ESTIMATING BIVARIATE TAILmath.univ-lyon1.fr/~mercadier/cirm2010/dibernardino.pdf · Estimating the tail of bivariate distributions Simulation Study Introduction This work deals with

Introduction: the univariate POT methodTwo-dimensional Pickands-Balkema-de Haan Theorem

Estimating the tail of bivariate distributionsSimulation Study

ESTIMATING BIVARIATE TAIL

Elena Di Bernardinoa, Véronique Maume-Deschampsa andClémentine Prieurb

aISFA, Université de Lyon, Université Lyon 1bUniversité Joseph Fourier

Workshop Spatio-temporal risk modeling, LuminyApril 26-30, 2010

Di Bernardino, Maume-Deschamps, Prieur ESTIMATING BIVARIATE TAIL, April 2010-CIRM Luminy



Introduction

This work deals with the problem of estimating the tail of a bivariatedistribution function.

We develop a general extension of the POT (Peaks-Over-Threshold)method, mainly based on a two-dimensional version of the

Pickands-Balkema-de Haan Theorem.

We construct a two-dimensional tail estimator and study itsasymptotic properties.

We also present a simulation study which illustrates our theoreticalresults.

Key words: Extreme Value Theory, Peaks Over Threshold method,Pickands-Balkema-de Haan Theorem, tail dependence.




Contents

1 Introduction: the univariate POT method

2 Two-dimensional Pickands-Balkema-de Haan TheoremUpper Tail dependence copulaTwo-dimensional Pickands Theorem: case FX 6= FY

3 Estimating the tail of bivariate distributionsConstruction of the bivariate estimatorConvergence results

4 Simulation StudyCase with identical marginal distributionsCase with di�erent marginal distributionsEstimation of threshold f 2(n)




Contents








Generalized Pareto distribution

The key idea of the Peaks-Over-Threshold method is the use of thegeneralized Pareto distribution (given by (1)) to approximate thedistribution of excesses over thresholds.

Vk,σ(x) :=

{1−

(1− kx

σ

) 1k , if k 6= 0, σ > 0,

1− e−xσ , if k = 0, σ > 0,

(1)

and x ≥ 0 for k ≤ 0 or 0 ≤ x < σkfor k > 0.

Let X1,X2, . . . be a sequence of i.i.d random variables with unknowndistribution function F .

Fix a threshold u. For x > u, decompose F as

F (x) = P[X ≤ x ] = (1− P[X ≤ u])Fu(x − u) + P[X ≤ u],

where Fu(x) = P[X − u ≤ x |X > u].




One-dimensional Pickands-Balkema-de Haan Theorem

De�nition (Maximum Domain of Attraction)

We say that F belongs to the Maximum Domain of Attraction of ageneralized extreme value distribution (F ∈ MDA(Hk)) if there exist asequence of positive numbers (an)n>0 and a sequence (bn)n>0 of realnumbers such thatlimn→∞ P

[(max{X1,X2, . . . ,Xn} − bn)(an)−1 ≤ x

]= Hk(x), where

x ∈ R and Hk(x) is GEV distribution.

We can now state a precise formulation of

Theorem (Pickands-Balkema-de Haan Theorem)

F ∈ MDA(Hk) ⇔ limu→xF sup0≤x<xF−u

∣∣Fu(x)− Vk,σ(u)(x)∣∣ = 0,

where Fu(x) = P[X − u ≤ x |X > u] and xF := sup{x ∈ R |F (x) < 1}.




Estimating the tail of univariate distributions

We use the Pickands-Balkema-de Haan Theorem

We estimate F (u) by the empirical distribution function F̂X (u)

and we obtain the univariate tail estimate

F ∗(x) = (1− F̂X (u))Vk,σ(x − u) + F̂X (u). (2)

Parameters k and σ of the GPD ⇒ MLE (k̂, σ̂) based on the excessesabove u.

We then write (2) as

F̂ ∗(x) = (1− F̂X (u))Vk̂,σ̂(x − u) + F̂X (u), for x > u.

(For example McNeil (1997), McNeil (1999) and references therein).




Upper Tail dependence copulaTwo-dimensional Pickands Theorem: case FX 6= FY

Contents









Modeling upper tail

De�nition (Upper-tail dependence copula)

Let X and Y be uniformly distributed on [0, 1]. Assume that the copulaassociated to (X ,Y ) is symmetric. For a threshold u ∈ [0, 1) satisfyingC∗(1− u, 1− u) > 0, we de�ne the upper-tail dependence copula at levelu ∈ [0, 1) relative to the copula C , ∀ (x , y) ∈ [0, 1]2 by

Cupu (x , y) := P[X ≤ Fu

−1(x),Y ≤ Fu

−1(y) |X > u,Y > u],

where Fu(x) := P[X ≤ x |X > u,Y > u] = 1− C∗(1−x∨u,1−u)C∗(1−u,1−u) .

Cupu (x , y) is a copula.

The asymptotic behavior of Cupu for u around 1 describes the

dependence structure of X ,Y in their upper tails.





Modeling upper tail

Theorem (Upper-tail Theorem; Juri and Wüthrich (2003))

Let C be a symmetric copula such that C∗(1− u, 1− u) > 0, for allu > 0. Furthermore, assume that there is a strictly increasing continuous

function g : [0,∞)→ [0,∞) such that

limu→1

C∗(x(1− u), 1− u)

C∗(1− u, 1− u)= g(x), x ∈ [0,∞).

Then, there exists a θ > 0 such that g(x) = xθg(1

x

)for all x ∈ (0,∞).

Further, for all (x , y) ∈ [0, 1]2

limu→1

Cupu (x , y) = x+y−1+G (g−1(1−x), g−1(1−y)) := C∗G (x , y), (3)

where G (x , y) := yθg(xy

), ∀ (x , y) ∈ (0, 1]2 and

G :≡ 0 on [0, 1]2 \ (0, 1]2.





Case symmetric copula C and FX 6= FY

Theorem (Two-dimensional Pickands Theorem)

Let X and Y be continuous real valued r.v., with marginal distributions

FX , FY , and symmetric copula C. Suppose that FX ∈ MDA(Hk1), FY∈ MDA(Hk2) and that C satis�es the hypotheses of the Upper-tail

Theorem for some g. Then

supA

∣∣∣∣P[X − u ≤ x ,Y − F−1Y (FX (u)) ≤ y∣∣X > u,Y > F−1Y (FX (u))

]−C∗G

(1−g(1−Vk1,a1(u)(x)), 1−g(1−Vk2,a2(F

−1Y (FX (u)))(y))

)∣∣∣∣−−−−→u→xFX

0,

where Vki ,ai (·) is the GPD with parameters ki , ai (·) de�ned in (1),xFX := sup{x ∈ R |FX (x) < 1}, xFY := sup{y ∈ R |FY (y) < 1} andA := {(x , y) : 0 < x ≤ xFX − u, 0 < y ≤ xFY − F−1Y (FX (u))}.





Case symmetric copula C and FX 6= FY

From (3),

C∗G(1− g(1− Vk1,a1(u)(x)), 1− g(1− Vk2,a2(F

−1Y (FX (u)))(y))

)= 1− g(1− Vk1,a1(u)(x))− g(1− Vk2,a2(F

−1Y (FX (u)))(y))

+ G(1− Vk1,a1(u)(x), 1− Vk2,a2(F

−1Y (FX (u)))(y)

),

where ai (u) as in one-dimensional Pickands Theorem, for i = 1, 2.




Construction of the bivariate estimatorConvergence results

Contents









A two-dimensional structure of dependence

Now we are interested to develop a two-dimensional extension of thePOT method.

We consider a two-dimensional structure of dependence as follows:

Continuous r. v. X and Y (in particular FX and FY are assumed tobe continuous).

FX and FY are a priori unknown, with regularity properties speci�edin the statement of our theorems.

The structure of dependence between X and Y is described by acontinuous and symmetric copula C , which is supposed to be knownor inferred from the data structure.





A new bivariate tail estimator

New tail estimator for the 2-dimensional distribution function F (x , y)(continuous symmetric copula C , FX 6= FY ).For x > u, y > F−1Y (FX (u)) := uY we de�ne

F̂ ∗(x , y) =

(1

n

n∑i=1

1{Xi>u,Yi>uY }

)(1− g(1− V

k̂X ,σ̂X(x − u))

−g(1−Vk̂Y ,σ̂Y

(y−uY )) +G(1−V

k̂X ,σ̂X(x−u), 1−V

k̂Y ,σ̂Y(y−uY )

))+ F̂ ∗

1(u, y) + F̂ ∗

2(x , uY )− 1

n

n∑i=1

1{Xi≤u,Yi≤uY },

where k̂X , σ̂X (resp. k̂Y , σ̂Y ) are MLE based on the excesses of X (resp.Y).





Description of the construction (1/2)

Distribution of excesses above u and uY :Fu,uY (x , y) := P[X − u ≤ x ,Y − uY ≤ y |X > u,Y > uY ].

So for x > u, y > uY ,

F (x , y) = (F (u, uY ))·Fu,uY (x−u, y−uY )+F (u, y)+F (x , uY )−F (u, uY ).

From 2-dimensional Pickands Theorem we can approximateFu,uY (x − u, y − uY ), for high thresholds u, uY withC∗G

(1− g(1− VkX ,σX (u)(x − u)), 1− g(1− VkY ,σY (uY )(y − uY ))

)We estimate F (u, u) and F (u, u) by

F̂ (u, uY ) =1

n

n∑i=1

1{Xi≤u,Yi≤uY }, F̂ (u, uY ) =1

n

n∑i=1

1{Xi>u,Yi>uY }.





Description of the construction (2/2)

We estimate F (u, y) and F (x , uY ) by

F̂ ∗1

(u, y) = C (F̂X (u), F̂ ∗Y (y)) and F̂ ∗2

(x , uY ) = C (F̂ ∗X (x), F̂Y (uY )),

where F̂X (u) (resp. F̂Y (uY )) are the empirical estimators of FX(resp. FY ), and

F̂ ∗X (x) = (1− F̂X (u))Vk̂X ,σ̂X

(x − u) + F̂X (u), for x > u.

F̂ ∗Y (y) = (1− F̂Y (uY ))Vk̂Y ,σ̂Y

(y − uY ) + F̂Y (uY ), for y > uY

are the one-dimensional tail estimators of the distribution functionsFX and FY above high thresholds.

N.B. We will propose an estimator for uY := F−1Y (FX (u)).





Maximum domain of attraction of Fréchet

We study one-dimensional convergence results, needed to deriveasymptotic properties of the bivariate tail estimator F̂ ∗(x , y).

Assume F ∈ MDA(Φα), MDA of Fréchet, for some α > 0.

Recall that F ∈ MDA(Φα) ⇔ F (x) = x−αL(x), for some slowlyvarying function L(x).

As in Smith (1987), we shall assume that L satis�es the followingcondition:

� SR2: L(tx)L(x) = 1 + k(t)φ(x) + o(φ(x)), ∀ t > 0, as x →∞,

where φ(x) > 0 and φ(x)→ 0 as x →∞. Excluding trivial cases, φ ∈ Rρ(with Rρ the set of ρ−regularly varying functions) for some ρ ≤ 0, and

k(t) = c hρ(t), with hρ(t) =∫ t1uρ−1du; (→ Smith (1987)).





Convergence results in univariate framework

Theorem (MLE Convergence Theorem, (Smith (1987))

Suppose L satis�es SR2. Let Y1, . . . , Ymn i.i.d from an unknown

distribution function Fumn where limn→∞mn =∞, limn→∞mnn

= 0. For

each mn we de�ne a threshold umn := f (mn) −−−→n→∞

∞ such that

√mn c φ(f (mn))

α− ρ−−−→n→∞

µ ∈ (−∞,∞).

We de�ne k = −α−1 and σmn = f (mn)α−1. Then there exists a local

maximum (σ̂mn , k̂mn ) of the GPD log likelihood function, such that

√mn

σ̂mnσmn− 1

k̂mn − k

d−−−→n→∞

N

µ(1−k)(1+2kρ)1−k+kρ

µ(1−k)k(1+ρ)1−k+kρ

;M−1

.





Convergence results in univariate framework

This theorem is written conditionally on N = mn. In practice we workwith some threshold u and N is considered as random. Therefore we givethe analogues of the MLE Convergence Theorem working unconditionallyon N.

Corollary

Suppose L satis�es SR2. Let n be the sample size and un := f (n) thethreshold, such that f (n) −−−→

n→∞∞. Let N = Nn denote the random

number of excesses above un. If

n(1− F (un)) −−−→n→∞

∞, (4)

√n(1− F (un))c φ(un) −−−→

n→∞µ(α− ρ), (5)

then the MLE Convergence Theorem holds also unconditionally on N.





Uniform Convergence result in univariate framework

We obtain a general result for the absolute error:

Theorem (Univariate Convergence Theorem)

Suppose F belongs to the maximum domain of attraction of Fréchet and

L satis�es SR2. Assume that the threshold un := f (n) −−−→n→∞

∞, then if

(4) and (5) hold we get

supx>f (n)

∣∣∣F (x)− F̂ ∗(x)∣∣∣ P−−−→

n→∞0.





Convergence results in bivariate framework

Let n be the sample size. We choose

u1 n := f 1(n) −−−→n→∞

∞ threshold for X ,

u2 n = f 2(n) = F−1Y (FX (f 1(n))) −−−→n→∞

∞ threshold for Y .

Theorem (Bivariate Convergence Theorem)

Suppose FX and FY belong to the maximum domain of attraction of

Fréchet and LX , LY satisfy the condition SR2. Assume that the copula C

is continuous and symmetric. Then under assumptions of the Upper-tail

Theorem and the Two-dimensional Pickands Theorem, if sequences

f 1(n), f 2(n), satisfy the conditions of the Univariate ConvergenceTheorem and if the conditions of the Two-dimensional Glivenko-Cantelli

Theorem hold then

supx > f 1(n), y > f 2(n)

∣∣∣F (x , y)− F̂ ∗(x , y)∣∣∣ P−−−→

n→∞0.




Case with identical marginal distributionsCase with di�erent marginal distributionsEstimation of threshold f 2(n)

Contents









Model

C (u, v) = u+v−1+[(1−u)−1+(1−v)−1−1]−1 (Survival Clayton copula),

FX (x) = 1−(1+x)−1, FY (y) = 1−(1+y)−1 (same Burr distributions).

Figure: Copula Survival Clayton.

Figure: Bivariate distributionfunction FX ,Y (x , y), with FX = FY ,for x > 0, y > 0.





Threshold

We choose f (n) = n13

3−−−→n→∞

∞.

then supx, y >f (n)

∣∣∣F (x , y)− F̂ ∗(x , y)∣∣∣ P−−−→

n→∞0.

We de�ne for each x > f (n), y > f (n), for i = 1, . . . , t:

ERRi, abs =∣∣∣F̂ ∗(x , y)− F (x , y)

∣∣∣ , ERRabs =1

t

t∑i=1

ERRi, abs ,

ERRi, rel =

∣∣∣∣∣ F̂ ∗(x , y)− F (x , y)

F (x , y)

∣∣∣∣∣ , ERRrel =1

t

t∑i=1

ERRi, rel .





Results

n ERRabs var(F̂ ∗(x0, y0))t ERRrel f (n) mean(Excesses)t

1000 0.0112 1.95e−04 0.0129 3.333 2332000 0.0086 8.87e−05 0.0099 4.199 3845000 0.0051 4.14e−05 0.0059 5.699 74510000 0.0034 2.16e−05 0.0039 7.181 1223

Table: Errors and empirical variance for (x0, y0) = (10, 10),t-simulations= 100, case with same marginals, (NB: both x0 and y0 are abovethe threshold f (n) for n = 1000, 2000, 5000 or 10000).

For n = 1000, t = 10, we discretize, with a grid of 62500 points, the set[f (n) + 1, 250]2 = [4.333, 250]2.On this grid: max(ERRabs) = 0.0171,max(ERRrel) = 0.0195,

max(var(F̂ ∗(x , y))) = 3.4e−04.





Model and Thresholds

C (u, v) = u+v−1+[(1−u)−1+(1−v)−1−1]−1 (Survival Clayton copula),

FX (x) = 1−(1+x)−1, FY (y) = 1−(1+y2)−1 (di�erent Burr distributions).

We choose f 1(n) = n13

3and f 2(n) = F−1Y (FX (f 1(n))) =

√n13

3.

then supx > f 1(n), y > f 2(n)

∣∣∣F (x , y)− F̂ ∗(x , y)∣∣∣ P−−−→

n→∞0.





Results

n ERRabs var(F̂ ∗(x0, y0))t ERRrel f 1(n), f 2(n) mean(Excesses)t

1000 0.0086 1.2e−04 0.0094 3.333, 1.825 2312000 0.0061 4.6e−05 0.0067 4.199, 2.049 3855000 0.0040 2.1e−05 0.0044 5.699, 2.387 74410000 0.0027 1.1e−05 0.003 7.181, 2.679 1221

Table: Errors and empirical variance for (x0, y0) = (10, 10) (withx0 > f 1(n), y0 > f 2(n)), t-simulations= 100, case with di�erent marginals.

For n = 1000, t = 10, we discretize, with a grid of 62500 points, the set[f 1(n) + 1, 250]× [f 2(n) + 1, 250] = [4.333, 250]× [2.825, 250].On this grid: max(ERRabs) = 0.0123,max(ERRrel) = 0.0143

max(var(F̂ ∗(x , y))) = 2.5e−04.





Model and Thresholds

Threshold for Y: f 2(n) = F−1Y (FX (f 1(n))). ⇒ FX and FY are unknown

so f 2(n) has to be estimated.

Model: For the simulations we keep the previous model.

Thresholds: We choose f 1(n) = n13

3and f̂ 2(n) = F̂−1Y (F̂X (f 1(n))).

From classical results (for instance Dekkers, de Haan (1989))[f̂ 2(n)− f 2(n)

]P−−−→

n→∞0.

So Bivariate Convergence Theorem is true when replacing f 2(n) by f̂ 2(n).





Results

n ERRabs var(F̂ ∗(x0, y0))t ERRrel f 1(n), m(f̂ 2(n))t m(Excesses)t

1000 0.0086 1.1e−04 0.009 3.333, 1.814 2292000 0.0068 5.3e−05 0.007 4.199, 2.054 3865000 0.0039 2.1e−05 0.004 5.699, 2.389 74710000 0.0031 1.4e−05 0.003 7.181, 2.676 1225

Table: Errors and the empirical variance calculate in (x0, y0) = (10, 10), in themodel with di�erent marginal distributions, estimated threshold f 2(n) andt-simulations= 100. (NB: both x0 and y0 are above the thresholds.)

For n = 1000, t = 10, we discretize, with a grid of 62500 points, the set

[f 1(n) + 1, 250]× [mean(f̂ 2(n))t + 1, 250] = [4.333, 250]× [2.814, 250].On this grid: max(ERRabs) = 0.0164,max(ERRrel) = 0.0195,

max(var(F̂ ∗(x , y))) = 3.9e−04.





Ideas for future developments

Estimation of copula /study of properties of g(x),G (x , y) in term ofdependence.

We can use F̂ ∗(x , y) to obtain estimation of bivariate upper-quantilecurves, for high levels α.

An intuitive and immediate measure of the risk, for a 2-dimensionalloss distribution function F , is represented by its α-level sets. Wecan estimate, for large α, the bi-dimensional Value-at-Risk as

VaRα(F̂ ∗) := {(x , y) ∈ (f 1(n),+∞)× (f̂ 2(n),+∞) : F̂ ∗(x , y) = α}

Starting from VaRα(F̂ ∗) we can propose a bivariate estimator for

the bivariate Conditional Tail Expectation: CTEα(F̂ ∗).





References

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References

Juri, A. and Wüthrich, M.V. (2003).

Tail Dependence From a Distributional Point of View.Extremes 6 (3), 213-246.

McNeil, A. J. (1999).

Estimating the Tails of loss severity distributions using extreme value theory.ASTIN Bulletin 27, 1117-1137.

McNeil, A. J. (1999).

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ESTIMATING BIVARIATE TAILmath.univ-lyon1.fr/~mercadier/cirm2010/dibernardino.pdf · Estimating the tail of bivariate distributions Simulation Study Introduction This work deals with