On the estimation of the second order parameter in
extreme-value theory
By
El Hadji DEME
Joint work
Laurent GARDES & Stéphane GIRARD
Strasbourg March 6th 2012
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Main results on extreme value theory
Let X1 , . . . , Xn be a sequence of independent copies of
a real random variable (r.v.) X with cumulative
distribution function F . The order statistics
associated to this sample are denoted by :
X1,n ≤ · · · ≤ Xn,n .
Fisher-Tippett-Gnedenko theorem
Under some conditions of regularity on F , there
exists a real parameter γ and two sequences
(an )n≥1 > 0 and (bn )n≥1 ∈ R such that for all x ∈ R,
lim P an−1 (Xn,n − bn ) ≤ x = EVγ (x),
n→∞
(
with
EVγ (x) =
where y+ = max(y , 0).
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“
”
−1/γ
exp −(1 + γx)+
`
´
exp −e −x
if γ 6= 0,
if γ = 0,
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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 be
heavy-tailed whenever, for some γ > 0, its survival
function is of the forme :
1 − F (x) = x −1/γ `F (x) ⇔ U(x) = x γ `U (x)
where U(x) = inf{y : F (y ) ≥ 1 − 1/x} is quantile
function 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 is
regulary varying at infinity with index −1/γ (denoted
1 − F (x) ∈ RV −1/γ ).
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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)
f (x) =
distribution
`
´ “ ”
Γ m+n
m ”−(m+n)/2
m m/2 m/2−1 “
`m´ 2 `n´
x
1+ x
x > 0; m, n > 0
n
n
Γ 2 Γ 2
is heavy-tailed with γ = 2/n and
`F (x) =
´ “ ”
`
„
«−(m+n)/2
Γ m+n
1
m m/2 m/2−1 m
`m´ 2 `n´
x
+
(1 + o(1))
n
n
x
Γ 2 Γ 2
for x → ∞.
Others : |t|, log-gamma, inverse gamma, Fréchet, Burr,...
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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 )
Zi,k = i (log Xn−i+1,n − log Xn−i,n )
the log-excesses
the rescaled log-spacings
Exemples of γ estimators
k
k
1X
1X
Wi,k =
Zi,k , Hill
k i=1
k i=1
0
”2 1−1
“
(1)
M
n,k
1B
C
(1)
γ̂n = Mn,k + 1 − @1 −
A
(2)
2
Mn,k
Hn,k =
Deker et al (1989) with M(α)
n,k =
Estimator, Hill (1975)
Moment Estmator,
k
1X α
Wi,k
k i=1
Others Estimators of γ, received a lot of attention Smith (1987), Csörgő
et al. (1985), Schultze and Steinebach (1996), Kratz and Resnick (1996),
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic behavior
Since F ∈ RV −1/γ , then if k → ∞ and k/n → 0 as n → ∞,
(α)
α
(1)
then Hn,k −→ γ, γ̂n −→ γ and Mn,k /µ(1)
α −→ γ with µα = Γ(α + 1)
P
P
P
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 order
parameter ρ ≤ 0 such that, for every x > 0,
lim
t→∞
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log U(tx) − log U(t) − γ log x
xρ − 1
=
.
A(t)
ρ
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Second Order Condition
Remarks
Under the regularly variation condition
log
U(tx)
→ γ log x
U(t)
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) = Cx γ (1 + Dx ρ + o(x ρ )), (x → ∞)
with C > 0, satisfies the second order condition with
A(x) = ρDx ρ .
Exemples : Frechet, Burr, Generalized Pareto (GP),
|t|,...
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic representation of γ’s estimators
Hill’s estmator
A(n/k)
γ
D
Hn,k = γ + √ Nk +
(1 + oP (1))
1−ρ
k
Moment’s estimator
(1)
(α) D
Mn,k = γ α µ(1)
α +
γ α σα (α)
√ Pk + αγ α−1 µ(2)
α A(n/k)(1 + oP (1))
k
(α)
where Nk and Pk
are asymptotically standard
normal random variables,
µ(2)
α =
p
α 1 − (1 − ρ)α
(1)
and σα
= Γ(2α + 1) − Γ2 (α + 1).
α
ρ (1 − ρ)
(1)
The Hill’s estimator Hn,k correspond to Mn,k
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic normality
√
kA(n/k) → λ ∈ R, then
„
«
λ
D
− γ) −→ N
, γ2
1−ρ
if k, n → ∞ with k/n → 0 and
√
k(Hn,k
and
√
D
(α)
k(Mn,k − γ α µ(1)
α ) −→ N
„
“
”2 «
α (1)
λαγ α−1 µ(2)
α , γ σα
Comment
ρ controls the bias of the estimators of γ
How to estimate the second order tail parameter ρ ?
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
New family of estimators for the second order parameter
The model
a random vector in Rd drawn from
the sample X1 , . . . , Xn .
The statistics can always be expanded as :
Tn = Tn (X1 , ..., Xn ) :
ωn−1 (Tn − χn I) −→ f (ρ)
P
where
I = t (1, . . . , 1) ∈ Rd ,
and ωn : random variables,
a random vector,
−
f : R → Rd : a function continuously differentiable
in a neighborhood of ρ (independent of γ).
χn
ξn ∈ Rd :
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
General approach
Notations
ψ : Rd → R such that
Invariance properties (Inv-prop)
ψ(x + λI) = ψ(x)
ψ(λx) = ψ(x)
for all x ∈ Rd and λ ∈ R,
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 is
a bijection from J0 to J.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
The estimator
Clearly
by the invariance and the regularity properties
ψ(ωn−1 (Tn − χn I)) = ψ(Tn ) −→ ψ(f (ρ)).
P
Zn = ψ(Tn ) ≈ ϕ(ρ).
Under the bijection property, our family of
estimators of the second order parameter is thus
defined by :
ρ̂n = ϕ−1 (Zn )1l{Zn ∈ J}.
1lA
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is the indicator function of the set A.
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic properties
Theorems
Suppose that Inv-prop, Reg-prop and Bij-prop hold
then
P
ρ̂n −→ ρ as n → ∞
if there exist a sequence vn → ∞, a function,
m : R− → Rd and a d × d matrix Σ such that
D
vn (ωn−1 (Tn − χn I) − f (ρ)) −→ Nd (m(ρ), Σ)
then
D
vn (ρ̂n − ρ) −→ N
2
mψ (ρ) σψ
(ρ)
, 0
0
ϕ (ρ) (ϕ (ρ))2
ϕ0 (ρ) = t f 0 (ρ)∇ψ(f (ρ)),
mψ (ρ) = t m(ρ)∇ψ(f (ρ)),
2
σψ
(ρ) = t ∇ψ(f (ρ)) Σ ∇ψ(f (ρ)).
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!
with,
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
1. Estimators based on rescaled log-spacings : log Xn−j+1 − log Xn−j
Kernel estimators
Rk (τ ) =
«
«
„
„
k
k
Xn−j+1,n
1X
1X
j
j
Hτ
j log
=
Hτ
Zi,k ,
k j=1
k +1
Xn−j,n
k j=1
k +1
1
Z
Hτ
is a kernel function such that
Hτ (u)du = 1.
0
This statistic is used for instance by Beirlant et
al., (Extremes, 1999) to estimate the extreme value
index γ and by Goegebeur et al. (JSPI, 2010) to
estimate the second order parameter ρ.
They proved asymptotic normality of these
estimators under a technical condition on the
kernel, denoted by (C1) hereafter and under.
For the asymptotic normality of ρ estimators they
use a third order condition.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Third Order Condition
Third
There
order
β≤0
Order Condition (T.O.C)
exist functions A(x) → 0 and B(x) → 0, a second
parameter ρ ≤ 0 and a third order parameter
such that, for every λ > 0,
lim
t→∞
log U(tx)−log U(t)−γ log x
A(t)
B(t)
−
x ρ −1
ρ
=
1
β
„
«
x ρ+β − 1
xρ − 1
−
,
ρ+β
ρ
where the functions |A| and |B| are regularly varying
with index ρ and β respectively.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Links with existing estimators
Link with our framework
Suppose the third order condition and (C1) hold. If
the sequence k satisfies
k → ∞, n/k → ∞, k 1/2 A(n/k) → ∞,
k 1/2 A2 (n/k) → λA and k 1/2 A(n/k)B(n/k) → λB ,
then the random vector
“
”
Tn(R) := (Rk (τi )/γ)θi , i = 1, . . . , d ,
P
satisfies the model i.e. ωn−1 (Tn(R) − χn I) −→
f (R) (ρ) with
χn = 1,
ωn = A(n/k)/γ(1 + oP (1)),
„ Z
f (R) (ρ) = θi
1
0
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Hτi (u)u −ρ du, i = 1, . . . , d
«
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
d = 8, ψδ : D 7→ R \ {0}
ψδ (x1 , . . . , x8 ) = ψeδ (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}.
„ «δ
y1 y4
ψeδ : R4 7→ R is given by : ψeδ (y1 , . . . , y4 ) =
.
y2 y3
(R)
Hτi , i = 1, ..., 8, the statistic Tn
depends on 16
parameters {(θi , τi ) ∈ (0, ∞)2 , i = 1, . . . , 8}.
Let θ̃ = (θ̃1 , . . . , θ̃4 ) ∈ (0, ∞)4 with θ̃3 6= θ̃4 and consider
{θi = θ̃di/2e , i = 1, . . . , 8} with δ = (θ̃1 − θ̃2 )/(θ̃3 − θ̃4 ) and
dxe = inf{n ∈ N|x ≤ n}.
τ1 < τ2 ≤ τ3 < τ4 , τ5 < τ6 ≤ τ7 < τ8
(R)
(R)
Tn involves 12 free parameters. Zn
(R)
ϕδ = ψδ ◦ f (R) .
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(R)
= ψδ (Tn ) and
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
(R)
Zn
does not depend on the unknown parameter.
We can thus define the following family of estimators :
(R)
ρ(R)
= ϕ−1
∈ JR }.
n
δ 1l{Zn
“
”
“
”
(R)
(R,i)
(R)
(R,i)
Let denote mA = mA , i = 1, ..., 4 , mB = mB , i = 1, ..., 4
“
”
and v (R) = v (R,i) , i = 1, ..., 4 with
(R,i)
mA
1
Z
= exp (θ̃i − 1)
ff
`
´ −ρ
Hτ2i−1 (u) + Hτ2i (u) u du ,
0
” 9
8 R1`
´“
< 0 Hτ2i−1 (u) − Hτ2i (u) u −(ρ+β) − u −ρ du =
(R,i)
mB = exp −
,
´
R1`
:
;
β 0 Hτ2i−1 (u) − Hτ2i (u) u −ρ du
( R1`
)
´
Hτ2i−1 (u) − Hτ2i (u) du
(R,i)
0
v
= exp R 1 `
.
´
Hτ2i−1 (u) − Hτ2i (u) u −ρ du
0
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
Suppose the third order condition and (C1) hold.
(R)
Since ϕδ is bjective and differentiable in ρ then
if the sequence k satisfies
k → ∞, n/k → ∞, k 1/2 A(n/k) → ∞,
k 1/2 A2 (n/k) → λA and k 1/2 A(n/k)B(n/k) → λB ,
we have
„
«
“
”
λA
D
(R)
(R)
k 1/2 A(n/k) ρ̂(R)
ABA (δ, ρ) − λB ABB (δ, ρ, β), AV (R) (δ, ρ)
n − ρ −→ N
2γ
with
(R)
(R)
ABA (δ, ρ) =
ϕδ (ρ)
(R)
[ϕδ ]0 (ρ)
(R)
and
(R)
AV (R) (δ, ρ) =
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(R)
(R)
log ψ̃δ (mA ), ABA (δ, ρ, β) =
γϕδ (ρ)
(R)
[ϕδ ]0 (ρ)
1
Z
(R)
ϕδ (ρ)
(R)
[ϕδ ]0 (ρ)
log2 ϕδ (v (R) (u))du
0
(R)
log ψ̃δ (mB )
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Exemples
Hτi = τi u τi −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
(R)
(R)
δ = 1 i.e. θ̃1 − θ̃2 = θ̃3 − θ̃4 . The rv Zn is denoted by Zn,1 .
(R)
(R)
ρ̂n,1 =
τ5 ω(1, θ̃) − τ1 Zn,1
(R)
ω(1, θ̃) − Zn,1
(R)
1l{Zn,1 ∈ ω(1, θ̃) • (1, ψe1 (τ4 , τ1 , τ4 , τ5 ))}.
(R)
δ = 0 i.e. θ̃1 = θ̃2 . The rv Zn
(R)
is thus denoted by Zn,2 :
(R)
(R)
ρ̂n,2 =
τ4 ω(0, θ̃) − τ1 Zn,2
ω(0, θ̃) −
(R)
Zn,2
(R)
1l{Zn,2 ∈ ω(0, θ̃) • (1, ψe0 (τ4 , τ1 , τ4 , τ5 ))}.
with ω(δ, θ̃) = ψeδ (θ̃1 (τ1 − τ2 ), θ̃2 (τ2 − τ4 ), θ̃2 (τ2 − τ4 ), θ̃4 (τ6 − τ4 ))
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
2. Estimators based on the log-excesses : log Xn−j+1 − log Xn−k
Sk (τ, α) =
«„
„
«α
k
Xn−j+1,n
1X
j
log
Gτ,α
, α > 0,
k j=1
k +1
Xn−k,n
Gτ,α is a positive function.
In the particular case where Gτ,α is constant, this statistic is used by
Dekkers 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.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
Suppose the third order condition, (C2) hold. If the sequence k
satisfies
k → ∞, n/k → ∞, k 1/2 A(n/k) → ∞,
k 1/2 A2 (n/k) → λA and k 1/2 A(n/k)B(n/k) → λB ,
then the random vector
Tn(S)
„
=
Sk (τi , αi )
γ αi
(S)
!
«θi
, i = 1, ..., d
satisfies the model i.e. ωn−1 (Tn − χn I) −→ f (S) (ρ) with χn = 1,
ωn = A(n/k)/γ(1 + oP (1)), and
„
«
Z 1
(S)
αi −1
f (ρ) = −θi αi
Gτi ,αi (u)(log(1/u))
K−ρ (u)du; i = 1, . . . , d ,
0
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P
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
d = 8, the function ψδ and ψ̃ 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
{θi αi = 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.
(S)
(S)
Tn involves 20 free parameters. Zn
(S)
ϕδ = ψδ ◦ f (S) .
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(S)
= ψδ (Tn ) and
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Link with our framework
(S)
Zn
does not depend on the unknown parameter.
We can thus define the following family of estimators :
−1
(S)
ρ(S)
∈ JS }.
n = ϕδ 1l{Zn
(S)
Using third order condition and (C2), since ϕδ
(S)
is bijective then ρn is asymptotically Gaussian
(consequence of our theorem)
exemple of weighted function
Consider the weighted function Gτ,α is given defined by :
Gτ,α (u) = R 1
0
gτ −1 (u)
gτ −1 (x)(− log x)α dx
, τ ≥ 0, α > 0
where g0 (x) = 1 and gτ −1 (x) = (τ )(1 − x τ −1 )/(τ − 1) for τ > 1.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
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
(S)
(S)
τ4 = α6 = 3. Denoting by Zn,4 the rv Zn , the estimator of ρ
is given by :
(S)
(S)
ρ̂n,4 =
6(Zn,4 + 2)
(S)
3Zn,4 + 4
(S)
1l{Zn,4 ∈ (−2, −4/3)}.
Consider ω ∗ , a function depends only on δ and ζ
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Exemples of explicit estimators
δ = 0, α1 = α3 = α4 = 1, τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3.
(S)
(S)
Denoting by Zn,5 the rv Zn ,
(S)
(S)
ρ̂n,5 =
(S)
(S)
ρ̂n,4 and ρ̂n,5 are estimators
2(Zn,5 − 2)
(S)
2Zn,5 − 1
(S)
1l{Zn,5 ∈ (1/2, 2)}.
Ciuperca and Mercadier , (Extremes,
2010).
δ = 1, α1 = α3 = α4 = 1, τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3
(S)
(S)
Zn,6 the rv Zn , a new estimator of ρ is given by
(S)
(S)
ρ̂n,6 =
3Zn,6 − 4ω ∗ (1, ζ)
(S)
Zn,8 − ω ∗ (1, ζ)
1l{Zn,6 ∈ ω ∗ (1, ζ) • (1/2, 2/3)}.
(S)
δ = 1 (i.e. ζ1 − ζ2 = ζ3 − ζ4 ), α1 = α2 = α3 = α4 = 1, τ1 = α5 = α8 = 2 and
(S)
(S)
τ4 = α6 = 3 denoting by Zn,7 the rv Zn , a nother new estimator of
ρ is given by :
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Link with existing estimators
Exemples of explicit estimators
(S)
(S)
ρ̂n,7 =
Zn,7 + 4/3ω ∗ (1, ζ)
(S)
2Zn,7 + 4/3ω ∗ (1, ζ)
1l{Zn,7 ∈ ω ∗ (1, ζ) • (−4/3, −2/3)}.
(S)
δ = 1 (i.e. ζ1 − ζ2 = ζ3 − ζ4 ), α1 = α3 = α4 = 1,
(S)
(S)
τ1 = τ4 = α2 = α5 = α8 = 2 and α6 = 3 denoting by Zn,8 the rv Zn , we
obtain a new estimator of ρ
(S)
(S)
ρ̂n,8 =
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3Zn,8 − 4ω ∗ (1, ζ)
(S)
Zn,8 − ω ∗ (1, ζ)
1l{Zn,8 ∈ ω ∗ (1, ζ) • (1/2, 2/3)}.
(S)
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic comparaison
Choice of the parameters
We use here the estimator of ρ based on rescaled
log-spacings.
Hτi (u) = (τi )u τi −1 , i = 1, ..., 8, τi > 1.
τ1 = 1.25, τ2 = τ3 = 1.75, τ4 = τ8 = 2, τ5 = 1.5, τ6 = τ7 = 1.75 and
θ̃1 = 0.01, θ˜3 = 0.02 θ̃4 = 0.04 and θ̃2 = θ̃1 + δ(θ̃4 − θ̃3 ), δ ≥ 0.
How to choose δ ?,
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1
minimization of the AMSE is impossible (depends on unknown
parameters).
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 as
a function of ρ.
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Asymptotic comparaison
δ =1.8
δ=1.5
0
2
4
δ
6
8
10
Choice of δ
−8
−6
−4
−2
ρ
Fig.: Optimal δ as a function of ρ
δ = 0, 1, 1.5, 1.8, +∞
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0
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Illustration on a Burr distribution
Survival function of Burr distribution
Burr(ζ,λ,η) :
1 − F (x) = (ζ/(ζ + x η ))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) = γx ρ /(1 − x ρ ) and B(x) = ρx ρ /(1 − x ρ ).
n = 5000, γ = ζ = 1, η = 1/λ, ρ = −η and k = 1, ..., 4995,
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
3.0
ρ = −1
3.0
ρ = − 0.25
2.5
2.0
1.5
AMSE
1.0
1.5
0.5
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
0.0
0.0
0.5
1.0
AMSE
2.0
2.5
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
0
1000
2000
3000
4000
5000
0
1000
k
2000
3000
k
(R)
Fig.: Asymptotic mean squared error of ρ̂n , ρ = −0.25; −1
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4000
5000
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
ρ = −3
2.5
2.0
ρ = − 2.5
2.0
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
AMSE
0.0
0.0
0.5
0.5
1.0
1.0
AMSE
1.5
1.5
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
0
1000
2000
3000
4000
5000
0
1000
k
2000
3000
k
(R)
Fig.: Asymptotic mean squared error of ρ̂n , ρ = −2.5; −3
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4000
5000
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
ρ = −5
20
10
ρ = −4
15
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
10
AMSE
0
0
2
5
4
AMSE
6
8
δ=0
δ=1
δ = 1.5
δ = 1.8
δ = +∞
0
1000
2000
3000
4000
5000
0
1000
k
2000
3000
k
(R)
Fig.: Asymptotic mean squared error of ρ̂n , ρ = −4; −5
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4000
5000
Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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 π
are 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.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
Main references
G. Ciuperca and C. Mercadier. Semi-parametric estimation for heavy tailed
distributions. Extremes, 13, 55–87, 2010.
A.L.M. Dekkers, J.H.J. Einmahl, and L. de Haan. A moment estimator for
the 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 of
semi-parametric estimators of the second order parameter. Portugaliae
Mathematica, 60(2), 193–213, 2003.
Y. Goegebeur, J. Beirlant, and T. de Wet. Kernel estimators for the
second order parameter in extreme value statistics. Journal of Statistical
Planning and Inference, 140, 2632–2652, 2010.
B.M. Hill. A simple general approach to inference about the tail of a
distribution, Annals of Statistics, 3, 1163–1174, 1975.
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Outline Extreme-Value Theory New family of estimators for the second order parameter Asymptotic properties Link with existing estimat
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