Estimation of second order parameters using probability weighted moments

DOI:10.1051/ps/2010017 www.esaim-ps.org

ESTIMATION OF SECOND ORDER PARAMETERS USING PROBABILITY WEIGHTED MOMENTS

Julien Worms

and Rym Worms

Abstract. The P.O.T. method (Peaks Over Threshold) consists in using the generalized Pareto dis- tribution (GPD) as an approximation for the distribution of excesses over a high threshold. In this work, we use a refinement of this approximation in order to estimate second order parameters of the model using the method of probability-weighted moments (PWM): in particular, this leads to the introduction of a new estimator for the second order parameter ρ, which will be compared to other recent estimators through some simulations. Asymptotic normality results are also proved. Our new estimator ofρlooks especially competitive when|ρ|is small.

Mathematics Subject Classification. 62G32, 60G70.

Received February 27, 2009. Revised April 16, 2010.

1. Introduction

In statistical extreme value theory, one is often interested by the estimation of the far tail of a distribution.

The quality of this estimation especially depends on knowledge about the so-called tail index γ = γ(F) of the underlying model F, which is the shape parameter of the generalized Pareto distribution (GPD) with distribution function (d.f.)

Gγ,σ(x) =

⎧⎨

⎩ 1−

1 + ^γx_{σ −γ}¹

, forγ= 0 1−exp

−^x_σ

, forγ= 0.

The GPD appears as the limiting d.f. of excesses over a high thresholdudefined forx≥0 by Fu(x) :=P(X−u≤x|X > u), whereX has d.f. F.

Keywords and phrases. Extreme values, domain of attraction, excesses, generalized Pareto distribution, probability-weighted moments, second order parameter, third order condition.

1 Universit´e de Versailles Saint-Quentin, Laboratoire de Math´ematiques de Versailles (CNRS UMR 8100), UFR de Sciences, Bˆat. Fermat, 45 Av. des Etats-Unis, 78035 Versailles Cedex, France. [email protected]

2 Universit´e Paris Est Cr´eteil, Laboratoire d’Analyse et de Math´ematiques Appliqu´ees (CNRS UMR 8050), 61 Av. du G^lde Gaulle, 94010 Cr´eteil Cedex, France. [email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2012

It was established in Pickands’ and Balkema and de Haan’s results (see [17] and [1]) thatF is in the domain of attraction of an extreme value distribution with shape parameterγ if and only if

u→lims₊(F) sup

0<x<s₊(F)−u

Fu(x)−Gγ,σ(u)(x)= 0

for some positive scaling function σ(u) depending onu, where s₊(F) = sup{x:F(x)<1}. Since the far tail of the unknown underlying distributionF is closely tied to the d.f. of excesses over a high threshold, accurate modelisation of the distribution of excesses is an important topic.

In what follows, we suppose thatF is twice differentiable and that its inverseF⁻¹ exists. LetV andA be the two functions defined by

V(t) =F⁻¹(e^−t) and A(t) = V(lnt) V(lnt) −γ.

We suppose the following first and second order conditions hold (RVρ below stands for the set of regularly varying functions with coefficient of variationρ):

limt→+∞A(t) = 0 (1.1)

A is of constant sign at∞and there exists ρ≤0 such that |A| ∈RVρ. (1.2) Under these assumptions, it is proved in [18] that if (un) is a sequence of thresholds such that un →s₊(F) as n→ ∞, then we have the following development

Fu_n(σny)− Gγ(y) = anDγ,ρ(y) + o(an), asn→+∞, (1.3) for ally, whereGγ(y) := 1−Gγ,1(y),

σn :=σ(un) =V

V⁻¹(un) , an:=A

e^V−1(un)

,

Dγ,ρ(y) :=

C₀,ρ(y), ifγ= 0, Cγ,ρ

1

γln(1 +γy)

ifγ= 0, and

Cγ,ρ(y) := e^−(1+γ)yIγ,ρ(y) and Iγ,ρ(y) :=

y 0 e^γu

u

0 e^ρsdsdu.

The idea of the present work is that, according to the result (1.3), Gγ,σ(u)(x) +anDγ,ρ(x/σ(u)) is a better approximation ofFu(x) thanGγ,σ(u)(x) alone: this is the starting point of our method for the estimation of the second order parametersan andρ.

The estimation ofρis of great importance (for instance for the determination of the optimal sample fraction needed in the estimation of the tail index or of high quantiles) and has been studied by several authors during the last 15 years. Many of the existing estimators ofρare based on functionals of the moment statisticsMn^(j)(kn) = kn−1k_n

i=1(lnXn−i+1:n−lnXn−k_n:n)^j, where Xi:n denotes the ith ascending order statistic associated to a sample (X₁, . . . , Xn) of d.f. F, andknis the number of excesses retained for the estimation (wherekn→ ∞but slower thann). We can cite those introduced in [3,7–9,13,14,16].

The estimation ofan can also be very useful. For instance, if we consider the estimation of the tail indexγ by the PWM estimator, it was proved in [4] that the main component of the bias of this estimator is of orderan. An estimation of the latter parameter could thus be used to reduce this bias. Moreover, it was proved in [21]

that, in the caseρ= 0, the GPDGγ+a_n,σ_n is a better approximation of the distribution of the excessesFuthan Gγ,σ_n; this is called the penultimate approximation, and the estimation of an is important in this framework.

In this work, we use the probability-weighted moments (PWM) techniques introduced by Hosking and Wallis in [15] to estimate the second order parameters ρ and an, as well as the scale parameter σn. The proposed estimators are based on an “external” estimation ofγ: a similar procedure was undertaken in [8].

Under conditions (1.1) and (1.2), it is known (see [18]) that

∀x∈R, lim

t→+∞

V(t+x)−V(t) V(t) −x

0 e^γsds

A(e^t) =Iγ,ρ(x). (1.4)

In order to achieve asymptotic normality results, we will need the following third order condition which specifies the rate of convergence in (1.4):

∀x∈R, lim

t→+∞

1 B(e^t)

⎛

⎝

V(t+x)−V(t) V(t) −x

0 e^γsds

A(e^t) −Iγ,ρ(x)

⎞

⎠=Rγ,ρ,β(e^x), (1.5)

where

Rγ,ρ,β(e^x) :=

x 0

e^γs s

0

e^ρz z

0

e^βydydzds

and the functionB tends to 0 and is of constant sign at∞and|B| ∈RVβ, for some β≤0. This condition has been introduced in [8] and studied in more details in [10].

Remark 1.1. We can choose, in our regular case,B(t) :=^tA_A(⁽_t^t₎⁾−ρ(F should then be three times differentiable).

In Section 2, we introduce the new model based on (1.3) and the associated probability-weighted moments and establish the asymptotic normality of their estimators. In Section3, we present our estimators forρ,anand σnand establish their asymptotic normality, first whenγ is supposed to be known and then for the unknownγ case. Section4 contains some simulations illustrating the behaviour of our new estimator ofρ, by comparison to two other recent estimators.

2. Estimators for the probability-weighted moments 2.1. Definition of the probability-weighted moments

In [15], Hosking and Wallis introduced the PWM method in order to define estimators of γ and σn based on a sample with d.f. supposed to be an exact GPD. These estimators were obtained through a substitution method based on the following quantities, the probability-weighted moments

νj =E(XG^j_γ,σn(X))

where j ∈ {0,1}and X has d.f. Gγ,σ_n. The results were generalized in [4] to the case where the sample was only supposed to be in the domain of attraction of a GPD.

In this work, more parameters are considered, and we noteθn= (γ, σn, an, ρ). According to the asymptotic result (1.3), we define our extended model by the distribution function

Bθ_n(x) =Gγ,σ_n(x)−anDγ,ρ

x σn

, for allx, and consider the corresponding first three PWM as follows, whereX has d.f. Bθ_n

˜

vj=E(XB^jθ_n(X)), forj∈ {0,1,2}.

It is easy to see that

˜ vj =

_+∞

0

(1−Bθ_n(x))^j+1 j+ 1 dx.

Note that for all the PWM and their estimators, the subscriptnis ommited in order to simplify the notations.

The following lemmas provide expressions of these PWM as functions of the parameters.

Lemma 2.1. Forj∈ {0,1,2}, ρ≤0 and−1< γ <1, νj= σn

(j+ 1)(j+ 1−γ)· Lemma 2.2.

˜

v₀=ν₀+an

_+∞

0 Dγ,ρ

x σn

dx:=v₀, and, for j∈ {1,2},

˜

vj =νj+an

_+∞

0 G^j_γ,σn(x)Dγ,ρ

x σn

dx+o(an) :=vj+o(an).

Lemma 2.3. Forj∈ {0,1,2}, ρ≤0 and−1< γ <1, we havev˜j=vj+o(an)where vj:= σn

(j+ 1)(j+ 1−γ)+anσn

uj

and uj:= (j+ 1)(j+ 1−γ)(j+ 1−γ−ρ).

In the sequel, we will use the quantities v₀, v₁, v₂ (rather than ˜v₀,˜v₁,v˜₂) in order to estimate ρ, an, σn, by a classical substitution method, relying on Lemma 2.3 above which gives the relations between the two triplets of parameters. The proof of lemmas 2.2 and 2.3 are given in Appendices A.1 and A.2 respectively. That of Lemma2.1 can be found in [4] forj= 0,1, the casej= 2 being similar.

2.2. Asymptotic behaviour of the estimators of the probability-weighted moments

Let (X₁, . . . , Xn) ben i.i.d. random variables with distribution functionF, and X_1:n, . . . , Xn:n denote the corresponding order statistics. For a given thresoldun, we introduceY₁,N_n, . . . , YN_n,N_ntheNn excesses overun, in ascending order,i.e.

Yj,N_n=Xn−N_n+j:n−un where Nn=n

i=1IX_i>u_n.

According to (1.3), the distribution Bθ_n is then likely to be a good approximation for the distributionFu_n of Y₁,N_n, . . . , YN_n,N_n. This method is of the peak-over-threshold (POT) type.

Remark 2.4. Note thatNn is binomial distributed with meann(1−F(un)) which will be chosen as going to infinity: consequently,Nn→ ∞andNn/(n(1−F(un))→1 in probability asn→ ∞.

Definition 2.5. Forj ∈ {0,1,2}, define the estimator ofvj by ˆ

vj :=

_+∞

0

(1−An,u_n(x))^j+1 j+ 1 dx, where,

An,u_n(x) := 1 Nn

N_n

i=1

I_{Y_i≤x}.

It follows that, conditionnally onNn =kn,

ˆ vj= 1

j+ 1

k_n

i=1

1−i−1 kn

j+1

−

1− i kn

j+1 Yi,k_n.

Letbn =B(e^V−1(un)), where functionB appears in the third order condition (1.5).

Theorem 2.6. Under assumptions (1.1),(1.2)and(1.5), with −1< γ <1/2, and if

nlim→∞

n(1−F(un))anbn = λ₁, λ₁∈R, (2.1)

nlim→∞

n(1−F(un))a²_n = λ₂, λ₂≥0, (2.2)

nlim→∞

n(1−F(un))an = ∞, (2.3)

we have, for almost all sequenceskn→+∞, conditionally on Nn=kn,

kn

⎛

⎜⎜

⎜⎜

⎜⎜

⎝ v₀ σn − v₀

σn

v₁ σn

− v₁ σn

v₂ σn − v₂

σn

⎞

⎟⎟

⎟⎟

⎟⎟

⎠

−→ Nd 3(λ₁C,Γ),

where

Γ =

⎛

⎜⎝

((1−2γ)(1−γ)²)⁻¹ ((2−γ)(1−γ)(2−2γ))⁻¹ ((3−γ)(1−γ)(3−2γ))⁻¹ ((2−γ)(1−γ)(2−2γ))⁻¹ ((3−2γ)(2−γ)²)⁻¹ ((2−γ)(3−γ)(4−2γ))⁻¹ ((3−γ)(1−γ)(3−2γ))⁻¹ ((2−γ)(3−γ)(4−2γ))⁻¹ ((5−2γ)(3−γ)²)⁻¹

⎞

⎟⎠ (2.4)

and

C=

⎛

⎝ c⁰_γ,ρ,β c¹_γ,ρ,β c²_γ,ρ,β

⎞

⎠ wherec^j_γ,ρ,β= ((j+ 1)(j+ 1−γ)(j+ 1−γ−ρ)(j+ 1−γ−ρ−β))⁻¹.

Proof of Theorem 2.6. Note that

An,u_n

=d Fu_n+√1 kn

αk_n◦Fu_n,

whereαk_n is the uniform empirical process based onkn i.i.d. random variables uniformly distributed on [0,1].

We have, forj∈ {0,1,2}, vj

σn

− vj

σn

= vj

σn

− νj

σn

−an

uj

= T_j,k¹ _n− 1

√kn

T_j,k² _n+ 1

√kn

T_j,k³ _n−an

uj

,

where, T_j,k¹ _n = 1

j+ 1 _+∞

0

(Fu_n(σny))^j+1−(Gγ(y))^j+1 dy Tj,k² _n =

_+∞

0 [αk_n◦Fu_n(σny)] (Fu_n(σny))^j dy Tj,k³ _n =

⎧⎨

⎩ _+∞

0

₁

0 (1−t)^√1_k

n[αk_n◦Fu_n]²(σny).j

Fu_n(σny)−^√t_k

nαk_n◦Fu_n(σny) j−1

dt dy ifj∈ {1,2}

0 ifj= 0.

This is indeed straightforward for j = 0, whereas for j ∈ {1,2} we use a Taylor expansion, as in the proof of Theorem 1 in [5] (p. 850), with power functions instead of their general weight functions, which have to be null at zero³.

The following lemma concerns the termsT_j,k¹ _n and will be proved in Appendix A.3.

Lemma 2.7. Under the assumptions of Theorem 2.6, for j∈ {0,1,2}, T_j,k¹ _n−an

uj

=c^j_γ,ρ,βanbn+o(anbn).

T₀²_,kn has been studied in [4]. The other terms T_j,k² _n and T_j,k³ _n, for j ∈ {1,2}, have been treated in [5] (see pp. 851–853), in a more general framework. The results are stated in the following lemma and the proofs remain valid under our slightly different assumptions (where the role of the condition√

knan→λis replaced here by

√kna²_n→λ).

Lemma 2.8. Under the assumptions⁴ of Theorem2.6, for j∈ {0,1,2}, as n→ ∞, T_j,k³ _n−→^P 0 and T_j,k² _n−→^d

₁

0 t^−γ−1t^jB(t) dt, whereBis a Brownian Bridge on[0,1]. Moreover, the vector of coordinates₁

0 t^−γ−1t^jB(t) dtwithj∈ {0,1,2}

has a multivariate normal distribution with mean0 and covariance matrix Γdefined by (2.4).

We deduce from these lemmas that kn

vj

σn

− vj

σn

=d c^j_γ,ρ,β

kn anbn+Z_n^j +o_P(1),

where, using [19] (p. 18), the vector of coordinates Zn^j (forj ∈ {0,1,2}) converges in distribution toN₃(0,Γ).

The statement of Theorem2.6follows by the assumption√

kn anbn→λ₁.

Remark 2.9. The third order condition is not used to prove Lemma2.8. This implies that the consistence of the vector of coordinates ˆvj/σn could be obtained under weaker assumptions.

3. Asymptotic normality of the PWM estimators of the parameters 3.1. Asymptotic normality for known γ

From now on we will use the following notations:

Vn= (v₀, v₁, v₂)^t, Vn = (v₀,v₁,v₂)^t.

3This fact exludes the casej= 0, where the weight function is identically equal to 1, from their study.

4The restrictionγ <¹₂ comes from the study ofT_0,k² _n.

The expressions of the probability weighted-moments as functions of the parameters ρ, an, σn are stated in Lemma2.3. Elementary calculus leads to the following equations (recall thatuj= (j+1)(j+1−γ)(j+1−γ−ρ) forj∈ {0,1,2}):

ρ=φ₁,γ(Vn), an=φ₂,γ(Vn), σn=φ₃,γ(Vn) where

φ₁,γ: (x, y, z)→ (1−γ)²x−4(2−γ)²y+ 3(3−γ)²z (1−γ)x−4(2−γ)y+ 3(3−γ)z

φ₂,γ: (x, y, z)→ 2((1−γ)x−2(2−γ)y)((1−γ)x−3(3−γ)z)(2(2−γ)y−3(3−γ)z)

((1−γ)x−4(2−γ)y+3(3−γ)z) (6(3−γ)z[(1−γ)x−2(2−γ)y]−2(2−γ)y[(1−γ)x−3(3−γ)z]) φ₃,γ: (x, y, z)→ 6(3−γ)z[(1−γ)x−2(2−γ)y]−2(2−γ)y[(1−γ)x−3(3−γ)z]

(1−γ)x−4(2−γ)y+ 3(3−γ)z ·

First assuming that the first order parameter γ is known (the case γ unknown will be handled in the next section), we can then define our estimators of the parametersρ,an, andσn as:

⎛

⎝ ρˆγ

ˆ an,γ

ˆ σn,γ

⎞

⎠:=

⎛

⎝ φ₁,γ( ˆVn) φ₂,γ( ˆVn) φ₃,γ( ˆVn)

⎞

⎠ hence

⎛

⎝ ρˆγ

ˆ an,γ

ˆ σn,γ/σn

⎞

⎠=

⎛

⎝ φ₁,γ( ˆVn/σn) φ₂,γ( ˆVn/σn) φ₃,γ( ˆVn/σn)

⎞

⎠.

Proving the asymptotic normality of these estimators by the delta-method (see [20] for instance) would be straightforward if the functionsφj,γ were well-defined at the limit

vγ := lim

n→∞

Vn

σn

=

(1−γ)⁻¹,(2(2−γ))⁻¹,(3(3−γ))^{−1 t}.

However this is not the case here, and the proof needs more care than it seems at first glance.

Proposition 3.1. Suppose that −1< γ <1/2is known, and assumptions(1.1),(1.2),(1.5), (2.1)–(2.3) hold.

Then for almost all sequenceskn→+∞, we have, conditionally on Nn=kn: knan

⎛

⎝ ρˆγ−ρ ˆ

an,γ/an−1 a⁻¹_n (ˆσn,γ/σn−1)

⎞

⎠ −→ N^d ₃(λ₁∇HγC,∇HγΓHγ∇^t) (3.1)

where∇ andHγ are3×3 matrices defined in the proof of this proposition, andλ₁ in (2.1).

Proof of Proposition 3.1. LetHγ denote the matrix

Hγ :=

⎛

⎝ 1−γ 0 0

0 2(2−γ) 0

0 0 3(3−γ)

⎞

⎠

and let us define the following functions

ψ₁,γ : (x, y, z) → (1−γ)x−2(2−γ)y+ (3−γ)z x−2y+z

ψ₂: (x, y, z) → 2(x−y)(x−z)(y−z) (x−2y+z)(2z(x−y)−y(x−z)) ψ₃: (x, y, z) → 2z(x−y)−y(x−z)

x−2y+z ·

If U denotes the subset of R³ on which ψ₂ is defined, we have φ₁,γ(u) =ψ₁,γ(Hγu),φ₂,γ(u) =ψ₂(Hγu) and φ₃,γ(u) =ψ₃(Hγu), for everyu= (x, y, z)^t∈U. In addition note that ife:= (1,1,1)^t, then for everyu∈U we have

ψ₁,γ(u+e) =ψ₁,γ(u), ψ₂(u+e) = ˜ψ₂(u)(1 +d₂(u)/d₁(u))⁻¹, ψ₃(u+e) =ψ₃(u) + 1, (3.2) whered₁(x, y, z) =x−2y+z,d₂(x, y, z) = 2z(x−y)−y(x−z) and

ψ˜₂: (x, y, z) → 2(x−y)(x−z)(y−z) (x−2y+z)² ·

The proof of the proposition relies on the introduction of the following modified probability-weighted moments V_n,γ =Vn/σn−vγ

an

and V =Vn/σn−vγ

an

= 1

u₀, 1 u₁, 1

u₂ t

(where theuj are defined in the statement of Lem.2.3). Defining nowVn,γ :=HγVn,γ and

V:=HγV = ((1−γ−ρ)⁻¹,(2−γ−ρ)⁻¹,(3−γ−ρ)⁻¹)^t (3.3) and noticing that Hγvγ = e, d₂(V) = 0 and d₁(V) = 0, it is now easy to prove the following identities using (3.2):

knan( ˆργ−ρ) =

knan(φ₁,γ(Vn/σn)−φ₁,γ(Vn/σn)) =

knan(ψ₁,γ(V_n,γ )−ψ₁,γ(V)) (3.4) knan(ˆan,γ/an−1) =

kn(φ₂,γ(Vn/σn)−φ₂,γ(Vn/σn)) =

knan( ˜ψ₂(V_n,γ )−ψ˜₂(V)) + Rn (3.5) kn(ˆσn,γ/σn−1) =

kn(φ₃,γ(Vn/σn)−φ₃,γ(Vn/σn)) =

knan(ψ₃(V_n,γ )−ψ₃(V)) (3.6) where

Rn=

knanψ˜₂(V_n,γ )

(1 +and₂(V_n,γ )/d₁(V_n,γ ))⁻¹−1

. (3.7)

The point is that the functionsψ₁,γ, ˜ψ₂ andψ₃and their derivatives are well-defined atVdefined by (3.3) (it was not the case for the functionsφj,γ at the limitvγ = limVn/σn). We have indeed

∇ψ1,γ(V) =

⎛

⎝ p(1−γ−ρ)/2

−p(2−γ−ρ) p(3−γ−ρ)/2

⎞

⎠, ∇ψ3(V) =

⎛

⎝ (1−γ−ρ)²/2

−(2−γ−ρ)² (3−γ−ρ)²/2

⎞

⎠,

∇ψ˜₂(V) =

⎛

⎝

12(1−γ−ρ)(−5 + 7γ+ 7ρ−2(γ+ρ)²) 2(2−γ−ρ)(4−4γ−4ρ+ (γ+ρ)²)

12(3−γ−ρ)(−9 + 9γ+ 9ρ−2(γ+ρ)²)

⎞

⎠

wherep= (1−γ−ρ)(2−γ−ρ)(3−γ−ρ).

Defining ∇as the Jacobian matrixJ_ψ˜

γ(V) where ψ˜γ : (x, y, z)→

ψ₁γ(x, y, z),ψ˜₂(x, y, z), ψ₃(x, y, z) t

,

relations (3.4) to (3.6) become knan

⎛

⎝ ρˆγ−ρ ˆ

an,γ/an−1 a⁻¹n (ˆσn,γ/σn−1)

⎞

⎠ =

knan( ˜ψγ(Vn,γ )−ψ˜γ(V)) +

⎛

⎝ 0 Rn

0

⎞

⎠

and thus the delta-method can be called upon to obtain relation (3.1) by combining Theorem 2.6 with the

following equality

knan(V_n,γ −V) = Hγ(

kn(Vn/σn−Vn/σn))

provided that Rn (defined in (3.7)) converges to 0 in probability. This is indeed the case, sinceV_n,γ →Vin probability asn→ ∞, and consequently

Rn=−

kna²nψ˜₂(Vn,γ )d₂(V_n,γ ) d₁(Vn,γ )

1 +and₂(Vn,γ )/d₁(Vn,γ ) ₋₁

vanishes to 0 as n→ ∞ (in probability) because ˜ψ₂(V) = 1, d₁(V) = 2/p, d₂(V) = 0, and using assump- tion (2.2) (which ensures that√

kna²_n has a real limit as n→ ∞).

3.2. Asymptotic normality for unknown γ

We can now define our final estimators of the parametersρ,an, andσn, by plugging-in an external estimator

ofγ. We set ⎛

⎝ ρˆ ˆ an

ˆ σn

⎞

⎠=

⎛

⎝ φ₁,ˆγ( ˆVn) φ₂,ˆγ( ˆVn) φ₃,ˆγ( ˆVn)

⎞

⎠

where ˆγ= ˆγn defines a sequence of estimators ofγ based on the ˜Nn upper excesses associated to a threshold

˜

un such that ˜un →s₊(F). Let ˜an=A(e^V−1(˜un)) and ˜λ, c, ddenote some real constants.

Theorem 3.2. Let the assumptions of Proposition3.1hold withρ <0and suppose that for some real constant

λ˜= 0,

n(1−F(˜un)) ˜an→λ˜ as n→ ∞. (3.8) If conditionally on N˜n= ˜kn

˜k_n^1/2(ˆγ−γ)−→ N(˜^d λc, d) as n→ ∞, (3.9) then for almost all sequences kn→ ∞ and ˜kn→ ∞ such that k˜n =o(kn), we have, conditionally on Nn =kn

andN˜n= ˜kn,

(˜kn)^1/2an

⎛

⎝ ρˆ−ρ ˆ

an/an−1 a⁻¹_n (ˆσn/σn−1)

⎞

⎠ = (˜kn)^1/2(ˆγ−γ)

⎛

⎝ c₁ c₂ c₃

⎞

⎠ + o_P(1) (3.10)

for some constantsc₁, c₂, c₃ depending on γ andρ(which expressions are given in the proof of the theorem).

Remark 3.3. The condition ˜kn=o(kn) means that we take less excesses for the estimation of the first-order parameterγthan for the estimation of the second-order parameterρ.

Remark 3.4. Proposition3.1is valid in the whole scopeρ≤0, whereas Theorem3.2excludes the caseρ= 0.

However, according to (3.10) and the expression ofc₁, it is worth noting that the asymptotic mean square error (AMSE) of ˆρtends to 0 when ρ→0, while this is not the case for many other estimators ofρstudied in the litterature, for which the AMSE goes to infinity whenρ→0 (in [9] or [3] for instance).

This advantage is however weakened by the fact that the rate of convergence in Theorem 2 is not√ knan, which is the rate attained by other estimators ofρin the litterature: therefore our estimator is asymptotically not as efficient as one would expect. For finitenthough, the performance of ˆρappears to be quite comptetitive in a number of models, particularly in situations where|ρ|is small, as it will be seen in the simulations below (Sect.4).

Remark 3.5. In our simulations, we used for ˆγthe PWM estimator defined by Hosking & Wallis in [15] (and studied in [4]).

Proof of Theorem 3.2. We keep using the notations previously introduced in the proof of Proposition 3.1 especially:

V_n,γ =Hγ

Vn/σn−vγ

an

=HγV_n,γ where vγ=

(1−γ)⁻¹,(2(2−γ))⁻¹,(3(3−γ))⁻¹ . We first study the deviation

ˆ

ρ−ρˆγ = ψ₁,ˆγ(Vn,ˆγ)−ψ₁,γ(Vn,γ )

= ψ₁,ˆγ(V_n,ˆγ)−ψ₁,γ(V_n,ˆγ) + ψ₁,γ(V_n,ˆγ)−ψ₁,γ(V_n,γ )

= γ−γˆ +ψ₁,γ(V_n,γˆ)−ψ₁,γ(V_n,γ ) (3.11) where we used the fact thatψ₁,γ(x, y, z) =−γ+ (x−4y+ 3z)/(x−2y+z). We thus have to concentrate on the second term. If we noteJ the 3×3 diagonal matrix with diagonal coefficients 1, 2 and 3, after some calculations we obtain the following essential development (remind thatanVn,γ = ˆVn/σn−vγ =o_P(1))

Vn,γˆ−Vn,γ = (Hγ_ˆ−Hγ)Vn,γ + Hγ_ˆ(Vn,γˆ−Vn,γ )

= (γ−ˆγ)JV_n,γ +Hγ_ˆ(vγ−vγ_ˆ)/an

= (γ−ˆγ)JV_n,γ +γ−ˆγ an

J vγ

= γ−ˆγ an

(J vγ+o_P(1)). (3.12)

Note that, according to assumptions (3.9), (3.8) (with ˜λ = 0) and the second order condition (1.2) (which ensures that ^˜a_aⁿ

n →0, whenρ <0), we have γ−γˆ

an

= ˜an

an

k˜¹n^/2(γ−ˆγ)

˜kn^1/2˜an

→0 asn→ ∞, in probability. (3.13)

SinceV_n,γ →V as n→ ∞,V_n,γ =HγV_n,γ converges in probability toV (defined in (3.3)), and consequently relations (3.12) and (3.13) imply that

nlim→∞V_n,γˆ= lim

n→∞V_n,γ =V in probability. (3.14)

Therefore, combining (3.11)–(3.14), there exists some sequenceWn converging to Vsuch that ˆ

ρ−ρ = (γ−γ) +ˆ a⁻¹_n (γ−ˆγ)<∇ψ1,γ(Wn), J vγ +o_P(1)> + ( ˆργ−ρ)

= a⁻¹_n (ˆγ−γ)(c₁+o_P(1)) + ( ˆργ−ρ) (3.15) where

c₁ := −(∇ψ₁,γ(V))^tJ vγ =−p 2

1−γ−ρ

1−γ −22−γ−ρ

2−γ +3−γ−ρ 3−γ

= pρ

(1−γ)(2−γ)(3−γ)· Note that the first term of the right-hand side of (3.15) makes it impossible to have√

knan as the rate for the asymptotic normality of ˆρ−ρbecause, in view of (3.13) and assumptions (2.2) and (2.3),√

knan(ˆγ−γ)→0 in probability but√

kn(ˆγ−γ) does not.

On the other hand, we have (see end of the proof for details on ˆan−ˆan,γ) ˆ

an−aˆn,γ = an( ˜ψ₂(V_n,ˆγ)−ψ˜₂(V_n,γ )) +a²_no_P(1) and σˆn

σn −σˆn,γ

σn

= an(ψ₃(V_n,ˆγ)−ψ₃(V_n,γ )). (3.16) Consequently, proceeding as for ˆρ−ρ and defining c₂ := −(∇ψ˜₂(V))^tJ vγ and c₃ := −(∇ψ3(V))^tJ vγ, it comes

(˜kn)^1/2an

⎛

⎝ ρˆ−ρ ˆ

an/an−1 a⁻¹_n (ˆσn/σn−1)

⎞

⎠ = (˜kn)^1/2(ˆγ−γ)

⎛

⎝ c₁+o_P(1) c₂+o_P(1) c₃+o_P(1)

⎞

⎠ +

⎛

⎝ 0 o_P((˜kn)^1/2a²_n)

0

⎞

⎠

+ (˜kn/kn)^1/2 knan

⎛

⎝ ρˆγ−ρ ˆ

an,γ/an−1 a⁻¹_n (ˆσn,γ/σn−1)

⎞

⎠ (3.17)

which, according to assumptions (3.8), (3.9) and Proposition3.1, yields the result of Theorem3.2.

We finish the proof by providing some details regarding (3.16). Settinge= (1,1,1)^tandh(u) =−ψ˜₂(u)(1 + d₂(u)/d₁(u))⁻¹(d₂(u)/d₁(u)), using (3.2) yields

ˆ

an−aˆn,γ = φ₂,ˆγ(Vn/σn−v_ˆγ+v_ˆγ)−φ₂,γ(Vn/σn−vγ+vγ)

= ψ₂(anV_n,γˆ+e)−ψ₂(anV_n,γ +e)

= ψ˜₂(anV_n,γˆ)−ψ˜₂(anV_n,γ ) +h(anV_n,ˆγ)−h(anV_n,γ )

= an( ˜ψ₂(V_n,ˆγ)−ψ˜₂(V_n,γ )) +a²_no_P(1)

where we used (3.14) and the following facts: d₂(V) = 0, ˜ψ₂(V) = 1, ˜ψ₂(αu) =αψ˜₂(u) and (d₂/d₁)(αu) = α(d₂/d₁)(u) for anyα= 0 andu∈U.

4. Simulation results

In this section, we shall present graphics obtained, concerning bias and mean square errors of our estimator of ρ, compared with two others, for two classes of underlying distributions and for values of γ in the scope of our theorem.

For the three estimators considered, the P.O.T. method we use consists in choosing a numberkn of excesses for the estimation ofρ, as well as a second number ˜kn for the preliminary estimation ofγ(only when necessary, since one of the estimators studied below does not rely on such an initial estimation of γ). Regarding our estimator, we take, for ˆγ= ˆγ(˜kn), the Hosking and Wallis’ estimator defined in [15] and studied in [4] and [5].

We compare our estimator, denoted in this section by ˆρP W M, with two others: the one presented in Fraga Alves et al.[9], which will be noted ˆρF GH and the one presented in Fraga Alveset al.[8], which will be noted

ˆ

ρF HL. They are defined as

ˆ

ρF GH =−

3(Tn^(τ)(kn)−1) Tn^(τ)(kn)−3

(see [9] for the definition ofTn^(τ)(kn)), with the tuning parameterτ equal to 0 whenever one expectsρto be in the range [−1,0) and equal to 1 otherwise (as suggested for instance in [2]), and

ˆ

ρF HL= 3−8ˆγ₋(˜kn) +6−12ˆγ₋(˜kn) Tn(kn)−3