On the estimation of the second order parameter in extreme-value …irma.math.unistra.fr/~gardes/SEMINAIRE/deme.pdf · 2012-03-02 · Strasbourg March 6th 2012. ... (C1) hereafter

On the estimation of the second order parameter inextreme-value theory

By

El Hadji DEME

Joint work

Laurent GARDES & Stephane GIRARD

Strasbourg March 6th 2012

Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimators Numerical results

1 Extreme-Value Theory

2 New family of estimators for the second order parameter

3 Asymptotic properties

4 Link with existing estimators

5 Numerical results

2 / 38


Main results on extreme value theory

Let X1, . . . ,Xn be a sequence of independent copies ofa real random variable (r.v.) X with cumulativedistribution function F. The order statisticsassociated to this sample are denoted by :X1,n ≤ · · · ≤ Xn,n.

Fisher-Tippett-Gnedenko theorem

Under some conditions of regularity on F, thereexists a real parameter γ and two sequences(an)n≥1 > 0 and (bn)n≥1 ∈ R such that for all x ∈ R,

limn→∞

P(a−1n (Xn,n − bn) ≤ x

)= EVγ(x),

with EVγ(x) =

(exp

“−(1 + γx)

−1/γ+

”if γ 6= 0,

exp`−e−x

íf γ = 0,

where y+ = max(y , 0).3 / 38


Definition :

� The parameter γ is the tail index, the primary parameter of

extreme events.

� EVγ is called the extreme value distribution and F is then said

to belong to the domain of attraction of EVγ (F ∈ DA(EVγ)).

Heavy-tailed models

� In statistics of Extremes, a model F is said to beheavy-tailed whenever, for some γ > 0, its survivalfunction is of the forme :

1− F (x) = x−1/γ`F (x)⇔ U(x) = xγÙ(x)

where U(x) = inf{y : F (y) ≥ 1− 1/x} is quantilefunction and `•(·) is a slowly varying function i.e.`•(λx)/`•(x)→ 1 as x →∞ for all λ ≥ 1.The present model is now often restated as the asumption of F isregulary varying at infinity with index −1/γ (denoted1− F (x) ∈ RV−1/γ).

4 / 38


Exemples of heavy-tailed models

� Strict Pareto distribution

1− F (x) = x−α, x > 1; α > 0.

is heavy-tailed with γ = 1/α and `F (x) = 1.

� F (m, n) distribution

f (x) =Γ`

m+n2

´Γ`

m2

´Γ`

n2

´ “m

n

”m/2

xm/2−1“

1 +m

nx”−(m+n)/2

x > 0; m, n > 0

is heavy-tailed with γ = 2/n and

`F (x) =Γ`

m+n2

´Γ`

m2

´Γ`

n2

´ “m

n

”m/2

xm/2−1

„m

n+

1

x

«−(m+n)/2

(1 + o(1))

for x →∞.

� Others : |t|, log-gamma, inverse gamma, Frechet, Burr,...

5 / 38


Inference statistic of γ for heavy-tailed model

In Statistics of extremes, inference is often based

�Wi,k = (log Xn−i+1,n − log Xn−k,n) the log-excesses

�Zi,k = i (log Xn−i+1,n − log Xn−i,n) the rescaled log-spacings

Exemples of γ estimators

Hn,k =1

k

kXi=1

Wi,k =1

k

kXi=1

Zi,k , Hill Estimator, Hill (1975)

γn =M(1)n,k + 1− 1

2

0B@1−

“M(1)

n,k

”2

M(2)n,k

1CA−1

Moment Estmator,

Deker et al (1989) with M(α)n,k =

1

k

kXi=1

W αi,k

Others Estimators of γ, received a lot of attention Smith (1987), Csorgoet al. (1985), Schultze and Steinebach (1996), Kratz and Resnick (1996),

6 / 38


Asymptotic behavior

� Since F ∈ RV−1/γ , then if k →∞ and k/n→ 0 as n→∞,

then Hn,kP−→ γ, γn

P−→ γ and M(α)n,k /µ

(1)α

P−→ γα with µ(1)α = Γ(α + 1)

� Asymptotic normality ?

The asymptotic distribution of estimators of γ is

obtained under a second order condition.

Second Order Condition (S.O.C)

There exist a function A(x)→ 0 and a second orderparameter ρ ≤ 0 such that, for every x > 0,

limt→∞

log U(tx)− log U(t)− γ log x

A(t)=

xρ − 1

ρ.

7 / 38


Second Order Condition

Remarks

Under the regularly variation condition

logU(tx)

U(t)→ γ log x for, t →∞

So the (S.O.C) specifies the rate of this convergence.

|A| is regularly varying with index ρ.

Exemple : Hall class of Heavy-tailed models

� Hall class (Hall and Welsh, (1985))

U(x) = Cxγ(1 + Dxρ + o(xρ)), (x →∞)

with C > 0, satisfies the second order condition with

A(x) = ρDxρ.

Exemples : Frechet, Burr, Generalized Pareto (GP),|t|,...8 / 38


Asymptotic representation of γ’s estimators

Hill’s estmator

Hn,kD= γ +

γ√kNk +

A(n/k)

1− ρ (1 + oP(1))

Moment’s estimator

M(α)n,k

D= γαµ(1)

α +γασ

(1)α√k

P(α)k + αγα−1µ(2)

α A(n/k)(1 + oP(1))

where Nk and P(α)k are asymptotically standard

normal random variables,

µ(2)α =

α

ρ

1− (1− ρ)α

(1− ρ)αand σ(1)

α =p

Γ(2α + 1)− Γ2(α + 1).

The Hill’s estimator Hn,k correspond toM(1)n,k

9 / 38


Asymptotic normality

if k, n→∞ with k/n→ 0 and√

kA(n/k)→ λ ∈ R, then

√k(Hn,k − γ)

D−→ N„

λ

1− ρ , γ2

«and √

k(M(α)n,k − γ

αµ(1)α )

D−→ N„λαγα−1µ(2)

α ,“γασ(1)

α

”2«

Comment

� ρ controls the bias of the estimators of γ

� How to estimate the second order tail parameter ρ ?

10 / 38


New family of estimators for the second order parameter

The model

Tn = Tn(X1, ...,Xn) : a random vector in Rd drawn fromthe sample X1, . . . ,Xn.The statistics can always be expanded as :

ω−1n (Tn − χnI)

P−→ f (ρ)

whereI = t(1, . . . , 1) ∈ Rd ,

χn and ωn : random variables,

ξn ∈ Rd : a random vector,

f : R− → Rd : a function continuously differentiablein a neighborhood of ρ (independent of γ).

11 / 38


General approach

Notations

ψ : Rd → R such that

� Invariance properties (Inv-prop)

ψ(x + λI) = ψ(x) for all x ∈ Rd and λ ∈ R,

ψ(λx) = ψ(x) for all λ ∈ R \ {0},

� Regularity properties (Reg-prop) ψ is continuously

differentiable in a neighborhood of f (ρ),

ϕ := ψ ◦ f is continuous in a neighborhood of ρ

� Bijection property (Bij-prop)

there exist J0 ⊆ R− and an open interval J ⊂ R such that ϕ isa bijection from J0 to J.

12 / 38


The estimator

Clearly

by the invariance and the regularity properties

� ψ(ω−1n (Tn − χnI)) = ψ(Tn)

P−→ ψ(f (ρ)).

� Zn = ψ(Tn) ≈ ϕ(ρ).

Under the bijection property, our family ofestimators of the second order parameter is thusdefined by :

ρn = ϕ−1(Zn)1l{Zn ∈ J}.

1lA is the indicator function of the set A.

13 / 38


Asymptotic properties

Theorems

Suppose that Inv-prop, Reg-prop and Bij-prop holdthen

ρnP−→ ρ as n→∞

if there exist a sequence vn →∞, a function,m : R− → Rd and a d × d matrix Σ such that

vn(ω−1n (Tn − χnI)− f (ρ))

D−→ Nd(m(ρ),Σ)

then

vn(ρn − ρ)D−→ N

mψ (ρ)

ϕ′(ρ),σ2ψ(ρ)

(ϕ′(ρ))2

!with,

� ϕ′(ρ) = t f ′(ρ)∇ψ(f (ρ)),

�mψ(ρ) = tm(ρ)∇ψ(f (ρ)),

� σ2ψ(ρ) = t∇ψ(f (ρ)) Σ ∇ψ(f (ρ)).

14 / 38


Link with existing estimators

1. Estimators based on rescaled log-spacings : log Xn−j+1 − log Xn−j

� Kernel estimators

Rk(τ) =1

k

kXj=1

Hτ

„j

k + 1

«j log

Xn−j+1,n

Xn−j,n=

1

k

kXj=1

Hτ

„j

k + 1

«Zi,k ,

Hτ is a kernel function such thatZ 1

0

Hτ (u)du = 1.

This statistic is used for instance by Beirlant etal., (Extremes, 1999) to estimate the extreme valueindex γ and by Goegebeur et al. (JSPI, 2010) toestimate the second order parameter ρ.

They proved asymptotic normality of theseestimators under a technical condition on thekernel, denoted by (C1) hereafter and under.For the asymptotic normality of ρ estimators theyuse a third order condition.

15 / 38


Third Order Condition

Third Order Condition (T.O.C)

There exist functions A(x)→ 0 and B(x)→ 0, a secondorder parameter ρ ≤ 0 and a third order parameterβ ≤ 0 such that, for every λ > 0,

limt→∞

log U(tx)−log U(t)−γ log xA(t)

− xρ−1ρ

B(t)=

1

β

„xρ+β − 1

ρ+ β− xρ − 1

ρ

«,

where the functions |A| and |B| are regularly varyingwith index ρ and β respectively.

16 / 38


Links with existing estimators

Link with our framework

Suppose the third order condition and (C1) hold. Ifthe sequence k satisfies

k →∞, n/k →∞, k1/2A(n/k)→∞,

k1/2A2(n/k)→ λA and k1/2A(n/k)B(n/k)→ λB ,

then the random vector

T (R)n :=

“(Rk(τi )/γ)θi , i = 1, . . . , d

”,

satisfies the model i.e. ω−1n (T

(R)n − χnI)

P−→ f (R)(ρ) withχn = 1,

ωn = A(n/k)/γ(1 + oP (1)),

f (R)(ρ) =

„θi

Z 1

0

Hτi (u)u−ρdu, i = 1, . . . , d

«

17 / 38




d = 8, ψδ : D 7→ R \ {0}

ψδ(x1, . . . , x8) = eψδ(x1 − x2, x3 − x4, x5 − x6, x7 − x8), where δ ≥ 0

D = {(x1, . . . , x8) ∈ R8; x1 6= x2, x3 6= x4, and (x5 − x6)(x7 − x8) > 0}.

eψδ : R4 7→ R is given by : eψδ(y1, . . . , y4) =y1

y2

„y4

y3

«δ.

Hτi , i = 1, ..., 8, the statistic T(R)n depends on 16

parameters {(θi , τi ) ∈ (0,∞)2, i = 1, . . . , 8}.

Let θ = (θ1, . . . , θ4) ∈ (0,∞)4 with θ3 6= θ4 and consider

� {θi = θdi/2e, i = 1, . . . , 8} with δ = (θ1 − θ2)/(θ3 − θ4) and

dxe = inf{n ∈ N|x ≤ n}.�τ1 < τ2 ≤ τ3 < τ4, τ5 < τ6 ≤ τ7 < τ8

T(R)n involves 12 free parameters. Z

(R)n = ψδ(T

(R)n ) and

ϕ(R)δ = ψδ ◦ f (R).

18 / 38




Z(R)n does not depend on the unknown parameter.

We can thus define the following family of estimators :

ρ(R)n = ϕ−1

δ 1l{Z (R)n ∈ JR}.

Let denote m(R)A =

“m

(R,i)A , i = 1, ..., 4

”, m

(R)B =

“m

(R,i)B , i = 1, ..., 4

”and v (R) =

“v (R,i), i = 1, ..., 4

”with

m(R,i)A = exp

(θi − 1)

Z 1

0

`Hτ2i−1 (u) + Hτ2i (u)

ú−ρdu

ff,

m(R,i)B = exp

8<:−R 1

0

`Hτ2i−1 (u)− Hτ2i (u)

´ “u−(ρ+β) − u−ρ

”du

βR 1

0

`Hτ2i−1 (u)− Hτ2i (u)

ú−ρdu

9=;,v (R,i) = exp

( R 1

0

`Hτ2i−1 (u)− Hτ2i (u)

´duR 1

0

`Hτ2i−1 (u)− Hτ2i (u)

ú−ρdu

).

19 / 38




Suppose the third order condition and (C1) hold.

Since ϕ(R)δ is bjective and differentiable in ρ then

if the sequence k satisfies

k →∞, n/k →∞, k1/2A(n/k)→∞,


we have

k1/2A(n/k)“ρ(R)

n − ρ”D−→ N

„λA

2γAB(R)

A (δ, ρ)− λBAB(R)B (δ, ρ, β),AV(R)(δ, ρ)

«with

AB(R)A (δ, ρ) =

ϕ(R)δ (ρ)

[ϕ(R)δ ]′(ρ)

log ψδ(m(R)A ), AB(R)

A (δ, ρ, β) =ϕ

(R)δ (ρ)

[ϕ(R)δ ]′(ρ)

log ψδ(m(R)B )

and

AV (R)(δ, ρ) =γϕ

(R)δ (ρ)

[ϕ(R)δ ]′(ρ)

Z 1

0

log2 ϕ(R)δ (v (R)(u))du

20 / 38



Exemples

Hτi = τiuτi−1, i = 1, ..., 8, τi > 1

New estimators of ρ (explicit or not), with Consistency and

Asymptotic normality (consequence of ours theorems)

Examples of explicit estimators

� δ = 1 i.e. θ1 − θ2 = θ3 − θ4. The rv Z(R)n is denoted by Z

(R)n,1 .

ρ(R)n,1 =

τ5ω(1, θ)− τ1Z (R)n,1

ω(1, θ)− Z(R)n,1

1l{Z (R)n,1 ∈ ω(1, θ) • (1, eψ1(τ4, τ1, τ4, τ5))}.

� δ = 0 i.e. θ1 = θ2. The rv Z(R)n is thus denoted by Z

(R)n,2 :

ρ(R)n,2 =

τ4ω(0, θ)− τ1Z (R)n,2

ω(0, θ)− Z(R)n,2

1l{Z (R)n,2 ∈ ω(0, θ) • (1, eψ0(τ4, τ1, τ4, τ5))}.

with ω(δ, θ) = eψδ(θ1(τ1 − τ2), θ2(τ2 − τ4), θ2(τ2 − τ4), θ4(τ6 − τ4))

21 / 38



2. Estimators based on the log-excesses : log Xn−j+1 − log Xn−k

Sk(τ, α) =1

k

kXj=1

Gτ,α

„j

k + 1

«„log

Xn−j+1,n

Xn−k,n

«α, α > 0,

Gτ,α is a positive function.

In the particular case where Gτ,α is constant, this statistic is used byDekkers et al. (Annals of statistics, 1989 ) to estimate γ and

by Fraga et al. . (Portugaliae Mathematica, 2003 ) to estimate

the second order parameter ρ

Ciuperca and Mercadier (Extremes, 2010 ) used the general

statistic to estimate the parameters γ and ρ.

They proved the asymptotic normality under a technical

condition on the function Gτ,α, denoted by (C2) hereafter.

22 / 38




Suppose the third order condition, (C2) hold. If the sequence ksatisfies

k →∞, n/k →∞, k1/2A(n/k)→∞,


then the random vector

T (S)n =

„Sk(τi , αi )

γαi

«θi

, i = 1, ..., d

!

satisfies the model i.e. ω−1n (T

(S)n − χnI)

P−→ f (S)(ρ) with χn = 1,ωn = A(n/k)/γ(1 + oP (1)), and

f (S)(ρ) =

„−θiαi

Z 1

0

Gτi ,αi (u)(log(1/u))αi−1K−ρ(u)du; i = 1, . . . , d

«,

23 / 38




d = 8, the function ψδ and ψ are the same as above

24 free parameters : {(θi , τi , αi ) ∈ (0,∞)3, i = 1, . . . , 8}Let (ζ1, . . . , ζ4) ∈ (0,∞)4 with ζ3 6= ζ4, such that

{θiαi = ζdi/2e, i = 1, . . . , 8} with δ = (ζ1 − ζ2)/(ζ3 − ζ4).

dxe = inf{n ∈ N|x ≤ n}.(τ2i−1, α2i−1) 6= (τ2i , α2i ), for i = 1, . . . , 4 and,

for i = 3, 4, (τ2i−1, α2i−1) ≤ (τ2i , α2i ) where

(x , y) 6= (s, t) means that x 6= s and/or y 6= t and (x , y) ≤ (s, t) means

that x ≤ s and y ≤ t.

T(S)n involves 20 free parameters. Z

(S)n = ψδ(T

(S)n ) and

ϕ(S)δ = ψδ ◦ f (S).

24 / 38




Z(S)n does not depend on the unknown parameter.

We can thus define the following family of estimators :

ρ(S)n = ϕ−1

δ 1l{Z (S)n ∈ JS}.

Using third order condition and (C2), since ϕ(S)δ

is bijective then ρ(S)n is asymptotically Gaussian

(consequence of our theorem)

exemple of weighted function

Consider the weighted function Gτ,α is given defined by :

Gτ,α(u) =gτ−1(u)R 1

0gτ−1(x)(− log x)αdx

, τ ≥ 0, α > 0

where g0(x) = 1 and gτ−1(x) = (τ)(1− xτ−1)/(τ − 1) for τ > 1.

25 / 38



exemples of estimators of ρ

New estimators of ρ (not necessarily explicit) with

Consistency and Asymptotic normality.

Exemples of explicit estimators

� δ = 0 (i.e. ζ1 = ζ2), α1 = α2 = α3 = α4 = 1, τ1 = α5 = α8 = 2

τ4 = α6 = 3. Denoting by Z(S)n,4 the rv Z

(S)n , the estimator of ρ

is given by :

ρ(S)n,4 =

6(Z(S)n,4 + 2)

3Z(S)n,4 + 4

1l{Z (S)n,4 ∈ (−2,−4/3)}.

Consider ω∗, a function depends only on δ and ζ

26 / 38




� δ = 0, α1 = α3 = α4 = 1, τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3.Denoting by Z

(S)n,5 the rv Z

(S)n ,

ρ(S)n,5 =

2(Z(S)n,5 − 2)

2Z(S)n,5 − 1

1l{Z (S)n,5 ∈ (1/2, 2)}.

ρ(S)n,4 and ρ

(S)n,5 are estimators Ciuperca and Mercadier , (Extremes,

2010).

� δ = 1, α1 = α3 = α4 = 1, τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3Z

(S)n,6 the rv Z

(S)n , a new estimator of ρ is given by

ρ(S)n,6 =

3Z(S)n,6 − 4ω∗(1, ζ)

Z(S)n,8 − ω∗(1, ζ)

1l{Z (S)n,6 ∈ ω

∗(1, ζ) • (1/2, 2/3)}.

� δ = 1 (i.e. ζ1 − ζ2 = ζ3 − ζ4), α1 = α2 = α3 = α4 = 1, τ1 = α5 = α8 = 2 and

τ4 = α6 = 3 denoting by Z(S)n,7 the rv Z

(S)n , a nother new estimator of

ρ is given by :

27 / 38




ρ(S)n,7 =

Z(S)n,7 + 4/3ω∗(1, ζ)

2Z(S)n,7 + 4/3ω∗(1, ζ)

1l{Z (S)n,7 ∈ ω

∗(1, ζ) • (−4/3,−2/3)}.

� δ = 1 (i.e. ζ1 − ζ2 = ζ3 − ζ4), α1 = α3 = α4 = 1,

τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3 denoting by Z(S)n,8 the rv Z

(S)n , we

obtain a new estimator of ρ

ρ(S)n,8 =

3Z(S)n,8 − 4ω∗(1, ζ)

Z(S)n,8 − ω∗(1, ζ)

1l{Z (S)n,8 ∈ ω

∗(1, ζ) • (1/2, 2/3)}.

28 / 38


Asymptotic comparaison

Choice of the parameters

We use here the estimator of ρ based on rescaled

log-spacings.

Hτi (u) = (τi )uτi−1, i = 1, ..., 8, τi > 1.

τ1 = 1.25, τ2 = τ3 = 1.75, τ4 = τ8 = 2, τ5 = 1.5, τ6 = τ7 = 1.75 and

θ1 = 0.01, θ3 = 0.02 θ4 = 0.04 and θ2 = θ1 + δ(θ4 − θ3), δ ≥ 0.

How to choose δ ?,

1 minimization of the AMSE is impossible (depends on unknownparameters).

2 We use an upper bound on the AMSE :AMSE ≤ c(γ, λA, λB)π(δ, ρ, β).

3 ρ = β, we minimize the function π in delta and the optimal δ asa function of ρ.

29 / 38


Asymptotic comparaison

Choice of δ

−8 −6 −4 −2 0

02

46

810

ρ

δ

δ=1.5δ =1.8

Fig.: Optimal δ as a function of ρ

δ = 0, 1, 1.5, 1.8,+∞

30 / 38


Illustration on a Burr distribution

Survival function of Burr distribution

Burr(ζ,λ,η) :

1− F (x) = (ζ/(ζ + xη))λ , x > 0, ζ, λ, η > 0,

is of heavy-tailed model with γ = 1/λη

Satisfies the third order condition with ρ = −1/λ and β = ρ,

A(x) = γxρ/(1− xρ) and B(x) = ρxρ/(1− xρ).

n = 5000, γ = ζ = 1, η = 1/λ, ρ = −η and k = 1, ..., 4995,

31 / 38


0 1000 2000 3000 4000 5000

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ = − 0.25

k

AM

SE

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

0 1000 2000 3000 4000 5000

0.0

0.5

1.0

1.5

2.0

2.5

3.0

ρ = − 1

kA

MS

E

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

Fig.: Asymptotic mean squared error of ρ(R)n , ρ = −0.25;−1

32 / 38


0 1000 2000 3000 4000 5000

0.0

0.5

1.0

1.5

2.0

ρ = − 2.5

k

AM

SE

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

0 1000 2000 3000 4000 5000

0.0

0.5

1.0

1.5

2.0

2.5

ρ = − 3

kA

MS

E

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

Fig.: Asymptotic mean squared error of ρ(R)n , ρ = −2.5;−3

33 / 38


0 1000 2000 3000 4000 5000

02

46

810

ρ = − 4

k

AM

SE

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

0 1000 2000 3000 4000 5000

05

1015

20

ρ = − 5

kA

MS

E

δ = 0δ = 1δ = 1.5δ = 1.8δ = + ∞

Fig.: Asymptotic mean squared error of ρ(R)n , ρ = −4;−5

34 / 38


Concluding Remarks

If ρ ≤ −4, the smallest AMSE is obtained with δ = 1.8.

If −3 ≤ ρ ≤ −2.5, the best AMSE is given by δ = +∞.

If ρ ≥ −1, the smallest AMSE is given by δ = 1.5.

� The values {1.5, 1.8,+∞} obtained by minimizing the function πare also of interest to minimize the asymptotic mean-squared

error.

� More generally, the minimization of π should permit to

determine optimal values for the parameters of any estimator of

ρ.

35 / 38


Main references

G. Ciuperca and C. Mercadier. Semi-parametric estimation for heavy taileddistributions. Extremes, 13, 55–87, 2010.

A.L.M. Dekkers, J.H.J. Einmahl, and L. de Haan. A moment estimator forthe index of an extreme-value distribution. Annals of Statistics, 17,1833–1855, 1989.

M.I. Fraga Alves, M.I. Gomes, and L. de Haan. A new class ofsemi-parametric estimators of the second order parameter. PortugaliaeMathematica, 60(2), 193–213, 2003.

Y. Goegebeur, J. Beirlant, and T. de Wet. Kernel estimators for thesecond order parameter in extreme value statistics. Journal of StatisticalPlanning and Inference, 140, 2632–2652, 2010.

B.M. Hill. A simple general approach to inference about the tail of adistribution, Annals of Statistics, 3, 1163–1174, 1975.

36 / 38


37 / 38

Documents

On the estimation of the second order parameter in extreme-value …irma.math.unistra.fr/~gardes/SEMINAIRE/deme.pdf · 2012-03-02 · Strasbourg March 6th 2012. ... (C1) hereafter