QR Oriented Orthant. C u x (Torres et al. 2015) D EFINITION Given - - PowerPoint PPT Presentation

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A DIRECTIONAL NOTION OF MULTIVARIATE EXTREME VALUE ANALYSIS Ral A. T ORRES D AZ Department of Statistics and Operation Research Universidad de Valladolid PhD. seminar in Mathematical Engineering, Universidad EAFIT joint work with: Elena Di

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A DIRECTIONAL NOTION OF MULTIVARIATE

EXTREME VALUE ANALYSIS

Raúl A. TORRES DÍAZ Department of Statistics and Operation Research Universidad de Valladolid

  • PhD. seminar in Mathematical Engineering, Universidad

EAFIT

joint work with: Elena Di Bernardino, CNAM Paris, Henry Laniado, EAFIT Medellín, Rosa E. Lillo, UC3M Madrid. May 2018

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 1 / 46

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INTRODUCTION AND MOTIVATION

1 Extreme Value Theory concerns to the limit behavior of the

sample extremes, (max or min in the univariate case) BUT

2 Multivariate analysis is mandatory because Extremes are gen-

erated by many variables acting jointly with different relation- ships

Asymptotic Independence & Asymptotic Dependence, (Pairs relations). Correlations, (Overall relation).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 2 / 46

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DIRECTIONAL PERSPECTIVE

1 Look at the data with different perspectives to improve the

identification and visualization of extremes BUT

There are infinite directions to analyze the data, how to select an interesting one?.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 3 / 46

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STATISTICAL TOOLS FOR EXTREMES

1 The classical tool for Extremes identification is the α−quantile

concept BUT

There is a lack of a total order in Rd. Conditioned to the α−level, there are 2 approaches of estimation, In − Sample and Out − Sample.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 4 / 46

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IMPORTANCE OF α IN THE ESTIMATION

In-Sample Vs. Out-Sample α > 1

n

Some Observations Available α ≤ 1

n

Non-Observations Available Standard Estimation Procedures Multivariate Extreme Value Theory

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 5 / 46

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OUTLINE

1 DIRECTIONAL NOTIONS 2 NON-PARAMETRIC OUT-SAMPLE ESTIMATION 3 REAL CASE STUDY 4 CONCLUSIONS

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 6 / 46

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Cu

x

≡ QR Oriented Orthant. (Torres et al. 2015)

DEFINITION Given x ∈ Rd and u ∈

  • z ∈ Rd : ||u|| = 1, ui = 0 for all i = 1, ..., d
  • , the

QR oriented orthant with vertex x and direction u is: Cu

x = {z ∈ Rd|Ru(z − x) ≥ 0},

where e =

1 √ d(1, ..., 1)′ and Ru is an unique orthogonal matrix obtained

by a QR decomposition, such that Ruu = e.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 7 / 46

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EXAMPLES OF ORIENTED ORTHANTS

Examples of oriented orthants in R2

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 8 / 46

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QX(α, u) ≡ Directional Multivariate Quantile (Laniado et al. 2012)

DEFINITION Given u ∈ Rd, ||u|| = 1 and a random vector X with distribution proba- bility P, the α-quantile curve in direction u is defined as: QX(α, u) := ∂{x ∈ Rd : P(C−u

x ) ≥ 1 − α},

where ∂ means the boundary and 0 ≤ α ≤ 1

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 9 / 46

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UX(α, u) ≡ Directional Multivariate Upper Level-Set LX(α, u) ≡ Directional Multivariate Lower Level-Set

DEFINITION Those sets are defined by: UX(α, u) := {x ∈ Rd : P

  • C−u

x

  • > 1 − α},

LX(α, u) := {x ∈ Rd : P

  • C−u

x

  • < 1 − α}.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 10 / 46

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DIRECTIONAL MULTIVARIATE LEVEL-SETS

u ∈ U =

  • − 1

√ 2 , − 1 √ 2

  • ,
  • − 1

√ 2 , 1 √ 2

  • ,

1 √ 2 , − 1 √ 2

  • ,

1 √ 2 , 1 √ 2

  • (A) Bivariate Uniform (B) Bivariate Exponential (C) Bivariate Normal

CLASSICAL DIRECTIONS

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 11 / 46

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DIRECTIONAL MULTIVARIATE LEVEL-SETS

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 Multivariate EVT May 2018 11 / 46

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DIRECTIONAL MULTIVARIATE QUANTILE PROPERTIES

Quasi-Odd Property: Q−X(α, u) = −QX(α, −u). Positive Homogeneity and Translation Invariance: QcX+b(α, u) = cQX(α, u) + b, for all c ∈ R+ and b ∈ Rd. Orthogonal Quasi-Invariance: Let w and u be two unit vectors. Then, an orthogonal matrix Q exists, such that, Qu = w and QX(α, u) = Q′QQX(α, w).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 12 / 46

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OUTLINE

1 DIRECTIONAL NOTIONS 2 NON-PARAMETRIC OUT-SAMPLE ESTIMATION 3 REAL CASE STUDY 4 CONCLUSIONS

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 13 / 46

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REVIEW ON MULTIVARIATE OUT-SAMPLE ESTIMATION

Optimization processes estimation (e.g., Girard and Stupfler (2015)) Estimation of level curves of joint density functions or depth functions (e.g., Cai et al. (2011), Einmahl et al. (2013), He and Einmahl (2017)) Estimation of level curve of either joint distribution or survival functions (e.g. De Haan and Huang (1995))

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 14 / 46

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NECESSARY BACKGROUND

ASSUMPTION 1, A1. The random vector X must be absolutely continuous. ASSUMPTION 2, A2. Given u, Ru X possesses positive upper-end points of the marginal distributions.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46

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NECESSARY BACKGROUND

DEFINITION X has first order multivariate regular variation with tail index γ, denoted by RV1/γ, if there exists a real-value function φ(t) > 0 that is regularly varying at infinity with exponent 1/γ and a non-zero measure µ(·) on the Borel σ−field ¯ Rd\{0} such that for every Borel set B, t P[(φ(t))−1X ∈ B]

v

→ µ(B), where

v

→ means vague convergence and t → ∞. ASSUMPTION 3, A3. X has 1st order multivariate regular variation with index γ.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46

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NECESSARY BACKGROUND

DEFINITION X has second order multivariate regular variation if there exist functions φ(·) ∈ RV1/γ and Λ(t) → 0, such that |Λ| ∈ RVπ, π ≤ 0; satisfying for all relatively compact rectangles B ∈ ¯ Rd\{0}, tP

  • (φ(t))−1X ∈ B
  • − µ(B)

Λ(φ(t)) → ψ(B), where ψ(B) is finite and not identically zero. ASSUMPTION 4, A4. X has 2nd order multivariate regular variation with indexes (γ, π).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 15 / 46

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DIRECTIONAL RESULTS

RESULT If X has 1st(2nd) order multivariate regular variation. Then, QX has 1st(2nd) order multivariate regular variation, for any orthogo- nal transformation Q. COROLLARY If X has 1st(2nd) order multivariate regular variation. Then the marginals of QX has 1st(2nd) order multivariate regular variation.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 16 / 46

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CHARACTERIZATION OF QX(α, u) AT HIGH LEVELS

Key tools A1-A3 and QX(α, u) = R′

uQRuX(α, e)

QRuX(α, e) ≈ QRuX(α, e, θ) where θ belongs to the unit d−dimensional ball and 0 ≤ θj ≤ 1 ASYMPTOTIC CHARACTERIZATION OF QX(α, u, θ) xu(α, θ) = (xu,1(α, θ), ..., xu,d(α, θ)) =

  • au,j(t)(ρu(θ)θj/tα)γ − 1

γ + bu,j(t); j = 1, . . . , d,

  • ,

QX(α, u, θ) = R′

uQRuX(α, e, θ).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 17 / 46

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HOW WAS THE CHARACTERIZATION POSSIBLE?

1 Pre-rotation of X through the orthogonal matrix Ru introduced in

the QR orthant definition. All the elements with subindex u refer to expressions related to RuX. For instance, Fu denotes the joint distribution and its marginals are Fu,j, j = 1, ..., d.

2 Key asymptotic result from the Multivariate Extreme Value Theory.

DISTRIBUTION OF CONVERGENCE OF THE SAMPLE MAXIMA There exist two sequences au(⌊t⌋), bu(⌊t⌋) such that, lim

t→∞ t (1 − Fu (au,j(⌊t⌋) xu,j + bu,j(⌊t⌋); j = 1, . . . , d)) = − ln (Gu(xu)) ,

where ⌊·⌋ is the floor function.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46

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HOW WAS THE CHARACTERIZATION POSSIBLE?

3 Marginal high level estimations through extreme value analysis.

HIGH LEVEL QUANTILES OF Fu,j, j = 1, ..., d xu,j(α) ≈ au,j(t)(1/tα)γ − 1 γ + bu,j(t).

4 Polar representation in Rd.

POLAR PARAMETRIZATION In Rd, any point x can be written in polar coordinates as x = ||x|| (x/||x||) = ρ(θ) θ, where ρ(θ) ∈ R+ and θ belonging to the unit d−dimensional ball.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46

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HOW WAS THE CHARACTERIZATION POSSIBLE?

5 Heuristic link between marginal quantiles of RuX and QX(α, u) at

high levels. BIVARIATE IDEAS FOUND IN DE HAAN AND HUANG (1995) The set of solutions to 1 − F(x1, x2) = α for a bivariate distribution F can be parametrized in polar coordinates as (ρ(θ)cos(θ), ρ(θ)sin(θ)), where ρ(θ) is a solution of the following equations, x1(α, θ) = a1(t)(ρ(θ)cos(θ)/tα)γ1 − 1 γ1 + b1(t), x2(α, θ) = a2(t)(ρ(θ)sin(θ)/tα)γ2 − 1 γ2 + b2(t).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46

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HOW WAS THE CHARACTERIZATION POSSIBLE?

Dimension 2 Dimension 3

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46

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HOW WAS THE CHARACTERIZATION POSSIBLE?

6 Deduction of the solution for the scalar function ρu.

SOLUTION IN TERMS OF THE TAIL FUNCTION Given that,

α = 1 − Fu(xu(α, θ)) ≈ t−1

  • − ln
  • Gu

xu,j(α, θ) − bu,j(t) au,j(t) ; j = 1, . . . , d

  • .

Then, ρu(θ) := − ln

  • Gu

θγ

j − 1

γ ; j = 1, . . . , d

  • .

Here − ln(Gu(z) is the tail function of the multivariate extreme value distribution where the distributions of the multivariate sample maxima converge.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 18 / 46

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ESTIMATION OF QX(α, u) AT HIGH LEVELS

PROPOSED ESTIMATOR FOR THE ELEMENTS IN QRuX(α, e, θ)

ˆ xu,j(α, θ, n/k) := ˆ au,j(n/k)     

  • k ˆ

ρu(θ) n α θj

ˆ

γ

− 1 ˆ γ      + ˆ bu,j(n/k), for all j = 1, . . . , d.

FINAL QX(α, u, θ) ESTIMATOR ˆ QX(α, u, θ, n/k) = R′

u ˆ

QRuX(α, e, θ, n/k).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 19 / 46

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ESTIMATION OF QX(α, u) AT HIGH LEVELS

PROPOSED ESTIMATOR FOR THE ELEMENTS IN QRuX(α, e, θ)

ˆ xu,j(α, θ, n/ k ) := ˆ au,j(n/ k )     

  • k

ˆ ρu(θ) n α

θj ˆ

γ

− 1 ˆ γ      + ˆ bu,j(n/ k ), for all j = 1, .., d

FINAL QX(α, u, θ) ESTIMATOR ˆ QX(α, u, θ, n/ k ) = R′

u ˆ

QRuX(α, e, θ, n/ k ).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 20 / 46

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ELEMENTS TO ESTIMATE QX(α, u) AT HIGH LEVELS

Marginal tail indexes ˆ γ (Dekkers et al. (1989)), ˆ γ := M(1)

k,j + 1 − 1

2

  • 1 −
  • M(1)

k,j

2 /M(2)

k,j

−1 , where, M(r)

k,j := 1

k

k−1

  • j=0

{ln([(RuX)j]n−i:n) − ln([(RuX)j]n−k:n)}r , r = 1, 2.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 21 / 46

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ELEMENTS TO ESTIMATE QX(α, u) AT HIGH LEVELS

The sequences ˆ au(n/k) and ˆ bu(n/k) (De Haan and Huang (1995)), ˆ au,j(n/k) := [(RuX)j]n−k:nM(1)

k,j max(1, 1 − ˆ

γ), ˆ bu,j(n/k) := [(RuX)j]n−k:n.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 21 / 46

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ELEMENTS TO ESTIMATE QX(α, u) AT HIGH LEVELS

The scalar function ˆ ρu(θ). Given that,

− ln (Gu (xu)) ≈ n k

  • 1 − Fu

xu,j − bu,j(n/k) au,j(n/k) ; j = 1, . . . , d

  • Then,

ˆ ρu(θ) := 1 k

n

  • i=1

1

d

  • j=1[(RuXi)j > ˆ

au,j(n/k)xu,j+ˆ bu,j(n/k)]

  • Torres Díaz, Raúl A.

Directional Multivariate EVT May 2018 21 / 46

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BOOTSTRAP-BASED METHOD TO SELECT k(n)

The multivariate tuning parameter selection based on the univari- ate procedure by Danielsson et al. (2001). ORTHANT ORDER IN (TORRES ET AL. 2015) x is said to be less than y in direction u if: x u y ≡ Cu

x ⊇ Cu y

≡ Rux ≤ Ruy.

STEP 1. Pre-rotate the sample to generate {Rux1, . . . , Ruxn} and center

that with respect to its mean.

STEP 2. Set m1 = ⌊n1−ǫ⌋ for some ǫ ∈ (0, 1/2).

Draw a large num- ber B1 of bootstrap samples of size m1 and order each of them according to the orthant order, dropping the observations with non-positive components.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 22 / 46

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BOOTSTRAP-BASED METHOD TO SELECT k(n)

STEP 3. Denote the bootstrap error of each marginal j = 1, . . . , d by,

Errj(m1, b1, kj) :=

  • M(2)

kj,j − 2

  • M(1)

kj,j

22 , b1 = 1, . . . , B1, where kj varies from 1 to m1 − 1. Then, determine the value kj(m1) that minimizes the mean sample error, 1 B1

B1

  • b1=1

Errj(m1, b1, kj).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 22 / 46

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BOOTSTRAP-BASED METHOD TO SELECT k(n)

STEP 4. Set m2 = ⌊m2

1/n⌋, and repeat Step 2 to obtain kj(m2).

STEP 5. Estimate the associated second order tail index π by

ˆ π = 1 d

d

  • j=1

log

  • kj(m1)

−2 log(m1) + 2 log(kj(m1))

  • ,

which is a consistent estimator, (see Qi (2008)).

STEP 6. The optimal selection for k = k(n) is given by,

ˆ k(n) := 1 d

d

  • j=1

kj(m1)2 kj(m2)

  • 1 − 1

ˆ π 1/(2ˆ

π−1)

.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 22 / 46

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ASYMPTOTIC NORMALITY FOR ˆ QX(α, u) AT HIGH

LEVELS

RESULT If RuX is a second order multivariate regularly varying random vector. Then, √ k

  • ˆ

xu,j(α, θ) − xu,j(α, θ) ˆ au,j(n/k) sn

1 tˆ γ−1(log t)dt; j = 1, . . . , d

  • ,

converges to a multivariate normal distribution. COROLLARY The asymptotic normality of the elements in ˆ QRuX(α, e, θ) implies the asymptotic normality of the elements in ˆ QX(α, u, θ).

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 23 / 46

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ILLUSTRATIVE EXAMPLE ≡ t−DISTRIBUTION

If X has a multivariate t−distribution, then holds A1-A4 and this distribution is closed under orthogonal transformations. Original Space Rotated Space in direction u µ Σ = σ2

1

σ1,2 σ1,2 σ2

2

  • ν

µu = Ruµ Σu = RuΣR′

u

ν Example ≡

  • µ = (0, 0)′, σ2

1 = 5, σ2 2 = 1, σ1,2 = 0.1, ν = 3

α = 1/n, u = e, FPC

  • Torres Díaz, Raúl A.

Directional Multivariate EVT May 2018 24 / 46

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DIRECTIONAL MULTIVARIATE QUANTILES

  • 80
  • 60
  • 40
  • 20

20 40 60 80

  • 80
  • 60
  • 40
  • 20

20 40 60 80

Theoretical DMQ for α ∈ {0.5, 0.3, 0.1} and u ∈ {e, FPC}

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 25 / 46

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BOOTSTRAP-BASED DISTRIBUTION OF k

1 12 13 14 15 16 17 18 19 1 160 180 200 220 240 260 280 300 320

(A) n = 500, α = 1

n

(B) n = 5000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 26 / 46

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BOXPLOTS OF THE RATIOS ˆ γu,1/γu,1

1

  • 1
  • 0.5

0.5 1 1.5 2 2.5 1

  • 1
  • 0.5

0.5 1 1.5 2 2.5

(A) n = 500, α = 1

n

(B) n = 5000, α = 1

n

Theoretical value γ = 1/ν, j = 1, 2

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 27 / 46

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ESTIMATION OF ρu

0.5 1 1.5 1.5 2 2.5 3 3.5 4 4.5 5 Theoretical Estimated 0.5 1 1.5 1.5 2 2.5 3 3.5 4 4.5 5

Theoretical Estimated

(A) n = 500, α = 1

n

(B) n = 5000, α = 1

n

Theoretical expression by Nikoloulopoulos et al. (2009)

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 28 / 46

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FINAL ESTIMATION IN THE CLASSICAL DIRECTION e

  • 80
  • 60
  • 40
  • 20

20 40 60 80

  • 80
  • 60
  • 40
  • 20

20 40 60 80 Confidence Quantile Region Simulated Data Theoretical Quantile Curve Theoretical Asymptotic Quantile Estimated Quantile

  • 80
  • 60
  • 40
  • 20

20 40 60 80

  • 80
  • 60
  • 40
  • 20

20 40 60 80 Confidence Quantile Region Simulated Data Theoretical Quantile Curve Theoretical Asymptotic Quantile Estimated Quantile

(A) n = 500, α = 1

n

(B) n = 5000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 29 / 46

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ESTIMATION IN THE FPC DIRECTION

  • 80
  • 60
  • 40
  • 20

20 40 60 80

  • 80
  • 60
  • 40
  • 20

20 40 60 80 Confidence Quantile Region Simulated Data Theoretical Quantile Curve Theoretical Asymptotic Quantile Estimated Quantile

  • 80
  • 60
  • 40
  • 20

20 40 60 80

  • 80
  • 60
  • 40
  • 20

20 40 60 80 Confidence Quantile Region Simulated Data Theoretical Quantile Curve Theoretical Asymptotic Quantile Estimated Quantile

(A) n = 500, α = 1

n

(B) n = 5000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 30 / 46

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3D EXAMPLE

Parameters of the t−distribution µ = (0, 0, 0)′ Σ =   5 2.44 −1.88 2.44 2.12 0.04 −1.88 0.04 2.36   ν = 4

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 31 / 46

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DIRECTIONAL MULTIVARIATE QUANTILES

Theoretical DMQ for α = 0.1 and u ∈ {e, FPC}

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 32 / 46

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BOOTSTRAP-BASED DISTRIBUTION OF k

1 15 16 17 18 19 20 1 1140 1150 1160 1170 1180 1190 1200 1210

(A) n = 500, α = 1

n

(B) n = 50000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 33 / 46

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ESTIMATION OF ρu

5 10 1.5 15 20 25 30 1 1.5 1 0.5 0.5 00 Theoretical Estimated 1.5 5 1.5 10 1 15 1 20 0.5 0.5 Theoretical Estimated

(A) n = 500, α = 1

n

(B) n = 50000, α = 1

n

Theoretical expression by Nikoloulopoulos et al. (2009)

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 34 / 46

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ESTIMATION IN THE FPC DIRECTION

  • 20
  • 20
  • 10
  • 10

20 10 10 10 20 20

  • 10
  • 20

Simulated Data Theoretical Quantile Surface Theoretical Asymptotic Quantile Estimated Quantile

(A) n = 500, α = 1

n

(B) n = 50000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 35 / 46

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RELATIVE ERROR OF ˆ QX(α, u, θ, n/k), WHERE θ = (1/ √ d, . . . , 1/ √ d)

1

  • 1
  • 0.5

0.5 1 1

  • 1
  • 0.5

0.5 1

(A) n = 500, α = 1

n

(B) n = 50000, α = 1

n

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 36 / 46

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apple uc3m Real case study

OUTLINE

1 DIRECTIONAL NOTIONS 2 NON-PARAMETRIC OUT-SAMPLE ESTIMATION 3 REAL CASE STUDY 4 CONCLUSIONS

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 37 / 46

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apple uc3m Real case study

REAL CASE STUDY

Portfolio

  • f Indices

≡ (S&P 500, FTSE 100, Nikkei 225) (USA, UK, Japan) Data from July 2nd, 2001 to June 29th, 2007 GARCH modeling to ensure i.i.d. Directional Analysis of the Filtered Losses

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 38 / 46

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SLIDE 50

apple uc3m Real case study

OVERALL ANALYSIS

6 2 5 4 3 6 4 2.5 8 3 2 10 1.5 2 1 1 0.5

Outlier criteria through Tukey depth trimming, α = 1/10000, (He and Einhmal (2017))

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 39 / 46

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apple uc3m Real case study

EVEN INVESTMENT (WEIGHTS = 1/ √ 3(1, 1, 1),

LEVERAGE BUT NOT OUTLIER)

2 4 6 8 2

  • 1.5

10 12

  • 1
  • 0.5

0.5

  • 2

1 1.5 Innovations Estimated Quantile Surface Big US Loss 2007/2/27

(A) Semi-parametric approach (B) Non-parametric approach Directional portfolio criteria, u = e and α = 1/1250.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 40 / 46

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SLIDE 52

apple uc3m Real case study

A CRITICAL INVESTMENT (WEIGHTS = (0.6,0.35,0.05),

IDENTIFIED OUTLIER)

2 4 6 8 2

  • 1.5

10 12

  • 1
  • 0.5

0.5

  • 2

1 1.5 Innovations Estimated Quantile Surface Big US Loss 2007/2/27

(A) Semi-parametric approach (B) Non-parametric approach Directional portfolio criteria, u = (0.6, 0.35, 0.05) and α = 1/1250.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 41 / 46

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SLIDE 53

apple uc3m Conclusions

OUTLINE

1 DIRECTIONAL NOTIONS 2 NON-PARAMETRIC OUT-SAMPLE ESTIMATION 3 REAL CASE STUDY 4 CONCLUSIONS

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 42 / 46

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SLIDE 54

apple uc3m Conclusions

CONCLUSIONS

Results that plug the directional approach into the multivariate value theory have been proved. A non-parametric procedure to perform out-sample estimation

  • f the directional multivariate quantiles has been developed.

A bootstrap-based method of selection for the tuning parame- ter k has been introduced. The asymptotic normality of the estimator has been shown. The performance of the estimation at high levels has been shown in a heavy tailed example. A real case study of a decision rule to determine the existence

  • f an outlier has been shown.

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 43 / 46

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SLIDE 55

apple uc3m Conclusions

Thanks

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 44 / 46

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apple uc3m Conclusions

BIBLIOGRAPHY

Torres R., Rosa E. Lillo and Laniado H., A Directional Multivariate Value at Risk. Insurance Math. Econom. 65, 111-123, 2015. http://dx.doi.org/10.1016/j.insmatheco.2015.09.002 Torres R., De Michele C., Laniado H. and Lillo R. Directional Multivariate Extremes in Environmental Phenomena. Environmetrics (In Press), 2016. De Haan L., and Huang X. (1995) Large Quantile Estimation in a Multivariate Setting, Journal of Multivariate Analysis, 53, 247-263. Danielsson J., de Haan L. and De Vries C., Using a Bootstrap Method to Choose the Sample Fraction in Tail Index Estimation, Journal of Multivariate Analysis, 76, 226-248, (2001). He, Y. and Einmahl, J. H. J., Estimation of extreme depth-based quantile regions. Journals of the Royal Statistical Society. Series B. Statistical Methodology, doi:10.1111/rssb.12163 (2017). Jessen A. and Mikosh T., Regularly varying functions. Publications de l’institute mathimatique 79, 2006. Laniado H., Lillo R. and Romo J., Extremality in Multivariate Statistics. Ph.D. Thesis, Universidad Carlos III de Madrid, 2012 Nikoloulopoulos A., Joe H. and Li H., Extreme value properties of multivariate t copulas. Extremes, 12, 129-148 (2009). Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 45 / 46

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apple uc3m Conclusions

Thanks

Torres Díaz, Raúl A. Directional Multivariate EVT May 2018 46 / 46