIONS AND FUTURE RESEARCH Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

12 Directional Basic Concepts C u x Oriented Orthant. DEFINITION Given x, u R n and u = 1, the orthant with vertex x and direction u is: C u x = {z R n R u (z x) 0}, where e = 1 n (1,..., 1) and R u is a matrix such that R u u = e. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

13 Directional Basic Concepts EXAMPLES OF ORIENTED ORTHANTS (A) Orthant in direction u = (0, 1) (B) Orthant in direction u = e Examples of oriented orthants in R 2 Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

14 Directional Basic Concepts Q X (α, u) Directional Multivariate Quantile DEFINITION Given u R n, u = 1 and a random vector X with distribution probability P, the α-quantile curve in direction u is defined as: Q X (α, u) := {x R n : P[C u x ] α}, where mans the boundary and 0 α 1 Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

15 Directional Basic Concepts U X (α, u) Directional Multivariate Upper Level-Set L X (α, u) Directional Multivariate Lower Level-Set DEFINITION Those sets are defined by: U X (α, u) := {x R n : P[C u x] < α}, L X (α, u) := {x R n : P[C u x ] > α}. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

16 Directional Basic Concepts D IRECTIONAL M ULTIVARIATE L EVEL -S ETS u U= 1 1, ,,,,,, (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal CLASSICAL DIRECTIONS Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

17 Directional Basic Concepts D IRECTIONAL M ULTIVARIATE L EVEL -S ETS u U = {(1, 0), (0, 1), ( 1, 0), (0, 1)} (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal CANONICAL DIRECTIONS Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

18 Non-Parametric Estimation NON-PARAMETRIC ESTIMATION X m := {x 1,, x m }, the sample data of the random vector X, P Xm [ ] is the empirical probability law of X m, ] } ˆQ h X m (α, u) := {x j : P Xm [C u xj α h the sample quantile curve with a slack h, avoiding an empty set of estimated quantiles. ] } ÛX h m (α, u) := {x j : P Xm [C u xj < α h the sample upperα level set with a slack h, ] } ˆL h X m (α, u) := {x j : P Xm [C u xj > α+h the sample lowe α level set with a slack h. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

19 Non-Parametric Estimation NON-PARAMETRIC ESTIMATION Input: u, α, h and the multivariate sample X m. for i = 1 to m P i = P Xm [ C u xi ], If P i α h x i ˆQ h X m (α, u), end If P i < α h x i Û h X m (α, u), end If P i > α+h x i ˆL h X m (α, u), end Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

20 Non-Parametric Estimation EXECUTION TIME Time in Seconds Dim\ Size In an Intel core i7 (3,4 GH) computer with 32 Gb RAM. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

21 Environmental EXTREMES THROUGH COPULAS (REVIEW) COPULA Roughly speaking, a n-copula C is a particular type of distribution with domain in the unit hyper-cube and uniform margins. Sklar s Theorem Let F be a n-dimensional distribution function with marginals F 1,, F n. Then there exists a n-copula C such that for all x R n, F(x 1,, x n ) = C(F 1 (x 1 ),, F n (x n )). If F 1,..., F n are continuous, then C is unique. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

22 Environmental {v [0, 1] n : C(v) = α} Copula Quantile Procedure Let C be the n copula of X and F i, i = 1,..., d its margins. Then for 0 < α < 1 the corresponding α quantile hyper-curve, upper and lower level sets are defined as: {x R n such that x i = F 1 X i (v i ); i = 1,..., n; v [0, 1] n : C(v) = α}, {x R n such that x i = F 1 X i (v i ); i = 1,..., n; v [0, 1] n : C(v) < α}, {x R n such that x i = F 1 X i (v i ); i = 1,..., n; v [0, 1] n : C(v) > α}. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

23 Environmental EXTREMES THROUGH COPULAS (REVIEW) Handle Copula Families & Marginal Distributions (G.E.V, etc) Sklar s Theorem Closed Multivariate Quantile Expressions Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

24 Environmental DRAWBACKS WORKING WITH COPULAS This approach is hard to manipulate because: The parametric nature generates complications and/or restrictions in high dimension, even using nested copula techniques. The models are quite rigid. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

25 Environmental DIRECTIONAL PROPOSAL OF EXTREMES IDENTIFICATION Directional Multivariate Level-Sets Directional Extremes Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

26 Environmental LOOKING THE DATA IN ANOTHER DIRECTION Why is useful a directional approach in environmental engineering? To solve this question we simulate data from the model in Salvadori et al. (2011), which refers to data of maximum annual flood peaks Q, volumes V and water levels L in the Ceppo Morelli dam, Italy. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

27 Environmental CEPPO MORELLI DAM EXAMPLE Q 40 V L Original dataset of the Ceppo Morelli dam Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

28 Environmental CEPPO MORELLI DAM EXAMPLE Copula model for the Ceppo Morelli dam Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

29 Environmental CEPPO MORELLI DAM EXAMPLE Simulated Sample from the model in Salvadori et al. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

30 Environmental CEPPO MORELLI DAM EXAMPLE Extremes with the non-parametric approach, in the classic direction e for α = 1% Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

31 Environmental LOOKING THE DATA IN ANOTHER DIRECTION Extremes with the non-parametric approach, in the first PCA direction for α = 1% Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

32 Environmental DETERMINISTIC PHYSICAL DAM-LEVEL BEHAVIOR Final level from the simulated occurrences of Q, V, L Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

33 Environmental MEASURES OF FALSE-POSITIVES AND TRUE-POSITIVES IN CLASSIC AND DIRECTIONAL APPROACHES FALSE-POSITIVES Classic Direction PCA Direction 90% 35% TRUE-POSITIVES Classic Direction PCA Direction 100% 100% Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

34 Environmental DIRECTIONAL MULTIVARIATE QUANTILES & COPULA APPROACH THEOREM Let u be fix, then the directional quantiles of a random vector X with regularity conditions are equivalent to those obtained by the copula procedure on the random vector R u X, where R u is the rotation matrix in the orthant definition. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

35 Environmental DIRECTIONAL MULTIVARIATE QUANTILES & COPULA APPROACH SKETCH OF THE PROOF Given α Q X (α, e) F 1 X (α)&q X(α, e) F 1 X (α) Under regularity conditions, Orthogonal Quasi-Invariance & Sklar s Theorem over R ±u X Directional approach Copula approach. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

36 Financial APPLICATION IN FINANCIAL RISK MEASURES Let X be a random variable representing loss, F its distribution function and 0 α 1. Then, VaR α (X) := inf{x R F(x) α}. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

37 Financial APPLICATION IN FINANCIAL RISK MEASURES Let X be a random variable representing loss, F its distribution function and 0 α 1. Then, VaR α (X) := inf{x R F(x) α}. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

38 Financial APPLICATION IN FINANCIAL RISK MEASURES Let X be a random variable representing loss, F its distribution function and 0 α 1. Then, VaR α (X) := inf{x R F(x) α}. α VaR α (X) Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

39 VALUE AT RISK (VaR) Financial The VaR has became in a benchmark for risk management. The VaR has been criticized by Artzner et al. (1999) since it does not encourage diversification. But defended by Heyde et al. (2009) for its robustness and recently by Daníelsson et al. (2013) for its tail subadditivity. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

40 MULTIVARIATE DRAWBACKS OF VaR Financial There is not a unique definition of a multivariate quantile. There are a lot of assets in a portfolio. (High Dimension) There is dependence among them. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

41 Financial REVIEW ON MULTIVARIATE VALUE AT RISK An initial idea to study risk measures related to portfolios X = (X 1,...,X n ), is to consider a function f : R n R and then: The VaR of the joint portfolio is the univariate-one associated to f(x). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

42 Financial REVIEW ON MULTIVARIATE VALUE AT RISK An initial idea to study risk measures related to portfolios X = (X 1,...,X n ), is to consider a function f : R n R and then: The VaR of the joint portfolio is the univariate-one associated to f(x). In Burgert and Rüschendorf (2006), f(x) = n i=1 X i or f(x) = max i n X i. Output: A NUMBER Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

43 Financial REVIEW ON MULTIVARIATE VALUE AT RISK Embrechts and Puccetti (2006) introduced a multivariate approach of the Value at Risk, Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

44 Financial REVIEW ON MULTIVARIATE VALUE AT RISK Embrechts and Puccetti (2006) introduced a multivariate approach of the Value at Risk, Multivariate lower-orthant Value at Risk VaR α (X) := {x R n F X (x) α}. Multivariate upper-orthant Value at Risk VaR α (X) := {x R n F X (x) 1 α}. Output: A SURFACE ON R n Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

45 Financial REVIEW ON MULTIVARIATE VALUE AT RISK Cousin and Di Bernardino (2013) introduced a multivariate risk measure related to the measure introduced by Embrechts and Puccetti (2006). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

46 Financial REVIEW ON MULTIVARIATE VALUE AT RISK Cousin and Di Bernardino (2013) introduced a multivariate risk measure related to the measure introduced by Embrechts and Puccetti (2006). Multivariate lower-orthant Value at Risk VaR α (X) := E[X F X (x) = α]. Multivariate upper-orthant Value at Risk VaR α (X) := E[X F X (x) = 1 α]. Output: A POINT IN R n Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

47 Financial DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR) DIRECTIONAL MVAR Let X be a random vector satisfying "the regularity conditions", then the Value at Risk of X in direction u and confidence parameter α is defined as ( VaR u α(x) = Q X (α, u) ) {λu+e[x]}, where λ R and 0 α 1. Output: A POINT IN R n Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

48 Financial DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR) (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal VaR e 0.7 (X) Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

49 Financial DIRECTIONAL MULTIVARIATE VALUE AT RISK (MVaR) (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal VaR e 0.3 (X) Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

50 Financial MVAR PROPERTIES Non-Negative Loading: If λ > 0, E[X] u VaR u α(x), where the order is given by PREORDER (LANIADO ET AL. (2010)) x is said to be less than y if: x u y C u x Cu y R u x R u y. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

51 Financial MVAR PROPERTIES Quasi-Odd Measure: VaR u α( X) = VaR u α (X). Positive Homogeneity and Translation Invariance: Given c R + and b R n, then VaR u α(cx+b) = cvar u α(x)+b. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

52 Financial MVAR PROPERTIES Orthogonal Quasi-Invariance: Let w and Q be an unit vector and a particular orthogonal matrix obtained by a QR decomposition such that Qu = w. Then, VaR w α (QX) = QVaRu α (X). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

53 Financial MVAR PROPERTIES Consistency: Let X and Y be random vectors such that E[Y] = cu+e[x], for c > 0 and X Eu Y. Then: VaR u α (X) u VaR u α (Y), where the stochastic order is defined by STOCHASTIC EXTREMALITY ORDER (LANIADO ET AL. (2012)) Let X and Y be two random vectors in R n, X Eu Y P[R u(x z) 0] P[R u(y z) 0] P X [C u z] P Y [C u z], for all z in R n. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

54 Financial MVAR PROPERTIES Non-Excessive Loading: For all α (0, 1) and u B(0), VaR u α(x) u R u sup{r u X(ω)}. ω Ω Subadditivity in the Tail Region: Let X and Y be random vectors, with the same mean µ and let (R u X, R u Y) be a regularly varying random vector. Then, VaR u α (X+Y) u VaR u α (X)+VaRu α (Y). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

55 Financial LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR RESULT Let X be a random vector and u a direction. Then for all 0 α 1, VaR u α (X) u VaR u 1 α (X). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

56 Financial LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR Then, analogously as Embrechts and Puccetti (2006) and Cousin and Di Bernardino (2013), we can define: Lower Multivariate VaR in the direction u as VaR u α(x), Upper Multivariate VaR in the direction u as VaR u 1 α (X). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

57 Financial LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR Lower Multivariate VaR = VaR e 0.3 (X) and Upper Multivariate VaR = VaR e 0.7 (X) Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

58 Financial LOWER AND UPPER VERSIONS OF DIRECTIONAL MVaR Lower Multivariate VaR = VaR ( 1, 2 ) (X) and Upper Multivariate VaR = VaR ( 1, 2 ) (X) Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

59 Financial RELATION BETWEEN THE MARGINAL VaR AND THE MVaR RESULT Let X be a random vector with survival function F quasi-concave. Then, for all α (0, 1): VaR 1 α (X i ) [VaR e α (X)] i, for all i = 1,..., n. Moreover, if its distribution function F is quasi-concave, then, for all α (0, 1), [ VaR e 1 α (X)] i VaR 1 α(x i ), for all i = 1,..., n. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

60 Financial RELATION BETWEEN THE MARGINAL VaR AND THE MVaR RESULT Let X be a random vector and u a direction. If the survival function of R u X is quasi-concave. Then, for all 0 α 1, VaR 1 α ([R u X] i ) [R u VaR u α (X)] i, for all i = 1,..., n. And if R u X has a quasi-concavity cumulative distribution, we have that [ Ru VaR u 1 α (X)] i VaR 1 α([r u X] i ), for all i = 1,..., n. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

61 Financial ROBUSTNESS We analyze the behavior of the MVaR when a sample is contaminated with different types of outliers. We use as a benchmark the measurement given by the multivariate VaR in Cousin and Di Bernardino (2013). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

62 Financial ROBUSTNESS We simulate 5000 observations of the following random vector: { X ω d X 1 with probability p = 1 ω, = X 2 with probability p = ω, d d where X 1 = N1 (µ 1,Σ 1 ), X 2 = N2 (µ 1 + µ,σ 1 + Σ ) and 0 ω 1. Specifically: ( ) µ 1 = [50, 50] , Σ 1 = Varying only the mean. Contaminating 2. Varying only the variances. 3. Varying all the parameters. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

63 Financial ROBUSTNESS To evaluate the impact of the contamination, we use: PV ω = Measure(Xω ) Measure(X 0 ) 2 Measure(X 0 ) 2, where Measure(X 0 ) is the sample with ω = 0% and Measure(X ω ) is the sample with level of contamination ω%, (ω = 1% 10%). Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

64 Financial ROBUSTNESS 1. Varying only the mean, µ 0, Σ = e VaR 0.1 (X) 0.06 e VaR (X) CB VaR 0.05 CB VaR ω ω (A) µ = (20, 20) (B) µ = (0, 50) Mean of PV ω Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

65 Financial ROBUSTNESS 2. Varying only the variances, µ = 0, Σ = [ ] 4.5 0, e VaR 0.1 (X) CB VaR ω Mean of PV ω Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

66 Financial ROBUSTNESS 3. Varying all the parameters, µ 0, Σ = [ ] , x 10-3 e VaR (X) x 10-3 e VaR (X) CB VaR 3.5 CB VaR ω ω (A) µ = (20, 20) (B) µ = (0, 50) Mean of PV ω Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

67 Estimation Research CURRENT RESEARCH ABOUT ESTIMATION RESULT Let X be a multivariate regularly varying random vector with tail index β. Then for all orthogonal transformation Q the random vector QX is regularly varying with tail index β. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

68 Estimation Research CURRENT RESEARCH ABOUT ESTIMATION Using as base the theory in De Haan and Huang (1995) for a bivariate estimation of quantile curves, we want to: Study the extension to higher dimensions. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

69 Estimation Research CURRENT RESEARCH ABOUT ESTIMATION Using as base the theory in De Haan and Huang (1995) for a bivariate estimation of quantile curves, we want to: Study the extension to higher dimensions. Link the directional notion into the theory by using previous result. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

70 Estimation Research CURRENT RESEARCH ABOUT ESTIMATION Using as base the theory in De Haan and Huang (1995) for a bivariate estimation of quantile curves, we want to: Study the extension to higher dimensions. Link the directional notion into the theory by using previous result. Study convergence and consistency, theoretically and practically. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

71 Estimation Research CURRENT RESEARCH ABOUT ESTIMATION Using as base the theory in De Haan and Huang (1995) for a bivariate estimation of quantile curves, we want to: Study the extension to higher dimensions. Link the directional notion into the theory by using previous result. Study convergence and consistency, theoretically and practically. Make comparisons between the previous non-parametric approach and the resultant by this research. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

72 Estimation Research MULTIVARIATE HEAVY TAILED EXAMPLE (A) n = 500, α = 1 n (B) n = 5000, α = 1 n Bivariate t distribution with ν = 3 Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

73 Estimation Research ESTIMATION IN THE FIRST PCA DIRECTION (A) n = 500, α = 1 n (B) n = 5000, α = 1 n Bivariate t distribution with ν = 3 Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

74 Conclusions CONCLUSIONS AND FUTURE RESEARCH We have introduced an extension of the multivariate extreme value analysis by introducing a directional notion. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

75 Conclusions CONCLUSIONS AND FUTURE RESEARCH We have introduced an extension of the multivariate extreme value analysis by introducing a directional notion. The directional approach allows to consider external intrinsic information of a system or management preferences in the analysis. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

76 Conclusions CONCLUSIONS AND FUTURE RESEARCH We have introduced an extension of the multivariate extreme value analysis by introducing a directional notion. The directional approach allows to consider external intrinsic information of a system or management preferences in the analysis. We provide arguments of the needing of these directional approach in practice. As well as, we present theoretical properties and results of this directional extension in the developed applications. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

77 Conclusions CONCLUSIONS AND FUTURE RESEARCH We have introduced an extension of the multivariate extreme value analysis by introducing a directional notion. The directional approach allows to consider external intrinsic information of a system or management preferences in the analysis. We provide arguments of the needing of these directional approach in practice. As well as, we present theoretical properties and results of this directional extension in the developed applications. Two important aspects have been studied. Asymptotic convergence and robustness in more general cases. Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

78 Thanks Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

79 BIBLIOGRAPHY Torres R., Lillo R.E., and Laniado H., A directional multivariate Value at Risk. Insurance Math. Econom., 2015 Accepted. Nelsen R.B., An Introduction to copulas, (2nd edn) Springer-Verlag: New York, Shaked M. and Shanthikumar J., Stochastic Orders and their. Associated Press, De Haan L., and Huang X. (1995) Large Quantile Estimation in a Multivariate Setting, Journal of Multivariate Analysis, 53, Arbia, G., Bivariate value at risk. Statistica LXII, , Artzner P., Delbae, F., Heath J. Eber D., Coherent measures of risk. Mathematical Finance, 3, , Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

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81 BIBLIOGRAPHY Daníelsson J. et al., Fat tails, VaR and subadditivity. Journal of econometrics 172-2, , Embrechts P. and Puccetti G., Bounds for functions of multivariate risks. Journal of Multivariate Analysis 97, , Fernández-Ponce J. and Suárez-Llorens A., Central regions for bivariate distributions. Austrian Journal of Statistics 31, , Fraiman R. and Pateiro-López B., Quantiles for finite and infinite dimensional data. Journal Multivariate Analysis 108, 1-14, Hallin M., Paindaveine D. and Šiman M., Multivariate quantiles and multiple-output regression quantiles: from L1 optimization to halfspace depth. Annals of Statistics 38, , Heyde C., Kou S. and Peng X., What is a good external risk measure: bridging the gaps between robustness, subadditivity, and insurance risk measure. (Preprint) Jessen A. and Mikosh T., Regularly varying functions. Publications de l institute mathématique 79, Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

82 BIBLIOGRAPHY Kong L. and Mizera I., Quantile tomography: Using quantiles with multivariate data. ArXiv preprint arxiv: v1, Laniado H., Lillo R. and Romo J., Extremality in Multivariate Statistics. Ph.D. Thesis, Universidad Carlos III de Madrid, 2012 Laniado H., Lillo R. and Romo J., Multivariate extremality measure. Working Paper, Statistics and Econometrics Series 08, Universidad Carlos III de Madrid, Nappo G. and Spizzichino F., Kendall distributions and level sets in bivariate exchangeable survival models. Information Sciences 179, , Rachev S., Ortobelli S., Stoyanov S., Fabozzi F., Desirable Properties of an Ideal Risk Measure in Portfolio Theory. International Journal of Theoretical and Applied Finance 11(1), 19-54, Serfling R., Quantile functions for multivariate analysis: approaches and applications. Statistica Neerlandica 56, , Zuo Y. and Serfling R., General notions of statistical depth function. Annals of Statistics 28(2), , Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57

83 Thanks Torres Díaz, Raúl A. Directional Extreme Value Analysis December / 57