Estimation of extreme quantiles from heavy and light tailed distributions

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Estimation of extreme quantiles from heavy and light tailed distributions

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Armelle Guillou

To cite this version:

Jonathan El Methni, Laurent Gardes, Stéphane Girard, Armelle Guillou. Estimation of extreme

quantiles from heavy and light tailed distributions. Journal of Statistical Planning and Inference,

Elsevier, 2012, 142 (10), pp.2735-2747. �10.1016/j.jspi.2012.03.025�. �hal-00627964v3�

Estimation of extreme quantiles from heavy and light tailed distributions

Jonathan El Methni

, Laurent Gardes

, St´ ephane Girard

& Armelle Guillou

(1)Team Mistis, INRIA Rhˆone-Alpes & LJK, Inovall´ee, 655, av. de l’Europe, Montbonnot, 38334 Saint-Ismier cedex, France

(2) Universit´e de Strasbourg & CNRS, IRMA, UMR 7501, 7 rue Ren´e Descartes, 67084 Strasbourg cedex, France

Abstract

In [18], a new family of distributions is introduced, depending on two parametersτ andθ, which encompasses Pareto-type distributions as well as Weibull tail-distributions. Estimators forθand extreme quantiles are also proposed, but they both depend on the unknown param- eterτ, making them useless in practical situations. In this paper, we propose an estimator of τwhich is independent ofθ. Plugging our estimator ofτ in the two previous ones allows us to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way.

The asymptotic distributions of our three new estimators are established and their efficiency is illustrated on a small simulation study and on a real data set.

AMS Subject Classifications: 62G05, 62G20, 62G30.

Keywords: Weibull tail-distributions, Pareto-type distributions, extreme quantile, maxi- mum domain of attraction, asymptotic normality.

1 Introduction

Let X₁, . . . , X_n be a sequence of independent and identically distributed random variables with a cumulative distribution function F and let X_1,n ≤ · · · ≤ X_n,n denote the order statistics as- sociated to this sample. The Gnedenko theorem [20] insures that for a large class of cumulative distribution functions, the maximumX_n,n (after proper renormalization) converges in distribution to an extreme-value distribution with shape parameter γ. Depending on its sign, three possible maximum domains of attraction forF are possible: Fr´echet (γ >0), Gumbel (γ= 0) and Weibull (γ <0). Since distributions in the Weibull maximum domain of attraction have a finite right tail, in most applications this maximum domain of attraction is not relevant. In this paper, we focus on the Fr´echet and Gumbel maximum domains of attraction.

Distributions in the Fr´echet maximum domain of attraction can be characterized through their survival functionF = 1−F asF(x) =x^−1/γL(x) whereγ >0 andLis a slowly varying function at infinityi.e. L(λx)/L(x)→1 asx→ ∞for allλ≥1.F is said to be regularly varying at infinity with index −1/γ. This property is denoted by F ∈ R_−1/γ, (see [6] for more details on regular variations theory) andF is called a Pareto-type distribution. To make inference on the distribution tail, most approaches consist of using thek_n upper order statisticsX_n−kn+1,n≤ · · · ≤X_n,n since the tail information is only contained in the extreme upper part of the sample. Here, (k_n) is an

intermediate sequence of integersi.e. such that

n→∞lim k_n =∞and lim

n→∞k_n/n= 0.

A large part of the extreme-value literature is devoted to the estimation of the tail-index γ >0, the most known estimator being the Hill estimator [24]. It can be equivalently defined in terms of log-excesses log(X_n−i+1,n/X_n−kn+1,n) or in terms of log-spacings log(X_n−i+1,n/X_n−i,n):

H_n(k_n) = 1 kn−1

kn−1

X

i=1

log

X_n−i+1,n Xn−k_n+1,n

= 1

kn−1

kn−1

X

i=1

ilog

X_n−i+1,n Xn−i,n

. (1)

At the opposite, there is no simple representation for distributions in the Gumbel maximum domain of attraction. However, an interesting subset of the Gumbel maximum domain of attrac- tion is the Weibull tail-distributions family. It encompasses for instance Weibull, Gaussian and Gamma distributions. Let us recall that a cumulative distribution functionF has a Weibull tail if F(x) = exp (−H(x)), whereH^←(t) := inf{x, H(x)≥t} ∈ Rβ. The functionH^←is the so-called generalized inverse of H. The tail of such distributions is driven by the shape parameter β > 0 called the Weibull tail-coefficient. Many papers are dedicated to the estimation of the Weibull tail-coefficient. A first family of approaches [2, 3, 7, 13] is based on the log-excesses while a second one relies on the log-spacings [5, 11, 16, 17, 19, 21, 22]. All these estimators are thus similar to the Hill statistic (1).

In order to understand the similarity between most estimators of the Weibull tail-coefficient and the Hill estimator, a new family of distributions has been proposed in [18]. These distributions depend on two parametersτ∈[0,1] andθ >0. More specifically, lettingK_x(y) =Ry

1 u^x−1duwhere x∈R, the considered family of survival distribution functions is given by:

(A1(τ, θ)) F(x) = exp(−K_τ^←(logH(x))) forx≥x_∗>0 withτ∈[0,1].

Here, H is an increasing function such thatH^←∈ Rθ whereθ >0. The parameterτ allows us to represent a large panel of distribution tails ranging from Weibull-type tails (in this caseτ= 0 and θcoincides with the Weibull tail-coefficientβ) to distributions belonging to the Fr´echet maximum domain of attraction (in this caseτ = 1 andθcorresponds to the tail indexγ).

In [18], an estimator ofθbased on the Hill statistic is also introduced:

θbn,τ(kn) = Hn(kn)

µτ(log(n/kn)), (2)

with, for allt >0,

µτ(t) = Z ∞

0

(Kτ(x+t)−Kτ(t)) e^−xdx,

and an estimator of the extreme quantile x_pn=F^←(p_n) withp_n →0 asn→ ∞is derived:

bx_p

n,bθ_n,τ =X_n−kn+1,nexp

bθn,τ(kn) [Kτ(log(1/pn))−Kτ(log(n/kn))]

. (3)

We refer to [14, 26, 15] for applications of extreme quantiles respectively in reliability, hydrology and finance. The asymptotic distributions of bθn,τ and bx_p

n,bθ_n,τ are established in [18]. Let us highlight that estimators (2) and (3) are only of theoretical interest since, in practical situationsτ is unknown, and therefore, they cannot be used.

This paper builds on the work of [18]. We propose an estimator bτn of τ, independent of θ.

Replacing τ by bτn in (2) and (3) yields two new estimators which can be computed in practical

situations. As a result, we are able to estimate extreme quantiles from Pareto-type and Weibull tail-distributions in an unified way. The asymptotic normality of our three new estimators is also established.

The paper is organized as follows. The estimators are defined in Section 2, their asymptotic properties are established in Section 3. The behavior of the extreme quantile estimator is illustrated on simulated data in Section 4 and on a real data set in Section 5. Proofs of the main results are presented in Section 6 while proofs of auxiliary results are postponed to the Appendix.

2 Definition of the estimators

Let us first describe the construction of an estimator ofτ. Let (kn) and (k⁰_n) withk⁰_n> kn be two intermediate sequences of integers such thatθb_n,τ(k_n)−→^P θandθb_n,τ(k_n⁰)−→^P θ. It straightforwardly follows that

θb_n,τ(k_n)

θbn,τ(k_n⁰)= H_n(k_n) Hn(k⁰_n)

µ_τ(log(n/k_n⁰)) µτ(log(n/kn))

−→P 1.

Introducing for all t > t⁰ > 0 the function defined by ψ(.;t;t⁰) : R → (−∞,exp(t−t⁰)) and ψ(x;t, t⁰) =µ_x(t)/µ_x(t⁰), it follows that

Hn(kn) H_n(k_n⁰)

∼P ψ(τ; log(n/k_n),log(n/k_n⁰)). (4) Moreover, it can be shown (see Lemma 3 in Section 6) thatψ(.; log(n/k_n),log(n/k⁰_n)) is a bijection from Rto (−∞, k_n⁰/kn). As a consequence, the following estimator ofτ is considered:

bτ_n =







ψ⁻¹_H

n(kn)

H_n(k⁰_n); log(n/kn),log(n/k⁰_n)

if ^H_H^n(kn)

n(k⁰_n) < ^k_k⁰ⁿ

n

u if ^H_H^n(kn)

n(k⁰_n) ≥ ^k_k⁰ⁿ

n,

(5) where uis the realization of a standard uniform distribution. In practice, only the first situation has to be considered, since Lemma 5 in Section 6 shows that, for nlarge enough,Hn(kn)/Hn(k_n⁰) is almost surely smaller thank_n⁰/k_n. It is thus possible to plugbτ_n in (2) to obtain a new estimator ofθ:

θb_n,

bτn(k_n) = Hn(kn) µ

bτ_n(log(n/k_n)). (6)

Similarly, replacingτandθb_n,τ by their estimates in (3) yields a new estimator of extreme quantiles:

xb_p

n,bθ_n,bτn =Xn−k_n+1,nexp

θbn,bτ_n(kn) [K

bτ_n(log(1/pn))−K

bτ_n(log(n/kn))]

. (7)

The asymptotic behavior of the three new estimators bτn,θbn,τb_n and bx_p

n,bθ_n,bτn is established in the next section.

3 Asymptotic properties

As a first result, we establish the consistency ofτb_n under the following assumption:

H^←(t) =t^θ`(t) =c t^θexp Z x

1

ε(u) u du

(8) withca positive constant andε(s)→0 ass→ ∞. Let us highlight that (8) amounts to supposing that H^←∈ Rθ and that the slowly varying function at infinity`is normalised, see [6], page 15.

Proposition 1 Suppose that (A1(τ, θ))holds with `(.) a normalised slowly varying function. If (k_n)and(k⁰_n)are two intermediate sequences of integers such thatk_n/k_n⁰ →0, then bτ_n−→^P τ.

Let us note that the consistency ofτbn is established for allθ >0 andτ ∈[0,1] in an unified way.

In this sense, the asymptotic behavior of this estimator is more a consequence of the log-spacings property than a tail behavior (which can be exponential forτ= 0 as well as polynomial forτ = 1).

Next, to establish the asymptotic normality of the three estimators (5), (6) and (7), a second-order condition on `is necessary:

(A2(ρ)) There existρ <0 andb(x)→0 such that uniformly locally onλ≥λ0>0 log

`(λx)

`(x)

∼b(x)Kρ(λ), whenx→ ∞, with|b|asymptotically decreasing.

It can be shown that necessarily |b| ∈ R_ρ. The second order parameter ρ < 0 tunes the rate of convergence of `(λx)/`(x) to 1. The closer is ρ to 0, the slower is the convergence. Condition (A2(ρ))is the cornerstone in all the proofs of asymptotic normality for extreme value estimators.

It is used in [4, 23, 24] to prove the asymptotic normality of several estimators of the extreme value index. Let log₂(.) = log(log(.)). The first theorem establishes the asymptotic normality ofτbn: Theorem 1 Suppose that (A1(τ, θ)) and (A2(ρ)) hold. If (kn) and (k_n⁰) are two intermediate sequences of integers such that kn/k⁰_n→0 and

pk⁰_nb(expK_τ(logn/k_n⁰))→0, (9) pkn(log₂(n/kn)−log₂(n/k_n⁰))→ ∞, (10) log(n/k⁰_n) (log₂(n/kn)−log₂(n/k_n⁰))→ ∞, (11) then

pkn(log₂(n/kn)−log₂(n/k_n⁰)) (τbn−τ)−→ N^d (0,1).

Condition (9) is standard in extreme-value theory. It imposes that the bias induced by the slowly varying function is asymptotically negligible. Condition (10) forces the speed of convergence ofτbn

to tend to infinity. Finally, (11) is of the same nature as (10): It imposes some minimal spacing between the two sequences (kn) and (k⁰_n). Besides, if τ = 0, conditions (9) and (10) imply that xb(x)→0 asx→ ∞. As a consequence, ifτ= 0 andρ >−1, it is not possible to choose sequences (k_n) and (k⁰_n) satisfying the above assumptions. In such a case, only the consistency ofbτ_n can be guaranteed.

In our next result, it is established thatθb_n,

bτn inherits from the asymptotic normality of bτ_n. Theorem 2 Suppose the assumptions of Theorem 1 hold. If, moreover,

(log₂(n/kn)−log₂(n/k⁰_n))/log₂(n/kn)→0, (12) pk_n(log₂(n/k_n)−log₂(n/k⁰_n))/log₂(n/k_n)→ ∞, (13)

then √

kn(log₂(n/kn)−log₂(n/k⁰_n)) log₂(n/kn)

θbn,bτ_n(kn)−θ _d

−→ N(0, θ²).

It appears that the estimation of τ has a cost in terms of rates of convergence. Condition (12) implies thatbθn,τb_n converges slower thanθbn,τ, see Lemma 7 in Section 6. As previously, (13) forces the speed of convergence of bθ_n,

bτ_n(k_n) to tend to infinity. Note that this condition implies (10) in Theorem 1. Similarly as for Theorem 1, it can be shown that, if τ = 0, (9) and (13) imply xlog(x)b(x)→ 0 as x→ ∞. Thus, again no sequences (k_n) and (k_n⁰) exist in case τ = 0 and ρ > −1. Now, if τ ∈ (0,1] or if τ = 0 and ρ < −1, a possible choice for the two intermediate sequences is log(kn) =aKτ(log(n)) and log(k⁰_n) =a⁰Kτ(log(n)) with the following restrictions on (a, a⁰)∈R:







0< a < a⁰<2ρ/(2ρ−1) if τ= 1 0< a < a⁰ <−2ρ if 0< τ <1 2< a < a⁰ <−2ρ if τ= 0.

(14)

Finally, in the case where τ= 0 andρ=−1, the existence of sequences (kn) and (k⁰_n) depends on the underlying distribution.

The last result is dedicated to the asymptotic distribution of the extreme quantile estimator.

Theorem 3 Suppose the assumptions of Theorem 2 hold. If, moreover,

pkn(log₂(n/kn)−log₂(n/k_n⁰)) /log₂(1/pn)→ ∞, (15) (log(n/kn))^1−τ[Kτ(log(1/pn))−Kτ(log(n/kn))]→ ∞, (16)

log₂(n/kn)[Kτ(log(1/pn))−Kτ(log(n/kn))]

,Z log(1/p_n) log(n/k_n)

log(u)u^τ−1du →0, (17)

then √

kn(log₂(n/kn)−log₂(n/k⁰_n)) Rlog(1/p_n)

log(n/kn)log(u)u^τ−1du

bx_p

n,bθ_n,bτn

xp_n

−1

!

−→ Nd (0, θ²).

A sufficient condition to verify (16) and (17) is log₂(1/pn)/log₂(n/kn) → ∞. This imposes an upper bound on the order pn of the extreme quantile. At the opposite, condition (15) provides a lower bound on pn. A possible choice for the order pn of the extreme quantile is given by log₂(1/pn) = [log₂(n)]^α for all α >1. The finite sample performances of xb_p

n,bθ_n,bτn are illustrated in the next section.

4 A small simulation study

In this section, our extreme quantile estimator is compared to the Moment estimator of Dekkers et al. [9] and the Peaks Over Threshold (POT), see for instance [8], on simulated data. The POT approach relies on the approximation of the excesses distribution, over a high threshold, by a Generalized Pareto Distribution (GPD). Among the numerous methods available for estimating the GPD parameters, we focus on the moments method which yields the best results in our simulations.

Let us emphasize that the Moment and POT estimators are designed to work in any domain of attraction. Let us recall that the extreme quantile estimator (7) requires the numerical inversion of the function ψ. This computation is achieved thanks to a dichotomy procedure since Lemma 3(i) ensures thatψis increasing.

The comparison is achieved on twelve different distributions:

• five Pareto-type distributions (τ = 1): the absolute value of the standard Cauchy distribution (θ = 1 andρ=−2), standard Pareto distribution (θ= 2 and ρ=−∞), the absolute value of the Student distribution with two degrees of freedom (θ= 1/2 andρ=−2) and two Burr distributions (θ = 1/2 and ρ ∈ {−1,−1/2}) with quantile function F^←(x) = (x^ρ−1)^−θρ, x∈(0,1),

• five Weibull tail-distributions (τ= 0): the absolute value of the standard Gaussian distribu- tion (θ = 1/2 and ρ= −1), a Weibull distribution with 2 as shape parameter (θ = 2 and ρ=−∞), a Gamma distribution with 2 as shape parameter (θ = 1 and ρ=−1) and two distributions with quantile functionF^←(x) = (−logx)^θ(1 + (ρ+θ)(−logx)^ρ),x∈(0,1) with θ= 1/2 andρ∈ {−1/2,−1/4},

• two log-Weibull tail distributions (τ = 1/2), see [18], paragraph 2.2: the standard lognormal distribution (θ = √

2/2 and ρ = 0) and the distribution with quantile function F^←(x) = exp{√

2[(−logx)^1/2−1]},x∈(0,1) for whichθ=√

2/2 andρ=−∞.

These distributions represent various situations (τ ∈ {0,1/2,1}, θ ∈ {1/2,√

2/2,1,2} and ρ ∈ {−∞,−2,−1,−1/2,−1/4,0}) in which the asymptotic normality of our estimator is not always established. In the following, we take pn = 10⁻³ and simulate N = 100 samples (Xn,j)j=1,...,N

of size n = 500. For each sample Xn,j, the estimator bx_p

n,bθ_n,bτn is computed for k_n⁰ = 3, . . . ,500 and kn =bck⁰_ncwith c = 0.1. This value has been chosen on the basis of intensive Monte-Carlo simulations. With such a choice, our estimator and the Moment and POT estimators depend on only one intermediate sequence of integers (k⁰_n). For all the considered distributions, the mean- squared errors associated to the estimators are computed as functions of k⁰_n and are reported on Figures 1–3. It appears that our estimator outperforms the Moment and POT estimators for almost all the values ofk_n⁰. Also fork_n⁰ ≥50, the mean-squared errors associated to our estimator are almost constant as a function ofk⁰_n, whatever the distribution is.

5 A real data set

The performance of our estimator (7) is illustrated through the analysis of extreme events on the Nidd river data set, which is common in extreme values studies, see for instance [8]. It consists of 154 exceedances of the levels 65m³s⁻¹ by the river Nidd (Yorkshire, England) during the period 1934-1969 (35 years). There is no general agreement on a maximum domain of attraction for this data set. In [10], a Fr´echet maximum domain of attraction is assumed and heavy tailed- distributions are considered as a possible model for such data. However, according to [25], the Nidd data may reasonably be assumed to come from a distribution in the Gumbel maximum domain of attraction. This result was in accordance with [12] where it has been shown that a Weibull tail-distribution could be considered for such a data. The estimation ofτ is therefore of great interest. The estimated values ofτ andθare depicted on Figure 4 as functions of k_n⁰ (recall that kn =b0.1k⁰_nc). It appears that the estimators become stable for k_n⁰ ≥80 with bτn ' 1 and θbn,τb_n(kn)'0.3. These results indicate that the data may be assumed to come from a distribution in the Fr´echet maximum domain of attraction. They are in accordance with the ones obtained by Bayesian methods [10], Figure 7 where the tail index is also estimated at 0.3.

The standard quantity of interest in environmental studies is theN-year return level, defined as the level which is exceeded on average once inN years. Here, we focus on the estimation of the 50- and 100- year return levels. In Figure 4, our extreme quantile estimator is compared to the Moment and POT estimators, plotting the associated N-year return level as a function of k_n⁰ forN = 50

andN = 100. It appears that our estimator and the Moment estimator yield similar curves. The POT sample path is different, but for all methods, choosing k_n⁰ '60, we obtain an estimation of the 50-year return level which belongs approximately to the interval [340m³s⁻¹,375m³s⁻¹], and an estimation of the 100-year return in [400m³s⁻¹,470m³s⁻¹]. Again, these results are in accordance with the the credibility intervals obtained in [10], Table 1.

6 Proofs

We first give some preliminary lemmas. Their proofs are postponed to the appendix.

6.1 Preliminary lemmas

In the following, C is a compact subset such that [0,1] ⊂ C ⊂ (−∞,2). The first lemma is a standard result on the behavior at infinity of the Laplace transform.

Lemma 1 Let x∈C andh_x∈C^∞(R⁺). Seti= min{j∈N/h^(j)x (0)6= 0}. If sup

x∈C y≥0

h⁽ⁱ⁺¹⁾_x (y) <∞,

then

t→∞lim sup

x∈C

tⁱ⁺¹ehx(t)−h⁽ⁱ⁾_x (0) = 0, withehx(t) =R+∞

0 exp(−tu)hx(u)du is the Laplace transform ofhx.

Lemma 1 is the key tool for establishing the uniform asymptotic expansions of µ_x and ∂µ_x/∂x given in Lemma 2:

Lemma 2 For all x∈C andt >0, we have (i) lim

t→∞sup

x∈C

µx(t) t^x−1 −1

= 0, (ii) lim

t→∞sup

x∈C

∂

∂xµ_x(t)−log(t)µ_x(t)

t^x−2 −1

= 0.

As a consequence of the above expansions, some important properties of ψcan be derived in the two next lemmas:

Lemma 3 For all t > t⁰ >0 andx∈R, we have (i)x→ψ(x;t, t⁰)is an increasing function, (ii) lim

x→∞ψ(x;t, t⁰) = exp(t−t⁰).

Lemma 4 For all x∈C andt > t⁰ >0, we have

∂

∂xψ(x;t, t⁰) = log(t/t⁰)ψ(x;t, t⁰)

1 +O

1 t⁰log(t/t⁰)

, as t⁰→ ∞.

The following lemma ensures that the estimator τbn is well-defined. In (5), the situation where Hn(kn)/Hn(k⁰_n)≥k⁰_n/kn is eventually impossible.

Lemma 5 Let (kn) and (k_n⁰) be two intermediate sequences of integers such that kn/k_n⁰ →0. If θb_n,τ(k_n)/bθ_n,τ(k⁰_n)−→^P 1 then

P

H_n(k_n) Hn(k_n⁰) ≥ k_n⁰

kn

→0asn→ ∞.

We now prove that θbn,τ(kn) is a consistent estimator for θ when τ is known and under general assumptions.

Lemma 6 Suppose that (A1(τ, θ)) holds with a normalised slowly varying function `(.). If (kn) is an intermediate sequence of integers, then θb_n,τ(k_n)−→^P θ.

The asymptotic normality of bθn,τ andxb_p

n,bθn,τ has already been established in the particular case whereτ is known by [18]. Let us quote two results from this work.

Lemma 7 (Theorem 1, [18]). Suppose that(A₁(τ, θ))and(A₂(ρ))hold. If (k_n)is an intermedi- ate sequence such that√

k_nb(expK_τ(log(n/k_n)))→0, then pkn

θb_n,τ(k_n) θ −1

!

−→ Nd (0,1). (18)

Lemma 8 (Theorem 2, [18]). Under the assumptions of Lemma 7 and if, moreover, (log(n/kn))^1−τ(Kτ(log(1/pn))−Kτ(log(n/kn)))→ ∞,

then,

√kn

K_τ(log(1/p_n))−K_τ(log(n/k_n))log bx_p

n,bθn,τ

x_pn

!

−→ Nd (0, θ²).

The next lemma establishes that θ can be replaced by θbn,τ(k_n⁰) in (18) without changing the asymptotic distribution.

Lemma 9 Suppose that (A1(τ, θ)) and (A2(ρ)) hold. Let (kn) and (k_n⁰) be two intermediate sequences of integers such that p

k_n⁰b(expKτ(logn/k_n⁰))→0 andkn/k_n⁰ →0. We have pk_n θb_n,τ(k_n)

θb_n,τ(k_n⁰)−1

!

−→ Nd (0,1).

The last lemma quantifies the effect of estimatingτ in θb_n,τ(k_n).

Lemma 10 Suppose that (A1(τ, θ)) and (A2(ρ)) hold. Let (kn) and (k_n⁰) be two intermediate sequences of integers such that

k_n/k_n⁰ →0, p

k_n(log₂(n/k_n)−log₂(n/k⁰_n))/log₂(n/k_n)→ ∞,

pk_n⁰b(expKτ(logn/k⁰_n))→0 and log(n/k_n⁰) (log₂(n/kn)−log₂(n/k⁰_n))→ ∞.

We have √

k_n(log₂(n/k_n)−log₂(n/k_n⁰)) log₂(n/kn)

θb_n,

bτn(k_n) θbn,τ(kn) −1

!

−→ Nd (0,1).

6.2 Proofs of the main results

Proof of Proposition 1−For allε >0, let us write

P(|bτn−τ|> ε) =P(bτn > τ+ε) +P(τbn< τ −ε) and consider the two terms separately. Clearly, one has

P(bτn > τ+ε) = P

{τbn> τ +ε}

Hn(kn) Hn(k⁰_n)≥ k⁰_n

kn P

Hn(kn) Hn(k⁰_n) ≥k⁰_n

kn

+ P

{τbn> τ +ε} ∩

H_n(k_n) Hn(k⁰_n) <k⁰_n

kn

≤ P

H_n(k_n) Hn(k⁰_n) ≥k_n⁰

kn

+P

H_n(k_n)

Hn(k⁰_n) ≥ψ(τ+ε; log(n/k_n),log(n/k⁰_n))

.(19) Focusing on the first probability of (19), note that,θb_n,τ(k_n) andθb_n,τ(k⁰_n) are consistent estimators forθ in probability by Lemma 6. Thusbθ_n,τ(k_n)/bθ_n,τ(k_n⁰)−→^P 1 and Lemma 5 entail

P

Hn(kn) H_n(k⁰_n) ≥k_n⁰

k_n

→0.

The second probability in (19) tends to 0 by (4) and Lemma 3(i). Similarly, we can prove that P(bτn < τ−ε)→0 which achieves the proof.

Proof of Theorem 1 − Letting v_n = √

k_n(log₂(n/k_n)−log₂(n/k_n⁰)), our goal is to prove that P(v_n(bτ_n−τ)≤s) → Φ(s), for all s ∈ R where Φ is the cumulative distribution function of the standard Gaussian distribution. To this aim, let us first remark that Lemma 9 yields θbn,τ(kn)/bθn,τ(k⁰_n)−→^P 1. Thus, introducingEn(s) ={vn(τbn−τ)≤s}, we have

P(En(s)) = P

En(s)∩

Hn(kn) Hn(k⁰_n) <k⁰_n

kn

+P

En(s)

Hn(kn) Hn(k_n⁰) ≥ k⁰_n

kn P

Hn(kn) Hn(k⁰_n)≥ k⁰_n

kn

=: T_n⁽¹⁾(s) +o(1),

by Lemma 5. From the definition of τb_n given in (5) and recalling that, from Lemma 3(i), ψ(.; log(n/k_n),log(n/k_n⁰)) is an increasing function, we obtain

T_n⁽¹⁾(s) = P

Hn(kn)

Hn(k_n⁰) ≤ψ(τ+s/vn; log(n/kn),log(n/k_n⁰))

∩

Hn(kn) Hn(k⁰_n) <k_n⁰

kn

= P

Hn(kn) Hn(k_n⁰) ≤min

ψ(τ+s/vn; log(n/kn),log(n/k_n⁰)),k_n⁰ kn

.

Lemma 3(ii) thus yields

T_n⁽¹⁾(s) = P

H_n(k_n)

Hn(k_n⁰) ≤ψ(τ+s/v_n; log(n/k_n),log(n/k_n⁰))

= P

Hn(kn)

H_n(k_n⁰) ≤ µ_τ+s/vn(log(n/kn)) µ_τ+s/vn(log(n/k⁰_n))

,

and, from (2) and the fact that ζn:=p

kn

θb_n,τ(k_n) θbn,τ(k⁰_n)−1

!

−→ Nd (0,1),

(see Lemma 9), we have T_n⁽¹⁾(s) = P

ζn ≤p

kn

µτ(log(n/k_n⁰)) µ_τ(log(n/k_n))

µ_τ+s/vn(log(n/kn)) µ_τ+s/vn(log(n/k_n⁰))−1

= P

ζn ≤p

kn

µτ(log(n/k⁰_n))

µ_τ(log(n/k_n))[ψ(τ+s/vn; log(n/kn),log(n/k_n⁰))−ψ(τ; log(n/kn),log(n/k_n⁰))]

.

A first order Taylor expansion leads to T_n⁽¹⁾(s) = P

ζ_n ≤ s

(log₂(n/kn)−log₂(n/k_n⁰))

µ_τ(log(n/k_n⁰)) µτ(log(n/kn))

∂

∂xψ(τ₀; log(n/k_n),log(n/k⁰_n))

, where τ0=τ+sη0/vn with η0 ∈(0,1). Since τ0 →τ, for nlarge enough, τ0 <2 and Lemma 4 entails

T_n⁽¹⁾(s) = P

ζ_n ≤sµ_τ0(log(n/k_n)) µτ₀(log(n/k⁰_n))

µ_τ(log(n/k_n⁰)) µτ(log(n/kn))

1 +O

1

log(n/k⁰_n) (log₂(n/kn)−log₂(n/k_n⁰))

= P

ζ_n ≤sµ_τ0(log(n/k_n)) µτ0(log(n/k⁰_n))

µ_τ(log(n/k_n⁰))

µτ(log(n/kn))(1 +o(1))

.

Besides, Lemma 2(i) entails that sµτ₀(log(n/kn))

µ_τ0(log(n/k_n⁰))

µτ(log(n/k_n⁰)) µ_τ(log(n/k_n)) =s

log(n/kn) log(n/k_n⁰)

^τ0−τ

(1 +o(1)) =sexp sη0

√kn

(1 +o(1)) −→

n→∞s, and thus

T_n⁽¹⁾(s) =P(ζn≤s(1 +o(1)))≤Φ(s) + sup

x∈R

|Φ(x)−P(ζn≤x)| →Φ(s) by [15], page 552. This achieves the proof of Theorem 1.

Proof of Theorem 2−On the first hand, Lemma 7 shows that θbn,τ(kn)−θ= ζnθ

√k_n where ζn

−→ Nd (0,1), and on the other hand, Lemma 10 entails

θb_n,

τbn(kn)

θb_n,τ(k_n) −1 = δ_nlog₂(n/k_n)

√k_n(log₂(n/k_n)−log₂(n/k_n⁰)) where δ_n −→ N^d (0,1). (20) Collecting these results, it follows that

θb_n,

τbn(k_n)−θ = θb_n,τ(k_n) θbn,bτ_n(kn) θb_n,τ(k_n) −1

! +

θb_n,τ(k_n)−θ

= θbn,τ(kn) δnlog₂(n/kn)

√k_n(log₂(n/k_n)−log₂(n/k⁰_n))+ ζnθ

√k_n. Consequently, one immediately has

√k_n(log₂(n/k_n)−log₂(n/k_n⁰)) log₂(n/kn)

θb_n,

bτn(k_n)−θ

= θb_n,τ(k_n)δ_n+ζ_nθlog₂(n/k_n)−log₂(n/k⁰_n) log₂(n/kn) and Theorem 2 follows.

Proof of Theorem 3−Let us consider the following expansion:

log xb_p

n,bθ_n,bτn

xp_n

!

= log xb_p

n,bθ_n,bτn

bx_p

n,bθ_n,τ

!

+ log bx_p

n,bθ_n,τ

xp_n

!

=:T_n⁽²⁾+T_n⁽³⁾. By (3), we have

T_n⁽²⁾ = θb_n,

bτn(kn)[K

bτn(log(1/pn))−K

τbn(log(n/kn))]−θbn,τ(kn)[Kτ(log(1/pn))−Kτ(log(n/kn))]

= [Kτ(log(1/pn))−Kτ(log(n/kn))]

θbn,bτ_n(kn)−θbn,τ(kn) +bθ_n,

τbn(k_n) [(K

bτn(log(1/p_n))−K

τbn(log(n/k_n)))−(K_τ(log(1/p_n))−K_τ(log(n/k_n)))]

=: T_n^(2,1)+T_n^(2,2).

Focusing on the first term, (20) leads to

T_n^(2,1)= log₂(n/kn)[Kτ(log(1/pn))−Kτ(log(n/kn))]

√k_n(log₂(n/k_n)−log₂(n/k_n⁰)) bθn,τ(kn)δn

which implies that

√kn(log₂(n/kn)−log₂(n/k⁰_n)) Rlog(1/p_n)

log(n/k_n)u^τ−1log(u)du

T_n^(2,1)=δnlog₂(n/kn)[Kτ(log(1/pn))−Kτ(log(n/kn))]

Rlog(1/p_n)

log(n/k_n)u^τ−1log(u)du

θbn,τ(kn) =o_P(1).

Let us now consider Tn^(2,2). Theorem 1 states that bτ_n=τ+ ξ_n

√kn(log₂(n/kn)−log₂(n/k⁰_n)) =:τ+ξ_nσ_n where ξ_n−→ N^d (0,1).

Replacing inTn^(2,2), we obtain T_n^(2,2)=bθ_n,

τbn(k_n)[(K_τ+σnξn(log(1/p_n))−K_τ+σnξn(log(n/k_n)))−(K_τ(log(1/p_n))−K_τ(log(n/k_n)))].

By definition

Kτ(log(1/pn))−Kτ(log(n/kn)) =

Z log(1/p_n) log(n/k_n)

u^τ−1du,

and therefore it immediately follows that T_n^(2,2)=bθ_n,

τbn(kn)

Z log(1/p_n) log(n/kn)

u^{τ+σnξn−1}−u^τ−1

du=θb_n,

bτn(kn)

Z log(1/p_n) log(n/kn)

u^τ−1 u^σnξn−1 du.

Lettingϕ(x) := exp(x)−1−x, we deduce that T_n^(2,2)=θbn,τb_n(kn)σnξn

Z log(1/p_n) log(n/k_n)

u^τ−1log(u)du+θbn,bτ_n(kn)

Z log(1/p_n) log(n/k_n)

u^τ−1ϕ(σnξnlog(u))du.

Now, there exists c > 0 such that x < log(c) implies |ϕ(x)| < ^c₂x². As a consequence, since σnlog₂(1/pn)→0 andσnlog₂(n/kn)→0, fornlarge enough, we have

Z log(1/p_n) log(n/kn)

u^τ−1ϕ(σnξnlog(u))du

≤

Z log(1/p_n) log(n/kn)

u^τ−1|ϕ(σnξnlog(u))|du

≤ c 2σ_n²ξ²_n

Z log(1/p_n) log(n/kn)

u^τ−1[log(u)]²du.

Thus,

Rlog(1/pn)

log(n/kn)u^τ−1ϕ(σ_nξ_nlog(u))du σ_nRlog(1/p_n)

log(n/kn)u^τ−1log(u)du

≤ cσnξ²_nRlog(1/pn)

log(n/k_n)u^τ−1[log(u)]²du 2Rlog(1/p_n)

log(n/kn)u^τ−1log(u)du

≤ cξ_n²log₂(1/p_n) 2√

kn(log₂(n/kn)−log₂(n/k⁰_n)) =o_P(1), and, replacing inTn^(2,2), we obtain

√k_n(log₂(n/k_n)−log₂(n/k_n⁰)) Rlog(1/pn)

log(n/k_n)log(u)u^τ−1du

T_n^(2,2)=ξ_nθb_n,

bτn(k_n) +o_P(1)−→ N^d (0, θ²).

Finally, Lemma 8 states that

T_n⁽³⁾= %n[Kτ(log(1/pn))−Kτ(log(n/kn))]

√k_n where %n

−→ Nd (0, θ²), and thus, under our assumptions,

√kn(log₂(n/kn)−log₂(n/k_n⁰)) Rlog(1/p_n)

log(n/kn)u^τ−1log(u)du

T_n⁽³⁾=o_P(1).

Combining the above results and using the delta method, Theorem 3 follows.

Acknowledgments

The authors are grateful to the referees for their helpful comments.

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Appendix: Proof of auxiliary results

Proof of Lemma 1−A (i+ 1)^thorder Taylor expansion leads to hx(u) =uⁱ

i!h⁽ⁱ⁾_x (0) + uⁱ⁺¹

(i+ 1)!h⁽ⁱ⁺¹⁾_x (ηu), withη∈(0,1) and consequently,

ehx(t) = Z ∞

0

exp(−tu)uⁱ

i!h⁽ⁱ⁾_x (0)du+ Z ∞

0

exp(−tu) uⁱ⁺¹

(i+ 1)!h⁽ⁱ⁺¹⁾_x (ηu)du=:T_x⁽¹⁾(t) +T_x⁽²⁾(t).

It follows that

T_x⁽¹⁾(t) = h⁽ⁱ⁾x (0) i!

Z ∞ 0

uⁱexp(−tu)du=h⁽ⁱ⁾x (0) tⁱ⁺¹ , and the change of variable v=tuyields

T_x⁽²⁾(t) = 1 (i+ 1)!

1 tⁱ⁺²

Z ∞ 0

exp(−v)vⁱ⁺¹h⁽ⁱ⁺¹⁾_x ηv

t

dv.

Therefore, we have, uniformly inx∈C:

|T_x⁽²⁾(t)| ≤sup

z∈Cy≥0

h⁽ⁱ⁺¹⁾_z (y)

1 (i+ 1)!

1 tⁱ⁺²

Z ∞ 0

exp(−v)vⁱ⁺¹dv= 1 tⁱ⁺² sup

z∈Cy≥0

h⁽ⁱ⁺¹⁾_z (y) =O

1 tⁱ⁺²

and thus

ehx(t) = h⁽ⁱ⁾x (0) tⁱ⁺¹ +O

1 tⁱ⁺²

= 1 tⁱ⁺¹

h⁽ⁱ⁾_x (0) +O 1

t

, which achieves the proof.

Proof of Lemma 2−(i) By definition, for all x∈Randt >0, we have µx(t) =

Z ∞ 0

(Kx(u+t)−Kx(t)) exp(−u)du= Z ∞

0

Z u+t t

y^x−1dy

exp(−u)du.

Using Fubini’s theorem, it follows µ_x(t) =

Z ∞ t

y^x−1 Z ∞

y−t

exp(−u)du

dy= exp(t) Z ∞

t

y^x−1exp(−y)dy, (21) and the change of variable u=y/t−1 yields

µx(t) =t^x Z ∞

0

exp(−tu)(u+ 1)^x−1du=t^xehx(t)

withhx(t) = (t+ 1)^x−1. Applying Lemma 1 withi= 0 concludes the proof of (i).

(ii) From (21) we obtain, for x∈Randt >0, µ_x(t) = exp(t)

Z ∞ t

y^x−1exp(−y)dy= exp(t)Γ(x, t), (22) where Γ(x, t) is the upper incomplete gamma function. We thus have (see for instance [1])

∂

∂xµx(t) = exp(t) Z ∞

t

exp(−y) log(y)y^x−1dy, (23)

and the change of variable u=y/t−1 yields

∂

∂xµx(t) = exp(t) Z ∞

0

exp(−tu) exp(−t)t^x−1(u+ 1)^x−1log(t(u+ 1))tdu

= t^x Z ∞

0

exp(−tu)(u+ 1)^x−1log(u+ 1)du+ log(t)t^x Z ∞

0

exp(−tu)(u+ 1)^x−1du

=: t^x Z ∞

0

exp(−tu)gx(u)du+ log(t)µx(t),

withgx(u) := (u+ 1)^x−1log(u+ 1). Applying Lemma 1 withi= 1, the conclusion follows.

Proof of Lemma 3 − (i) First of all, remark thatψ(x;t, t⁰) >0 for allx∈ Rand t > t⁰ >0.

Besides, routine calculations show that

∂

∂xψ(x;t, t⁰) = ψ(x;t, t⁰)

∂/∂x(µx(t))

µx(t) −∂/∂x(µx(t⁰)) µx(t⁰)

(24)

=: ψ(x;t, t⁰) (Qx(t)−Qx(t⁰)),

where, by (21) and (23),

Qx(z) = R∞

z log(y)y^x−1exp(−y)dy R∞

z y^x−1exp(−y)dy . Let us remark that Qxis an increasing function on (0,∞) since

Q⁰_x(z) = z^x−1exp(−z)R∞

z y^x−1exp(−y) log(y/z)dy R∞

z y^x−1exp(−y)dy² >0.

As a consequence t > t⁰ implies ∂/∂x(ψ(x;t, t⁰)) =ψ(x;t, t⁰)(Qx(t)−Qx(t⁰))> 0 and concludes the first part of the proof.

(ii) From (22), we have 1

ψ(x;t, t⁰)

exp(−t⁰)

exp(−t) −1 = µx(t⁰) exp(−t⁰) µ_x(t) exp(−t) −1 =

Rt

t⁰y^x−1exp(−y)dy R∞

t y^x−1exp(−y)dy =:Mx(t, t⁰).

Besides, considering the following inequalities, 0<

Z t t⁰

y^x−1exp(−y)dy < t^x−1(exp(−t⁰)−exp(−t)),

and Z ∞

t

y^x−1exp(−y)dy >

Z ∞ 2t

y^x−1exp(−y)dy >(2t)^x−1exp(−2t), it follows that

M_x(t, t⁰)<2^1−xexp(−t⁰)−exp(−t) exp(−2t) , and thusMx(t, t⁰)→0 asx→ ∞. The conclusion follows.