Quoc-Hung Vu

Jean-Michel Pereira

Anh Minh Tang Navier

Water retention curve of clayey sands determined from pore structure by using various methods

Besides standard approaches, the soil water retention curve (SWRC) can be estimated either from pore size distribution obtained by mercury intrusion porosimetry or from the soil freezing characteristic curve. These methods are based on simplified laws such as Young-Laplace for capillary suction or Clausius-Clapeyron for cryo-suction. These laws might not be valid for clayey sands where the presence of clay particles would induce other water retention mechanisms. This study aims to assess the clay content's effect on the validity of these two methods in determining the SWRC of clayey sands. Clay sands were prepared at different clay contents by mixing pure sand with kaolin clay prior to compaction at the Proctor maximum dry density. Five clay contents (dry mass of clay divided by dry mass of soil) were considered (0, 5, 10, 15, and 20%). The reference SWRC was obtained by standard suction measurements. The results show that SWRCs estimated from mercury intrusion porosimetry and soil freezing characteristic curve are generally in agreement with the reference SWRC. However, the results obtained by the soil freezing characteristic curve are limited in terms of suction range (500 to 5 MPa) which is not appropriate for clayey sands, while those from mercury intrusion porosimetry show significant discrepancy because of the structure heterogeneity obtained by Proctor compaction.

Introduction

The soil water retention curve (SWRC) describes the relationship between water content and suction. This curve plays an important role in modelling the coupled hydro-mechanical behaviour of unsaturated soils [START_REF] Abed | A study on how to couple thermo-hydro-mechanical behaviour of unsaturated soils: Physical equations, numerical implementation and examples[END_REF][START_REF] Sun | Elastoplastic prediction of hydro-mechanical behaviour of unsaturated soils under undrained conditions[END_REF][START_REF] Zhou | An advanced hydro-mechanical constitutive model for unsaturated soils with different initial densities[END_REF]. SWRC is conventionally determined from suction measurement [START_REF] Duong | Development of a Large-Scale Infiltration Column for Studying the Hydraulic Conductivity of Unsaturated Fouled Ballast[END_REF][START_REF] Toll | Advances in suction measurements using high suction tensiometers[END_REF] or control [START_REF] Blatz | Vapour Equilibrium and Osmotic Technique for Suction Control[END_REF][START_REF] Rouf | Unified measurement system with suction control for measuring hysteresis in soil-gas transport parameters[END_REF] following drying and/or wetting processes.

Soil suction (𝑠) generally includes two components (matric and osmotic) and matric suction is generally the most dominant component in sandy soils in the case without salt presence. By using the Young-Laplace equation [START_REF] Young | An essay on the cohesion of fluids[END_REF][START_REF] Laplace | Supplement to book 10 of Mecanique Celeste, Crapelet, Courcier[END_REF], matric suction in porous media can be estimated from pore radius. For this reason, SWRC can also be estimated from the pore size distribution (PSD), which is usually determined by mercury intrusion porosimetry (MIP) [START_REF] Lindquist | Pore and throat size distributions measured from synchrotron X-ray tomographic images of Fontainebleau sandstones[END_REF][START_REF] Wang | Feng Comparison of Pore-Size Distribution of Soils Obtained by Different Methods[END_REF], by using Laplace's equation [START_REF] Delage | Propriétés de rétention d'eau et microstructure de différents géomatériaux[END_REF][START_REF] Turturro | Mercury intrusion porosimetry and centrifuge methods for extended-range retention curves of soil and porous rock samples[END_REF]. However, SWRC obtained from MIP and SWRC obtained by conventional methods generally do not agree for fine-grained soils [START_REF] Muñoz-Castelblanco | The water retention properties of a natural unsaturated loess from northern France[END_REF] probably due to pore trapping effect or water and dissolved salts effect on clay fabric compared to the process of mercury intrusion [START_REF] Romero | Water permeability, water retention and microstructure of unsaturated compacted Boom clay[END_REF].

Besides, soil matric suction can be determined from freezing temperature by using Clausius-Clapeyron equation [START_REF] Edlefsen | C Anderson Thermodynamics of soil moisture[END_REF]. However, the consistency between SWRC obtained from soil freezing characteristic curve and that obtained by conventional methods remains questionable because of the complexity of the mechanisms associated with freezing-thawing compared to drying-wetting, such as supercooling, different contact angles [START_REF] Bittelli | A thermodielectric analyzer to measure the freezing and moisture characteristic of porous media[END_REF], accuracy of water content measurement [START_REF] Yoshikawa | Comparing unfrozen water content measurements of frozen soil using recently developed commercial sensors[END_REF] and assumptions behind Clausius-Clapeyron equation [START_REF] Ren | Comparison of Soil-Freezing and Soil-Water Characteristic Curves of Two Canadian Soils[END_REF][START_REF] Santoyo | A review of the existing data on soil-freezing experiments and assessment of soilfreezing curves derived from soil-water retention curves[END_REF].

Among factors such as dry density, grain nature, grain size distribution, etc., fines content influences significantly SWRC [START_REF] Bicalho | Favero Evaluation of the Soil-Water Retention Curves of Different Unsaturated Silt-Sand Soil Mixtures[END_REF][START_REF] Su | Soil-water retention behaviour of fine/coarse soil mixture with varying coarse grain contents and fine soil dry densities[END_REF]. Several studies investigated SWRC of sandy soils with different clay contents [START_REF] Vanapalli | The influence of soil structure and stress history on the soil-water characteristics of a compacted till[END_REF][START_REF] Baetens | On the influence of coarse fragments on soil water retention[END_REF][START_REF] Gallage | Effects of Dry Density and Grain Size Distribution on Soil-Water Characteristic Curves of Sandy Soils[END_REF][START_REF] Benson | Estimating van Genuchten Parameters α and n for Clean Sands from Particle Size Distribution Data[END_REF] and found that the SWRC of soil with higher clay content has a lower steepness, a higher air entry suction and a higher residual suction. In addition to the aforementioned effects, increasing clay content from 30% to 100% by dry weight may enlarge the hysteresis of SWRC [START_REF] Raghuram | Effect of Fines Content on the Hysteretic Behavior of Water-Retention Characteristic Curves of Reconstituted Soils[END_REF]. For soils with low clay content (0% to 8%), SWRC shows a clear bimodal shape, which is strongly dependent on fines content [START_REF] Zhang | Prediction of Soil-Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents[END_REF].

This study aims at assessing the effect of clay content on the validity of using mercury intrusion porosimetry and soil freezing characteristic curve in determining SWRC of clayey sands. Clayey sands were prepared at different clay contents by mixing pure sand with kaolin clay prior to the compaction at the Proctor maximum dry density. Five clay contents (dry mass of clay divided by dry mass of soil) were considered (0, 5, 10, 15, and 20%). The reference SWRC was obtained by standard suction measurements. SWRC obtained by the three methods were compared. The results were finally analysed to identify the main mechanisms explaining the effect of fines content on the soil water retention curve. 1 and Table 2, respectively) were mixed at various clay contents in dry mass (see Table 3 and Fig. 1). Each mixture was wetted at optimal water content [START_REF] Boussaid | Sols intermédiaires pour la modélisation physique : application aux fondations superficielles[END_REF] and packed in a plastic bag for at least 24 h for moisture distribution homogenization prior to compaction in a rigid metallic cylindrical mould (150 mm in height and 150 mm in diameter).

Material and experimental methods

Table 2. Physical properties of Speswhite Kaolin clay.

Property Value

Liquid limit, LL (%) 55 To determine the reference SWRC following the suction measurement method [START_REF] Tarantino | Benchmark of experimental techniques for measuring and controlling suction[END_REF], after soil compaction in a mould, three sensors were carefully inserted inside the sample (Fig. 2 and Fig. 3): (i) a tensiometer T8 (1 kPa in accuracy) to measure matric suction (𝑠) [START_REF] Duong | Development of a Large-Scale Infiltration Column for Studying the Hydraulic Conductivity of Unsaturated Fouled Ballast[END_REF]; (ii) a time domain reflectometry probe ThetaProbe ML2x (1% in accuracy) to measure volumetric water content (𝜃 9 ); (iii) and a KD2-Pro probe to measure the soil thermal conductivity (whose data were not analysed in this study). Afterwards, the sample was saturated by injecting water from its bottom (during 0.5 to 2 days depending on fines content) until a layer of water of 10 mm was visible on the top of the specimen. To dry the specimen, two holes created on the top cover were opened to allow evaporation from the top surface (Fig. 3). After a certain duration, the two holes were covered by two lids to prevent moisture exchange with the atmosphere. The equilibrium was reached after several hours prior to the next step; this procedure is similar to that used by Nguyen et al. [START_REF] Nguyen | Water retention and thermal conductivity of a natural unsaturated loess[END_REF]. Once suction excesses the capacity of the tensiometer (100 kPa), one small piece of soil was extracted from the sample to determine SWRC within a drying-wetting path by using a chilled-mirror dew-point hygrometer (WP4C) following the procedure used by Wang et al. [START_REF] Wang | Shrinkage behaviour of a compacted limetreated clay[END_REF]. For MIP tests, small samples (less than 1 g) were extracted from the compacted specimen using a cutter and freeze-dried by the procedure proposed by Delage and Pellerin [START_REF] Delage | Influence de la lyophilisation sur la structure d'une argile sensible du Québec[END_REF] prior to conducting mercury intrusion.

For thawing tests, after soil compaction, sensors were installed as shown in Fig. 4. Beside tensiometer and KD2-Pro probe, whose data were not analysed in this study, a ThetaProbe and a temperature sensor (PT100, accuracy ±0.03 °C) were used. Afterwards, soil specimen was saturated prior to being installed in the temperature-controlled bath. To perform the test, soil temperature (T) was first decreased down to -2 °C or -3 °C to freeze soil water. It was afterwards increased in steps of 0.2 °C until 0 °C to thaw the frozen soil.

Results

Similar pattern can be observed on SWRC obtained by conventional method (Fig. 5), data for pure sand (S0) was taken from the study of [START_REF] Doussan | Prediction of unsaturated soil hydraulic conductivity with electrical conductivity[END_REF]. From saturated state (degree of saturation 𝑆 ; = 100%), suction increase induced a first decrease of 𝑆 ; at 𝑠 ~ 5 kPa. Afterward, 𝑆 ; decreased with a lower rate. When suction exceeded 1000 kPa, 𝑆 ; decreased again with a higher rate. For suction higher than 5000 kPa, 𝑆 ; decreased with a smaller rate. At a given suction, 𝑆 ; was higher for a higher clay content. In addition, for the same soil at the same suction, 𝑆 ; was slightly smaller during the wetting path than during the drying path. The hysteresis effect of SWRC can be mostly attributed to pore blocking and capillarity. The fact of adding fines in sandy soil changed the arrangement of pores. The latter made clearer hysteresis effect of SWRC in soils with higher fines content. The wetting path was stopped at suction of around 500 kPa because WP4C is less accurate at suction lower than this value. This bimodal shape has been observed by Zhang et al. [START_REF] Zhang | Prediction of Soil-Water Retention Curves of Road Base Aggregate with Various Clay Fine Contents[END_REF] on well-graded gravel mixed with 0 -8% of Kaolin clay and explained by the dual porosity (micropores between clay particles and macropores between grains). Actually, results of MIP tests, Fig. 6, show clearly two families of pores: macropores (entrance pore diameter 𝑑 = 20 -100 µm) and micropores (𝑑 = 0.1 -1 µm). The volume of macropores was higher at a lower clay content. Note that tests with S10, S15, and S20 were replicated showing a certain discrepancy. In addition, MIP test could not be performed on pure sand (S0). The bimodal SWRC (Fig. 5) can thus be clearly explained by the double porosity observed with MIP test (Fig. 6). The results of this study evidence the effect of fines content on water retention in the high suction range (> 500 kPa) which corresponds to micropores. The discrepancy between MIP results (Fig. 6) can be explained by the material preparation method. Small samples (less than 1 g) were extracted from large specimens mixed and compacted using the Normal Proctor method. In addition, the mixing order (sand-clay-water) and the water content at compaction in the present study could lead to heterogeneous microstructure after Yin et al. [START_REF] Yin | The Influence of Mixing Orders on the Microstructure of Artificially Prepared Sand-Clay Mixtures[END_REF].

All the results of the thawing path obtained during the freezing/thawing tests (Fig. 7) show an increase of 𝜃 9 when 𝑇 increased, indicating the progressing thawing process. Thawing occurred significantly at temperature close to 0 °C for all soils. At a given temperature, soil with higher clay content had higher 𝜃 9 .

Results of MIP and thawing tests were then used to estimate SWRC. For MIP test, Laplace equation that links the capillary pressure to the interfacial properties of the fluids-solid system with entrance pore entrance diameter 𝑑 was applied to obtain the relationship between matric suction and mercury intrusion pressure 𝑢 BC [START_REF] Muñoz-Castelblanco | The water retention properties of a natural unsaturated loess from northern France[END_REF]:

𝑠 = 0.196 𝑢 BC (1)
Water content was calculated from the change of intruded mercury void ratio 𝑒 J by assuming that water content is null at the highest suction (40 MPa).

Clausius-Clapeyron equation was used to estimate suction during thawing:

𝑠 = 𝐿 L 𝜌 9 NON P N P (2)
where 𝜌 9 = 1000 kg/m 3 is the density of water, 𝐿 L = 334 kJ/kg is the latent heat of fusion of water, 𝑇 L = 273.15 𝐾 is the freezing point of water. Fig. 8 shows SWRCs derived from the three methods for each soil. Generally, the results of MIP should be compared with the drying path while the results of the thawing test should be compared with the wetting path. For clean sand S0, data on the wetting path were not available. Anyway, results obtained by thawing are in good agreement with the drying path (which should be similar to the wetting path for this material in the range of suction available with the thawing method, i.e. 500 kPa -3 MPa). For S5, results from MIP is in good agreement with the drying path for the whole range of suction while results from the thawing test are in good agreement with the wetting path (both are limited to suctions higher than 500 kPa). For S10 and S20, observations similar to S5 can be made even if the agreement between MIP and drying path is less obvious, which is consistent with the conclusions by [START_REF] Muñoz-Castelblanco | The water retention properties of a natural unsaturated loess from northern France[END_REF]. For S15, a notable discrepancy between the MIP results of can be observed. The difference between SWRCs obtained by the various methods remains smaller than this discrepancy.

For all soils, results of MIP tests show generally a smaller water content (compared to the other methods) at suction higher than 1000 kPa. This difference is more notable at higher clay content. This phenomenon can be explained by the role played by water adsorption on clay particles on the water retention properties of the soil, which cannot be predicted by MIP data and its interpretation based on capillary suction effects.

Conclusions

SWRCs of clayey sands (having clay content varying from 0 to 20%) were obtained by three methods. The results show that SWRCs estimated from mercury intrusion porosimetry and soil freezing characteristic curve are generally in agreement with the reference SWRC.

MIP tests can provide SWRC covering a wide range of suction (1 kPa to 40 MPa) which is appropriate for clayey sands. However, it underestimates water content at suction higher than 1000 kPa for sands with high clay content because it neglects the presence of adsorbed water on clay particles. In addition, results obtained by MIP test on sands with high clay content show significant discrepancy because of soil structure heterogeneity obtained by Proctor compaction.

Soil water characteristic curve can be used to estimate SWRC. However, this method is limited to a small suction range (500 kPa to 3 MPa in this study), which is not appropriate for clayey sands where a wider range is necessary to cover both macropores related to the arrangement of sand particles and micropores related to clay particles.

45 Fig. 1 .

 451 Fig. 1. Grain size distribution curves of tested soils.

Fig. 2 .

 2 Fig. 2. Schematic view of the experimental device used for suction measurement. (1) Soil; (2) Metallic cylindrical cell; (3) Insulating cover; (4) Tensiometer; (5) Thermal conductivity probe; (6) Soil water content sensor; (7) Data logging system.

Fig. 3 .

 3 Fig. 3. View from the top of the experimental device used for suction measurement.

Fig. 4 .

 4 Fig. 4. Schematic view of the experimental setup used for freezing-thawing tests. (1) Temperature-controlled bath; (2) Soil specimen; (3) Temperature controlling system; (4) Temperaturecontrolled liquid (30% ethylene glycol + 70% water); (5) Metallic cylindrical cell; (6) Insulating cover; (7) Temperature sensor; (8) Tensiometer; (9) Soil water sensor; (10) Thermal conductivity probe; (11) Data logger system.

Fig. 5 .

 5 Fig. 5. SWRC obtained by the conventional method.

Fig. 6 .

 6 Fig. 6. Results of MIP tests: (a) Intruded mercury void ratio and (b) Pore size density function as a function of the entrance pore diameter.

Fig. 7 .

 7 Fig. 7. Results obtained on the thawing path of freezing/thawing tests: volumetric unfrozen water content as a function of the temperature.

Fig. 8 .

 8 Fig. 8. SWRCs determined using the three methods for soils: (a) S0; (b) S5; (c) S10; (d) S15; (e) S20.

Table 1 .

 1 Physical properties of Fontainebleau sand.

	Property	Value
	Median grain size, 𝑫 𝟓𝟎 (mm)	0.21
	Uniformity coefficient, 𝑪 𝑼	1.52
	Minimum void ratio, 𝒆 𝒎𝒊𝒏	0.54
	Maximum void ratio, 𝒆 𝒎𝒂𝒙	0.94
	Particle density, 𝝆 𝐬 (Mg/m 3 )	2.65
	Minimum dry density, 𝝆 𝐝,𝐦𝐢𝐧 (Mg/m 3 )	1.37
	Maximum dry density, 𝝆 𝒅,𝒎𝒂𝒙 (Mg/m 3 )	1.72

Table 3 .

 3 Physical properties of samples.

	Soil	Clay	Water	Dry	Porosity
		content	content 𝒘	density	𝒏
		(%)	(%)	𝝆 𝐝	(-)
				(Mg/m 3 )
	S0	0	0.21	1.70	0.36
	S5	5	0.19	1.77	0.33
	S10	10	0.14	1.93	0.27
	S15	15	0.13	1.99	0.25
	S20	20	0.13	1.96	0.26

E3S Web of Conferences 382, 09006 (2023) https://doi.org/10.1051/e3sconf/202338209006 UNSAT 2023


Jonathan El Methni

Stéphane Girard

Jonathan El Methni

A refined extreme quantiles estimator for Weibull tail-distributions

Keywords: extreme quantile, bias reduction, Weibull tail-distribution, extreme-value statistics, asymptotic normality. Mathematics Subject Classification: 60G70, 62G32, 62G20

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Let X 1 , X 2 , . . . , X n be independent and identically distributed random variables with cumulative distribution function F and let X 1,n ≤ . . . ≤ X n,n denote the associated order statistics. We consider the case where F belongs to the family of Weibull tail-distributions [START_REF] Broniatowski | On the estimation of the Weibull tail coefficient[END_REF]:

(A.1) F is twice differentiable and F (•) = 1-exp(-H(•)) with V (t) := H ← (t) = t θ (t) for all t > 0, where θ > 0 is called the Weibull tail-coefficient and where is a (positive) slowly-varying function i.e. (tx)/ (x) → 1 as x → ∞ for all t > 0.

Here, and in the following, Φ ← (•) = inf{x ∈ R, Φ(x) > •} denotes the generalized inverse of an increasing function Φ. The inverse cumulative hazard function V is said to be regularly-varying at infinity with index θ and this property is denoted by V ∈ RV θ , see [START_REF] Bingham | Regular variation[END_REF] for a detailed account on this topic. The shape parameter θ is referred to as the Weibull tail-coefficient. Weibull tail-distributions are part of the Gumbel maximum domain of attraction, i.e. with extreme-value index γ = 0, see [START_REF] Gardes | Weibull tail-distributions revisited: a new look at some tail estimators[END_REF]Proposition 2(ii)], and as such, are light-tailed distributions. They include for instance exponential (θ = 1), Gamma (θ = 1), logistic (θ = 1), Normal (θ = 1/2) and Weibull distributions (θ is the inverse of the shape parameter), see [START_REF] Girard | A Hill type estimate of the Weibull tail-coefficient[END_REF]Table 1]. We refer to [START_REF] Beirlant | Modeling large claims in non-life insurance[END_REF] for an application to the modeling of large claims in non-life insurance and to [START_REF] Vladimirova | Sub-Weibull distributions: generalizing sub-Gaussian and sub-Exponential properties to heavier-tailed distributions[END_REF] for an analysis of neural networks distributional properties. Dedicated methods have been proposed to estimate the Weibull tail-coefficient θ since the relevant information is localised in the extreme upper part of the sample. Most approaches rely on the k n upper order statistics X n-kn+1,n , . . . , X n,n where k n → ∞ as n → ∞. Note that, since θ is defined through a tail behavior, the associated estimator should only use the extreme-values of the sample and thus the extra condition k n /n → 0 is required. More specifically, recent estimators are based on the log-spacings between the k n upper order statistics [START_REF] Beirlant | Practical analysis of extreme values[END_REF][START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF][START_REF] Gardes | Weibull tail-distributions revisited: a new look at some tail estimators[END_REF][START_REF] Girard | A Hill type estimate of the Weibull tail-coefficient[END_REF] or on the mean excess function [START_REF] Beirlant | Semiparametric lower bounds for tail index estimation[END_REF][START_REF] Beirlant | The mean residual life function at great age: Applications to tail estimation[END_REF][START_REF] Beirlant | A new estimation method for Weibull-type tails based on the mean excess function[END_REF]. The introduction of kernel based weights has been investigated for both approaches, see [START_REF] Gardes | Estimation of the Weibull tail-coefficient with linear combination of upper order statistics[END_REF] for the log-spacings case and [START_REF] Goegebeur | A weighted mean excess function approach to the estimation of Weibull-type tails[END_REF] for the mean excess function framework. A bias reduction method adapted to the estimation of the Weibull tail-coefficient is proposed in [START_REF] Diebolt | Bias-reduced estimators of the Weibull tail-coefficient[END_REF] and the adaptation to random censoring is achieved in [START_REF] Worms | Estimation of extremes for Weibull-tail distributions in the presence of random censoring[END_REF].

We address the problem of estimating extreme quantiles of Weibull tail-distributions. Recall that an extreme quantile q(α n ) of order α n is defined by q(α n ) = F ← (1 -α n ) with nα n → 0 as n → ∞. The latter condition implies that q(α n ) is almost surely asymptotically larger than X n,n , the sample maximum. It is shown in [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF] that classical extreme-value estimators of such large quantiles are numerically outperformed by estimators dedicated to Weibull tail-distributions [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF], see also Lemma 2 in the Appendix for a theoretical argument. The latter methods estimate q(α n ) by combining two ingredients: an order statistic X n-kn+1,n and an estimator of the Weibull tailcoefficient θ used to extrapolate from this anchor point.

In this work, we show that the biases associated with the previous extrapolation method and the estimator of θ may asymptotically cancel out in the extreme quantile estimator thanks to an appropriate tuning of the number of upper order statistics involved in the Weibull tail-coefficient estimator. The construction of the resulting estimator is presented in Section 2 and an asymptotic normality result is provided, emphasizing that the proposed extreme quantile estimator is asymptotically less biased than the original one [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF]. Its performances are illustrated on simulated data in Section 3 and compared to three state-of-the-art competitors [START_REF] Beirlant | A new estimation method for Weibull-type tails based on the mean excess function[END_REF][START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF][START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF]. An illustration on a real data set of daily wind measures is provided in Section 4. Finally, a small conclusion is proposed in Section 5 and the proofs are postponed to the Appendix.

2 A refined estimator of the extreme quantile

Extreme quantile estimators

Weibull-tail estimators of the extreme quantile q(α n ) rely on an intermediate quantile q(k n /n) where (k n ) is an intermediate sequence of integers i.e. such that k n ∈ {1, . . . , n -1}, k n → ∞ and k n /n → 0 as n → ∞, see for instance [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF][START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF]. Indeed, in view of (A.1), one has

q(α n ) q(k n /n) = V (log(1/α n )) V (log(n/k n )) log(1/α n ) log(n/k n ) θ =: τ θ n , (1)
as n → ∞, where τ n = log(1/α n )/log(n/k n ) is the (logarithmic) extrapolation factor. From an intuitive point of view, an extreme quantile can thus be approximated by multiplying an intermediate quantile by an appropriate extrapolation term: q(α n ) q(k n /n)τ θ n . The intermediate quantile q(k n /n) can then be estimated by its empirical counterpart X n-kn+1,n while the extrapolation term depends on the tail heaviness through θ which has to be estimated as well. Following the ideas of [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF], we propose a refined Weissman [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF] type estimator:

qn (α n , k n , k n ) = X n-kn+1,n log(1/α n ) log(n/k n ) θn(k n ) = X n-kn+1,n τ θn(k n ) n , (2)
with θn (k n ) an estimator of θ depending on another intermediate sequence (k n ). Let us focus on the estimator introduced in [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]:

θRSH n (k n ) = 1 µ(log(n/k n )) 1 k n k n i=1 (log X n-i+1,n -log X n-k n +1,n ), (3)
with, for t > 0, µ(t)

= +∞ 0 log 1 + x t e -x dx = e t E 1 (t)
, where E 1 is the Exponential integral function [START_REF] Abramowitz | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]Page 228]. This estimator is motivated by the remark that, in view of (1), the logspacings between two quantiles are approximately proportional to θ. This property is also used in the real data application (see the top-right panel of Figure 4) to visually check the Weibull-tail assumption. Clearly, θRSH n (•) can be interpreted as a rescaled Hill estimator since

θRSH n (k n ) = γH n (k n ) µ(log(n/k n )) , (4)
where γH n (•) is the well-known Hill estimator [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] of the extreme-value index γ. Let us note, when k n = k n , one recovers the extreme quantile estimator for Weibull taildistributions introduced in [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF]. In the next paragraph, the asymptotic normality of qn (α n , k n , k n ) is established, and it is shown that choosing k n = k n can yield better results from an asymptotic point of view. A similar phenomenon occurs in the estimation of the endpoint of a distribution in the Weibull maximum domain of attraction, see [START_REF] Aarssen | On the maximal life span of humans[END_REF] for details. We also refer to [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF] for the estimation of the tail-index in the Fréchet maximum domain of attraction.

Asymptotic analysis

The study of the limit distribution of qn (α n , k n , k n ) requires a second-order condition on the slowlyvarying function introduced in (A.1):

(A.2) There exist ρ ≤ 0 and b(t) → 0 as t → ∞, with ultimately constant sign, such that uniformly locally on x ≥ 1,

lim t→∞ 1 b(t) log (tx) (t) = K ρ (x) := x 1 u ρ-1 du.
It can be shown that necessarily |b| ∈ RV ρ . The second-order Weibull parameter ρ ≤ 0 tunes the rate of convergence of the ratio (tx)/ (t) to 1. The closer ρ is to 0, the slower is the convergence. Condition (A.2) is the cornerstone in all proofs of asymptotic normality for extreme-value estimators. Again, we refer to [20,

(i) k n b(log(n/k n )) → λ ∈ R, (ii) log(n/k n )/ log(n/k n ) → β ≥ 1, (iii) τ n → τ > β.
Then, as n → ∞,

k n qn (α n , k n , k n ) q(α n ) -1 d -→ N λ(log(τ ) -β -ρ K ρ (τ )), (θ log τ ) 2 . ( 5
)
Let us first remark that condition (i) implies log(n/k n ) ∼ log(n) as n → ∞ (see [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]Lemma 5.1]), then condition (ii) yields log(n/k n ) ∼ log(n)/β and therefore condition (iii) can be rewritten as

log(1/α n ) ∼ (τ /β) log(n) as n → ∞.
As a consequence, the condition τ > β in (iii) implies nα n → 0 as n → ∞ which, in turns, implies that q(α n ) is an extreme quantile. It follows from (5) that the asymptotic bias associated with qn (α n , k n , k n ) is given by

log τ -β -ρ K ρ (τ ) b(log(n/k n )) ∼ (β ρ log(τ ) -K ρ (τ )) b(log(n/k n )) =: B(β, τ, ρ)b(log(n/k n )),
since |b| ∈ RV ρ . It appears that each choice of k n yields an associated constant β in (ii) and thus a corresponding bias factor B(β, τ, ρ) = β ρ log(τ ) -K ρ (τ ). From the theoretical point of view, two cases can be considered.

• The usual choice k n = k n yields β = 1 and one recovers [17, Corollary 3.1] as a particular case of Theorem 1. Moreover, for all τ > 1, ρ ≤ 0,

B(1, τ, ρ) = log(τ ) -K ρ (τ ) ≥ 0, (6)
which is the (positive) bias factor associated with the extreme quantile estimator qn (α n , k n , k n ) investigated in [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]. Note that ρ → B(1, τ, ρ) is a decreasing function such that B(1, τ, 0) = 0 which is an unusual situation in extreme-value theory. For instance, the bias factor associated with the Weissman estimator [START_REF] Weissman | Estimation of parameters and large quantiles based on the k largest observations[END_REF] dedicated to heavy-tailed distributions is proportional to 1/(1 -ρ) and increases with ρ, see [22, Theorem 3.2.5 and Theorem 4.3.8].

• The choice β (τ, ρ) := (K ρ (τ )/ log(τ )) 1/ρ yields B(β (τ, ρ), τ, ρ) = 0.

The associated intermediate sequence is given by k n (τ, ρ) = n(k n /n) β (τ,ρ) and therefore the extreme quantile estimator qn (α n , k n , k n (τ, ρ)) is asymptotically unbiased. Note that this estimator cannot be used in practice since the second-order Weibull parameter ρ is unknown. Up to our knowledge, there is no estimator of the second-order Weibull parameter in the statistical literature. In practice, one can replace the true unknown value of ρ by a misspecified value y ≤ 0 in the above β (τ, ρ) leading to

β (τ, y) = (K y (τ )/ log(τ )) 1/y , (8)
k n (τ, y) = n(k n /n) β (τ,y) , (9)
B(β (τ, y), τ, ρ) = β (τ, y) ρ log(τ ) -K ρ (τ ) (10)
= (K y (τ )/ log(τ )) ρ/y log(τ ) -K ρ (τ ),
with ρ ≤ 0 and τ > 1. This misspecification technique has been used both to deal with Paretotype distributions (γ > 0), see for instance [START_REF] Feuerverger | Estimating a tail exponent by modelling departure from a Pareto distribution[END_REF], and Weibull tail-distributions (γ = 0) [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF]. Some properties of the intermediate sequence k n (τ, y) are given in the next Lemma.

Lemma 1. Let β (τ, y) and k n (τ, y) be defined by ( 8) and (9) respectively. Then, for all τ > 1:

(i) β (τ, y) → 1 as y → -∞ and β (τ, •) can be extended by continuity by setting β (τ, 0) := √ τ .

(ii) 1 < β (τ, y) < τ for all y ≤ 0.

(iii) For all y ≤ 0, k n (τ, y) is an increasing function of k n , k n (τ, y) ≤ k n and k n (τ, y)/k n → 0 as n → ∞.

(iv) k n (τ, y) is a decreasing function of y ∈ (-∞, 0).

In particular, it appears in (iii) that the number of upper order statistics k n (τ, y) used in the Weibull tail-coefficient estimator should be asymptotically small compared to k n for all finite values of y. From (iv), this is all the more true as y is large. When y → -∞, meaning that one does not take into account the bias, (i) shows that k n (-∞, τ ) = k n is recovered as a limit case. These properties are illustrated on the left panel of Figure 1, where k n is drawn as a function of k n for several values of y. The next Corollary shows that these choices indeed lead to a bias reduction in the estimation of the extreme quantile.

Corollary 1. Assume (A.1) and (A.2) hold. Let c > 0, τ > 1, y ≤ 0, λ = 0 such that λb(•) is ultimately positive, and β (τ, y) be defined as in [START_REF] Beirlant | A new estimation method for Weibull-type tails based on the mean excess function[END_REF]. y) and define k n (τ, y) as in [START_REF] Beirlant | Modeling large claims in non-life insurance[END_REF].

Let α n = c n -τ /β (τ,y) , k n = n{λ 2 /(nb 2 (log n))} 1/β (τ,
(i) Then, as n → ∞, k n (τ, y) qn (α n , k n , k n (τ, y)) q(α n ) -1 d -→ N λ(log(τ ) -(K y (τ )/ log(τ )) -ρ/y K ρ (τ )), (θ log τ ) 2 .
: Graphs of k n ∈ {2, . . . , 500} → k n (τ n , y) for y ∈ {-∞, -2, -1, -1/2, 0} respectively in {black, blue, red, violet, green}. Right: graphs of k n ∈ {2, . . . , 500} → B(1, τ n , ρ) (dotted lines) and k n ∈ {2, . . . , 500} → B(β (τ n , ρ # = -1), τ n , ρ) (solid lines)
given in Equations ( 6) and ( 10), with ρ ∈ {-2, -1, -1/2} respectively in {blue, red, violet}. On both panels:

τ n = log(1/α n )/log(n/k n ) with α n = 1/n and n = 500.
(ii) Moreover, for all τ > 1, y ≤ 0:

|B(β (τ, y), τ, ρ)| < B(1, τ, ρ) for all ρ < 0, B(β (τ, y), τ, 0) = B(1, τ, 0) = 0.
Let us first highlight that k n (τ, y) ∼ λ/b(log n) as n → ∞ (see the proof of Corollary 1 in the Appendix) which is the (logarithmic) rate of convergence of usual extreme quantile estimators dedicated to Weibull tail-distributions, see for instance [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF]Theorem 1]. In contrast, the rate of convergence of extreme quantile estimators is a power function of n in the Fréchet maximum domain of attraction, see [START_REF] De Haan | Extreme value theory: an introduction[END_REF]Theorem 4.3.8 and Equation (3.2.10)] for Weissman estimator and [4, Corollary 2] for the associated refined version. This may be seen as a consequence of Lemma 2 in the Appendix where it is established that the second-order parameter associated with Weibull distributions is ψ = 0. Surprisingly, as a consequence of Corollary 1(ii), the extreme quantile estimator qn (α n , k n , k n (τ, y)) computed with k n (τ, y) defined in ( 8) and ( 9) has a smaller asymptotic bias than the usual one qn (α n , k n , k n ) whatever the chosen value y ≤ 0. Let us recall that, from [START_REF] Beirlant | The mean residual life function at great age: Applications to tail estimation[END_REF], the theoretical best choice would be y = ρ. In practice, we use y = ρ # = -1 leading to β (τ, -1) = τ log(τ )/(τ -1). This "canonical" choice is also used in [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF], see Section 3.2 hereafter. Let us stress that the use of a similar bias reduction method in the Fréchet maximum domain attraction [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF] is not based on such a misspecification technique but requires the estimation of ρ.

Corollary 1(ii) is illustrated on the right panel of Figure 1 through the graphical comparison of the bias factors associated with β = β (τ, -1) (refined Weibull-tail estimator) and β = 1 (usual Weibull-tail estimator [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]). It clearly appears that, from the theoretical point of view, the first choice yields smaller bias factors in absolute value than the second one. The performance of the refined estimator in practice is assessed on simulated data in the next Section.

Validation on simulated data

The refined extreme quantile estimator is compared on simulated data to the original estimator [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF] and to two other competitors described hereafter.

Experimental design

Let us consider the class of D(ζ, η, a)-distributions which is an adaptation of Hall's class [START_REF] Hall | On some simple estimates of an exponent of regular variation[END_REF][START_REF] Hall | Adaptive estimates of parameters of regular variation[END_REF] to the Weibull-tail framework. In this family, the inverse cumulative hazard function is defined for all x > 0 by

V (x) := x 1/ζ 1 + a η x -η ,
(x) = log(x)/(4x)), the Gamma distribution G(υ = 1, λ) (θ = 1, ρ = -1 and b(x) = (1 -υ) log(x)/x) and the Weibull distribution W(υ, λ) (θ = 1/υ, ρ = -∞ and b(x) = 0)
, it is thus possible to obtain D-distributions with arbitrary Weibull tail-coefficient θ > 0 and second-order Weibull parameter ρ ∈ [-θ, 0).

In the following, we set θ ∈ {1/2, 3/4, . . . , 5/2}, ρ ∈ {-5/2, -2, . . . , -1/2}, a = 10 and focus on the only 25 situations of the D-distribution where ρ ≥ -θ to fulfill the constraint ζη ≤ 1, see Table 1. We also consider 5 situations from the (absolute) Normal distribution N (µ, σ = 1) with µ ∈ {1, 3, 5, 7, 9}, 4 situations from the Gamma distribution G(υ, λ = 1) with υ ∈ {4, 6, 8, 10} and 2 situations from the Weibull distribution W(υ, λ) with υ = λ ∈ {1/2, 2}. In each case, N = 1, 000 replications of a data set of n = 500 i.i.d. realisations are simulated from the 25 + 5 + 4 + 2 = 36 considered parametric models. Finally, the same two cases as in [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF] are investigated for the order of the extreme quantile: α n ∈ {1/n 2 , 1/n 4 }. Summarizing, this experimental design includes

36 × 2 = 72 configurations. (ρ, θ) 1/2 3/4 1 5/4 3/2 7/4 2 9/4 5/2 -∞ W 1 W 1 -5/2 D 1 -2 D 1 D 1 D 1 -3/2 D 1 D 1 D 1 D 1 D 1 -1 N 5 G 4 /D 1 D 1 D 1 D 1 D 1 D 1 D 1 -1/2 D 1 D 1 D 1 D 1 D 1 D 1 D 1 D 1 D 1
Table 1: All considered configurations for (ρ, θ). The letter stands for the distribution and the subscript for the number of investigated situations. As an example, N 5 corresponds to the (absolute) Normal distribution N (µ, σ = 1) where five cases are considered µ ∈ {1, 3, 5, 7, 9}.

Competitors

The refined estimator dedicated to the estimation of extreme quantiles for Weibull tail-distributions is compared to three competitors. All three estimators share the same structure and rely on three quantities, i.e. the order statistic X n-kn+1,n , an extrapolation term and an estimator of the Weibull tail-coefficient. Let us first consider the estimator (3) of the Weibull tail-coefficient introduced in [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]. The extreme quantile estimator proposed in [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF] can be interpreted as a particular case of (2) with

k n = k n and θn (•) = θRSH n (•), see (4): qRSH n (α n , k n ) = X n-kn+1,n τ θRSH n (kn) n . ( 11
)
More recently, an estimator of the Weibull tail-coefficient based on the mean excess function t →

m(t) = E(X -t | X > t)
has been introduced in [START_REF] Beirlant | A new estimation method for Weibull-type tails based on the mean excess function[END_REF]. In practice, the authors estimate m(X n-j,n ) for all j ∈ {1, . . . , k n } by its empirical counterpart:

mn (X n-j,n ) = 1 j j i=1 X n-i+1,n -X n-j,n ,
which leads to the following estimator of θ based on log-spacings between the mean excesses: 3) yields the following estimator of the extreme quantile:

θMEF n (k n ) =   1 - 1 γH n (k n ) 1 k n kn j=1 log mn (X n-j,n ) -log mn (X n-kn-1,n )   -1 . Letting k n = k n and θn (•) = θMEF n (•) in (
qMEF n (α n , k n ) = X n-kn+1,n τ θMEF n (kn) n . ( 12
)
Up to our knowledge there exists only one bias-reduced extreme quantile estimator dedicated to Weibull tail-distributions. This estimator [START_REF] Diebolt | Bias-reduced extreme quantile estimators of Weibull tail-distributions[END_REF] is based on a least-squares approach and involves a bias-reduced estimator of the Weibull tail-coefficient proposed by the same authors [START_REF] Diebolt | Bias-reduced estimators of the Weibull tail-coefficient[END_REF]:

θLSE n (k n ) = Ȳkn -b(log(n/k n ))x kn ,
where

Ȳkn = 1 k n kn j=1 Y j with Y j = j log(n/j)(log X n-j+1,n -log X n-j,n ), xkn = 1 k n kn j=1 x j with x j = log(n/k n )/ log(n/j),
and where

b(log(n/k n )) = kn j=1 (x j -xkn )Y j kn j=1 (x j -xkn ) 2 .
The associated extreme quantile estimator is defined as

qLSE n (α n , k n ) = X n-kn+1,n τ θLSE n (kn) n exp b(log(n/k n ))K ρn (τ n ) . (13)
The authors suggest to choose in practice ρ n = ρ # = -1. This estimator features two bias corrections: a first one in the estimator of the Weibull tail-coefficient and a second one in the extrapolation term. This estimator is built under the assumption that x|b(x)| → ∞ as x → ∞ leading to the constraint ρ ≥ -1. The latter assumption is fulfilled by the class of D(ζ, η, a)distributions when η ≤ 1.

Finally, recall that our estimator is given by

qRWT n (α n , k n , k n (τ n , -1)) = X n-kn+1,n τ θRSH n (k n (τn,-1)) n , ( 14
)
where 1) and β (τ n , -1) = τ n log(τ n )/(τ n -1). For the sake of simplicity, the above extreme quantile estimators ( 11)-( 14) are respectively referred to as RSH, MEF, LSE and RWT in the sequel.

k n (τ n , -1) = n(k n /n) β (τn,-

Selection of the intermediate sequence

All four considered extreme quantile estimators (RWT, RSH, LSE, MEF) depend on the intermediate sequence k n . The selection of k n is a crucial point which has been widely discussed in the extreme-value literature. A new algorithm for the selection of k n is proposed in [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF], basing on a bisection method inspired from random forests. The objective is to find the region with the smallest variance in a given series {Z 1 , . . . , Z m }. The proposed method starts by randomly splitting the series into two parts, computes the variance in each sub-region and repeats the action in the one with smallest variance until getting a final single point, see [4, Algorithm 2]. The above procedure is embedded in a bootstrap technique, see [4, Algorithm 1], and the final k † n is selected by taking the median across the T = 10, 000 bootstrap samples. In the simulations, Z j = q(α n , k j,n ), an estimator (RWT, RSH, LSE or MEF) of the extreme quantile at level α n computed with the intermediate sequence k j,n ∈ {15, 16, . . . , 3n/4}.

Results

The performance of the four extreme quantile estimators is assessed using the Mean absolute relative error:

MARE (q n (α n )) = 1 N N i=1 q(i) n (α n , k (i, †) n ) q n (α n ) -1 , (15)
where

q(i) n (α n , k (i, †) n
) denotes the estimator computed on the ith replication, i ∈ {1, . . . , N = 1, 000} with the intermediate sequence k (i, †) n selected using the above described procedure. The computed MAREs are provided in Table 4 and Table 6 for α n = 1/n 2 and in Table 5 and Table 7 for

α n = 1/n 4 .
The results are summarized in Table 2: We start by remarking that, as expected, the smaller the order α n of the extreme quantile is, i.e. the more one extrapolates, the larger the error is. This is true for all four considered estimators. The proposed RWT estimator is the most accurate one in average since it provides the best results in 48% of cases. Let us remark that, since we fixed ρ # = -1, the RWT estimator performs best overall when ρ is close to -1. The second most accurate estimator is LSE which provides the best results in 26% of the considered cases (19 out of 72 situations). As expected, and similarly to the RWT estimator, it performs well when ρ = -1. The RSH estimator shares similar performances with 25% of best results. It is remarkably efficient when ρ = -∞ (all 4 situations) and more surprisingly when ρ = -1/2 where it obtains 14 best results. RSH performs well in this difficult case despite the fact that it does not benefit from a bias reduction. This unexpected performance may be explained by the relatively small bias factor, see the graph of B(1, •, -1/2) in the right panel of Figure 1. The four cases where RSH fails to obtain the best results when ρ = -1/2 correspond to a Weibull tail-coefficient θ smaller than 1. Finally, MEF yields very poor estimations (even in the strict Weibull case), with less than 2% of best results (only 1 situation). In particular, it does not give acceptable results (with MARE ≥ 1) in 22% of the considered situations. 4567. Best estimator of the extreme quantiles q(α n = 1/n 2 ) & q(α n = 1/n 4 ) computed on simulated data from Weibull tail-distributions. The situations in bold are illustrated for the D(ζ = 1/θ, η = -ρ, a = 10)-distribution on Figure 2 and Figure 3.

(ρ, θ) 1/2 3/
As an illustration, the median and MARE associated with the RWT, RSH and LSE estimators computed on a D(ζ, η, a = 10)-distribution for α n = 1/n 2 are depicted on Figure 2 and Figure 3 as functions of k n . In Figure 2, the Weibull tail-coefficient is fixed to θ = 3/2 and ρ ∈ {-1/2, -1, -3/2} decreases (from top to bottom), while, in Figure 3, the second-order Weibull parameter is fixed to ρ = -1 and θ ∈ {1, 3/2, 2} increases (from top to bottom). In most of these situations, the RWT estimator has the smallest bias and thus the minimum value of the MARE is reached for larger values of k n than RSH and LSE. To conclude, it appears on these experiments that, overall, the RWT estimator performs the best within the four considered estimators. One of its main competitors is LSE, which, similarly to RWT, considers the two sources of bias (associated with the Weibull tail-coefficient estimator and the extrapolation term).

Illustration on a real data set

We study a data set of daily wind measures (in m/s) at Reims (France) from 01/01/1981 to 04/30/2011. For seasonality reasons, only the months from October to March are considered, resulting in n = 5, 371 measures, see the top-left panel of Figure 4 for an histogram of the considered data. It is shown in [START_REF] Albert | Asymptotic behavior of the extrapolation error associated with the estimation of extreme quantiles[END_REF] that the Weibull tail model represents fairly well the upper tail of these data. The goal is to estimate the extreme quantile q(1/n) (with 1/n 1.86 • 10 -4 ) and to compare it to the maximum of the sample x n,n = 42.26 m/s.

To this end, the Weibull tail-coefficient is estimated first by θRWT

n (k † n ) = θRSH n ( k n ) = 0.5597, where k † n = 2
, 877 has been selected following the procedure introduced in [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF] and sketched in Subsection 3.3. This yields k n = k n (τ n , ρ # ) = 961 where we set ρ # = -1. As a visual check, a Weibull quantile-quantile plot of the log-excesses (log X n-i+1,n -log X n-k n +1,n ) versus (log log(n/i)log log(n/ k n )) for i ∈ {1, . . . , k n } is drawn on the top-right panel of Figure 4. The relationship appearing in this plot is approximately linear, which constitutes an empirical evidence that the Weibull-tail assumption makes sense and that k n = 961 is a reasonable choice for the estimation of the Weibull tail-coefficient. A line with the estimated value θRWT n (k † n ) = 0.5597 as slope is added to the quantile-quantile plot highlighting the linear relationship. The function k n → θRSH n (k n ) is plotted on the bottom-left panel of Figure 4, it features as a nice stability for all k n ∈ {100, . . . , 4000}. A similar procedure is carried out for the other three estimators to select k † n . The two Weibull tail-coefficient estimators RSH and MEF that do not benefit from a bias reduction provide respectively the smallest and the largest estimation:

θRSH n (k † n ) = 0.

Conclusion

As a conclusion, the RWT estimator is an efficient tool for estimating extreme quantiles from Weibull tail-distributions. It relies on the ideas of [START_REF] Allouche | A refined Weissman estimator for extreme quantiles[END_REF], consisting in selecting carefully two intermediate sequences to reduce the asymptotic bias of a Weissman type estimator. In contrast to this previous work, the proposed approach does not rely on a preliminary estimate of the second-order parameter; Any negative value may be used, and does yield an asymptotic bias reduction, as shown in our theoretical results. Other surprising features of Weibull tail-distributions can be found in [START_REF] Asimit | Pitfalls in using Weibull tailed distributions[END_REF]. The proposed method provides satisfying results in our numerical experiments and outperforms all its competitors in half of the considered situations. This work could be extended by investigating the adaptation of this bias reduction principle to other estimators of extreme quantiles from Weibull tail-distributions. (k † n ) = 0.5597 is superimposed in red. Bottom-left: Estimates of the Weibull-tail coefficient RSH (blue), LSE (green) and MEF (red) as functions of k n . The range is limited to k n ∈ {100, . . . , 4000} for the sake ok readability. The pair (k † n , θn (k † n )) associated with the selected value k † n is emphasized by a circle. The RWT estimate is represented by an orange circle on the RSH curve. Bottom-right: Estimates of the extreme quantile q(α n = 1/n) by RWT (orange), RSH (blue), LSE (green) and MEF (red) as functions of k n with their associated k † n emphasized by a circle. The sample maximum x n,n is depicted by a black dashed line.

Recalling that, from [START_REF] Gardes | Comparison of Weibull tail-coefficient estimators[END_REF]Lemma 5.3], µ(t) ∼ 1/t as t → ∞, the above expansion can be simplified as

k n log qn (α n , k n , k n ) q(α n ) = θ log(τ )ξ n + k n /k n log(n/k n ) θξ n + k n {log(τ )b(log(n/k n ))(1 + o(1)) -K ρ (τ )b(log(n/k n ))(1 + o(1))} + O P k n k n log 2 (n/k n ) .
Finally, remark that assumption (ii) implies k n ≤ k n eventually and b(log

(n/k n )) ∼ β -ρ b(log(n/k n ))
as n → ∞ so that

k n log qn (α n , k n , k n ) q(α n ) = θ log(τ )ξ n + k n b(log(n/k n )) log(τ ) -β -ρ K ρ (τ ) + o(1) (1 + o(1)
)

+ o P (1)
.

Assumption (i) then yields k n log qn (α n , k n , k n ) q(α n ) d -→ N λ(log τ -β -ρ K ρ (τ )), (θ log τ ) 2
and a first order Taylor expansion proves the result.

Proof of Lemma 1. (i) Remarking that K y (τ ) ∼ -1/y as y → -∞ for all τ > 1 yields β (y, τ ) → 1 as y → -∞. The result β (τ, 0) := √ τ follows from a second-order Taylor expansion. (ii) First, Lemma 4(iii) implies that, for all τ > 1 and y < 0, h y (τ ) > 1 and thus β (τ, y) = h y (τ ) -1/y > 1. Second, Lemma 4(iii) implies that, for all τ > 1 and y < 0, h y (τ ) < τ -y/2 < τ -y when y < 0. This straightforwardly implies that K y (τ )/ log(τ ) > τ y which is equivalent to β (τ, y) < τ . In the particular case where y = 0, from (i), one can take β (τ, 0) := √ τ < τ since τ > 1 and the result is proved. (iii) Let us first consider kn (τ, y) = n(k n /n) β (τ,y) such that k n (τ, y) = kn (τ, y) . Clearly, kn (τ, y)/k n = (k n /n) β (τ,y)-1 and β (τ, y) > 1 in view of (ii). As a consequence, for all y ≤ 0, kn (τ, y) is an increasing function of k n , kn (τ, y) ≤ k n and kn (τ, y)/k n → 0 as n → ∞. These properties can be extended to k n (τ, y) without difficulty since the integer part is an increasing function and k n (τ, y) ≤ kn (τ, y). (iv) Routine calculations give for all τ > 1 and y < 0,

∂ ∂y log(β * (τ, y)) = 1 y 2 (τ y -1) τ y log(τ y ) -τ y + 1 -τ y log τ y -1 log(τ y ) + log τ y -1 log(τ y ) =: 1 y 2 (τ y -1)
ϕ(τ, y).

Letting x := τ y ∈ (0, 1) yields

φ(x) := ϕ(τ, y) = x log(x) -x + 1 -x log x -1 log(x) + log x -1 log(x) ,
and differentiating, one gets

φ (x) = -log 1 - 1 x 1 log(x) + 1 - 1 x 1 log(x) -1 = -log(u(x)) + u(x) -1,
where u(x) := (1 -1/x) / log(x) > 0. It thus appears that φ (x) ≥ 0 for all x ∈ (0, 1) since -log(u) + u -1 ≥ 0 for all u > 0. As a consequence, φ(•) is an increasing function on (0, 1). Moreover, taking account of φ(x) → 0 as x → 1 -shows that φ(x) ≤ 0 for all x ∈ (0, 1). Finally, τ y -1 < 0 and ϕ(τ, y) ≤ 0 for all τ > 1 and y < 0 imply that β * (τ, y) is an increasing function of y which in turns shows that k n (τ, y) is a decreasing function of y.

Proof of Corollary 1. (i) To prove the convergence in distribution, it is sufficient to show that conditions (i), (ii) and (iii) of Theorem 1 hold. Let y ≤ 0 and τ > 1. First, one can easily check that k n (τ, y)b 2 (log n) → λ 2 as n → ∞ and thus k n (τ, y)b(log n) → λ in view of the sign assumption on λ. Besides,

log(n/k n (τ, y)) = log n + 2 log |b(log n)| -2 log |λ| + o(1) ∼ log n, (19)
since b(•) is regularly-varying so that b(log(n/k n (τ, y))) ∼ b(log n) and thus

k n (τ, y)b(log(n/k n (τ, y))) → λ,
as n → ∞. Theorem 1(i) is thus proved. Second, observe that

τ n = log(1/α n ) log(n/k n ) = log(1/c) log(n/k n ) + τ β (τ, ρ) log n log(n/k n ) and log(n/k n ) = 1 β (τ, ρ) (log n + 2 log |b(log n)| -2 log |λ|) + o(1) ∼ 1 β (τ, ρ) log n, (20)
as n → ∞. It is thus clear that τ n → τ as n → ∞, which is Theorem 1(ii). Third, Theorem 1(iii) is a straightforward consequence of ( 19) and ( 20).

(ii) Proposition 1 concludes the proof.

A.2 Auxiliary results

Let us begin with a Lemma that establishes that the strict Weibull distribution belongs to the Gumbel maximum domain of attraction (γ = 0), and more importantly, with a second-order parameter ψ = 0. This result illustrates why inference on Weibull-tail distributions may be difficult since the situation γ = ψ = 0 is the most complicated one for classical extreme-value estimators. Let us also recall that, in contrast, the second-order Weilbull parameter is ρ = -∞, see [20, Table 1] and therefore strict Weibull distributions are an easy situation for dedicated Weibull-tail estimators.

Lemma 2. Suppose F is the cumulative distribution function of a strict Weibull distribution with shape parameter υ > 0, υ = 1 and scale parameter λ > 0. Then, the associated tail quantile function

U (•) := F ← (1 -1/•) verifies the second-order condition 1 A(t) U (tx) -U (t) a(t) -K γ (x) → x 1 s γ-1 K ψ (s)ds,
as t → ∞, for all x > 0, see [START_REF] De Haan | Extreme value theory: an introduction[END_REF]Equation (3.4.5)], with γ = 0, ψ = 0, a(t) = (λ/υ)(log t) 1/υ-1 and A(t) = (1 -υ)/(υ log t).

The following three analytical results are used to prove Proposition 1 below.

Lemma 3. Let us define for all (u, β) ∈ (0, 1] × [0, 1/2], g β (u) := log(u)(1 + u β ) -2(u -1). Then, ∀β ∈ [0, 1/2] one has g β (u) < 0 if u ∈ (0, 1) and g β (1) = 0. Then, (i) h y (•) can be extended by continuity letting h y (1) = 1 for all y ≤ 0.

(ii) h y (•) is an increasing function on [1, ∞) for all y < 0 (and h 0 (•) is a constant function).

(iii) 1 < h y (τ ) < τ -y/2 for all τ > 1 and y < 0 (all three quantities coincide at y = 0). (iii) 0 < f y (τ ) < 1/2 for all τ ≥ 1 and y < 0 (the last two quantities coincide at y = 0).

The next Proposition establishes two unexpected results. First, the bias associated with the refined estimator of extreme quantiles is strictly smaller than the bias associated with the original one, even though a misspecification of the second-order Weibull parameter is used. Second, both (asymptotic) biases vanish at ρ = 0.

Proposition 1. For all τ > 1, y ≤ 0 and ρ < 0, |B(β (τ, y), τ, ρ)| < B(1, τ, ρ), see (6) and [START_REF] Beirlant | Practical analysis of extreme values[END_REF].

Besides, B(β (τ, y), τ, 0) = B(1, τ, 0) = 0 for all τ > 1 and y ≤ 0.

A.3 Proofs of auxiliary results

Proof of Lemma 2. Let θ = 1/υ and x > 0. A second-order Taylor expansion yields, as t → ∞,

U (tx) -U (t) = λ(log t) θ 1 + log x log t θ - 1
= λθ(log x)(log t) θ-1 1 + θ -1 2 log x log t (1 + o(1)) .

Remarking that K 0 (x) = log x and letting a(t) = λθ(log t) where A(t) := (θ -1)/ log t, the result is thus proved. with h y (τ ) = -y 2 τ y-1 log(τ ) < 0, for all y < 0 and τ > 1. As a consequence hy (•) is a decreasing function for all y < 0. Besides, since hy (1) = 0, it follows that hy (τ ) < 0 and a y (τ ) > 0 for all y < 0 and τ > 1. This proves that h y (•) is an increasing function for all y < 0. (iii) h y (τ ) > 1 for all τ > 1 and y < 0 is a direct consequence of Lemma 4(i,ii). To prove the second inequality, let ∆ y (τ ) := h y (τ )τ y/2 -1 for all τ > 1 and y < 0. Differentiating three times yields ∆ y (τ ) = yτ y/2-1 (τ y -1) 2 τ y -1 -

Figure 1 :

 1 Figure1: Left: Graphs of k n ∈ {2, . . . , 500} → k n (τ n , y) for y ∈ {-∞, -2, -1, -1/2, 0} respectively in {black, blue, red, violet, green}. Right: graphs of k n ∈ {2, . . . , 500} → B(1, τ n , ρ) (dotted lines) and k n ∈ {2, . . . , 500} → B(β (τ n , ρ # = -1), τ n , ρ) (solid lines) given in Equations (6) and[START_REF] Beirlant | Practical analysis of extreme values[END_REF], with ρ ∈ {-2, -1, -1/2} respectively in {blue, red, violet}. On both panels: τ n =

  with a, ζ, η > 0 and ζη ≤ 1. Under these conditions, the above class of distributions fulfills assumptions (A.1) and (A.2) with Weibull tail-coefficient θ = 1/ζ, second-order Weibull parameter ρ = -η, slowly-varying function (x) = 1 + (a/η)x -η and b(x) = -ax -η . Unlike classical distributions such as the (absolute) Normal distribution N (µ, σ) (θ = 1/2, ρ = -1 and b

Figure 2 :

 2 Figure 2: Illustration on simulated data sets of size n = 500 from a D(ζ = 1/θ = 2/3, η = -ρ, a = 10)-distribution with ρ ∈ {-1/2, -1, -3/2} (from top to bottom) computed on N = 1000 replications. Medians (left panel) and MAREs (right panel) as functions of k n ∈ {2, . . . , n -1}, associated with estimators RWT (orange), RSH (blue) and LSE (green) of the extreme quantile q(α n = 1/n 2 ) (black dashed line).

Figure 3 :

 3 Figure 3: Illustration on simulated data sets of size n = 500 from a D(ζ = 1/θ, η = -ρ = 1, a = 10)distribution with θ ∈ {1, 3/2, 2} (from top to bottom) computed on N = 1000 replications. Medians (left panel) and MAREs (right panel) as functions of k n ∈ {2, . . . , n -1}, associated with estimators RWT (orange), RSH (blue) and LSE (green) of the extreme quantile q(α n = 1/n 2 ) (black dashed line).

Figure 4 :

 4 Figure 4: Illustration on the daily wind data set. Top-left: Histogram of the data set. Top-right: Weibull quantile-quantile plot (horizontally: (log log(n/i) -log log(n/ k n )), vertically: (log X n-i+1,n -log X n-k n +1,n ) for i ∈ {1, . . . , k n = 961}). A line with slope θRWT n

Lemma 4 .

 4 Let us define, for all τ > 1 and y ≤ 0,h y (τ ) := K 0 (τ ) K y (τ ) = y log(τ ) τ y -1if y < 0 and h 0 (τ ) := 1 otherwise.

Lemma 5 .

 5 Let us define, for all τ > 1 and y ≤ 0,f y (τ ) := -log(h y (τ )) y log(τ ) if y < 0 and f 0 (τ ) = 1/2 otherwise.Then, (i) f y (•) can be extended by continuity letting f y (1) = 1/2 for all y ≤ 0.(ii) f y (•) is a decreasing function on [1, ∞) for all y < 0 (f 0 (•) is a constant function).

0 Figure 5 :

 05 Figure 5: Variations of the function gβ (•), β ∈ (0, 1/2] studied in the proof of Lemma 3.

1 2 y

 12 log τ (τ y + 1) =: yτ y/2-1 (τ y -1) 2 ∆y (τ ), ∆ y (τ ) = y 2τ (τ y (1 -y log τ ) -1) =:y 2τ ≈ ∆ y (τ ), ≈ ∆ y (τ ) = -y 2 (log τ )τ y-1 < 0.It thus appears that≈ ∆ y (•) is a decreasing function on (1, ∞) with ≈ ∆ y (1) = 0 so that ≈ ∆ y (τ ) < 0 for all τ > 1.As a consequence ∆ y (τ ) > 0 and ∆y (•) is an increasing function on (1, ∞) with ∆y (1) = 0 so that ∆y (τ ) > 0 for all τ > 1. Finally, ∆ y (τ ) < 0 and ∆ y (•) is thus a decreasing function on (1, ∞) with ∆ y (1) = 0 so that ∆ y (τ ) < 0 for all τ > 1. The result is proved.Proof of Lemma 5. (i) A second-order Taylor expansion shows that f y (•) can be extended by continuity letting f y (1) = 1/2 for all y ≤ 0. (ii) Let y < 0 and τ > 1. Differentiating twice, one has:f y (τ ) = -1 τ y log(τ ) 2 (1 -log h y (τ ) -τ y h y (τ )) =: -fy (τ ) τ y log(τ ) 2 , f y (τ ) = h 2 y (τ )τ y -1 τ log τ < 0,in view ofLemma 4(iii). This implies that fy (•) is a decreasing function on (1, ∞). Besides fy (1) = 0 since h y (1) = 1 from Lemma 4(i), and therefore fy (τ ) < 0 and f y (τ ) < 0 for all τ > 1 which implies

Table

  1] for ρ parameters associated with usual Weibull taildistributions. Our first result is a refinement of [17, Corollary 3.1]. It provides an asymptotic normality result for the extreme quantile estimator (2) based on two intermediate sequences (k n ) and (k n ). Assume (A.1) and (A.2) hold. Let (k n ) and (k n ) be two intermediate sequences and introduce (α n ) a probability sequence such that α n → 0 as n → ∞. Suppose, as n → ∞,

	Theorem 1.

Table 2 :

 2 Summary of results obtained in Tables

			4	1	5/4	3/2	7/4	2	9/4	5/2
	-∞	RSH						RSH
	-5/2									LSE
	-2							RWT	RWT	RWT
	-3/2					RWT	RWT	RWT	RWT	RWT
	-1	RWT/LSE		RWT/LSE	RWT	RWT	LSE/MEF	LSE	LSE	LSE
	-1/2	RWT	LSE	RSH	RSH	RSH	RSH	RSH	RSH	RSH

Table 3 :

 3 5017 and θMEF These results are reported in Table3with the corresponding estimated extreme quantiles qn (1/n). The estimates of the extreme quantile provided by RSH and MEF seem to respectively underestimate and overestimate q(1/n) with respectively RSH(1/n) = 33.89 m/s and MEF(1/n) = 49.53 m/s while the sample maximum is x n,n = 42.26 m/s. It appears that LSE(1/n) = 37.08 is significantly smaller than the sample maximum. Let us stress that the proposed refined estimator gives the closest estimate to the maximum value of the sample: RWT(1/n) = 41.00 m/s. Note that the behaviour of the RWT estimate is stable with respect to the choice of ρ # : RWT(1/n) ∈ {37.62, 41.00, 39.84} when ρ # ∈ {-2, -1, -1/2} even though ρ # = -1 seems to be the best option. Finally, both sample paths k n → RWT(1/n) and k n → LSE(1/n) enjoy a stable behaviour in a large neighbourhood of k † n , see the bottom-right panel of Figure4. As a conclusion, according to RWT(1/n) estimate, one can expect a daily wind larger than 41.00 m/s to occur in average once every 30 years during the October to March period. Comparison of the four estimators on the daily wind data set: Estimates of the Weibull tail-coefficient θ and extreme quantile q(1/n). The selected intermediate sequence k † n is also given for each estimator.

	n	(k † n ) = 0.6693,

Table 4 :

 4 MAREs associated with the four estimators of the extreme quantile q(α n = 1/n 2 ) computed on simulated data from the D(ζ = 1/θ, η = -ρ, a = 10)-distribution. The best result is emphasized in bold. MAREs larger than 1 are not reported.

		RWT	RSH	LSE	MEF
			θ = 1/2
	ρ = -1/2	0.0133	0.0441	0.0203	0.0969
			θ = 3/4
	ρ = -1/2	0.0702	0.0459	0.0412	0.2543
			θ = 1
	ρ = -1	0.1125	0.3449	0.1814	-
	ρ = -1/2	0.1317	0.0640	0.0715	0.3015
			θ = 5/4
	ρ = -1	0.1430	0.3386	0.1846	0.5964
	ρ = -1/2	0.1937	0.0857	0.1029	0.2283
			θ = 3/2
	ρ = -3/2	0.2116	0.7095	0.3844	-
	ρ = -1	0.1874	0.3374	0.1900	0.4198
	ρ = -1/2	0.2517	0.1090	0.1332	0.2076
			θ = 7/4
	ρ = -3/2	0.2470	0.6989	0.3831	-
	ρ = -1	0.2442	0.3330	0.1986	0.2705
	ρ = -1/2	0.3154	0.1349	0.1663	0.3809
			θ = 2
	ρ = -2	0.3236	0.8833	0.4406	-
	ρ = -3/2	0.2869	0.6945	0.3833	-
	ρ = -1	0.2934	0.3311	0.2136	0.3168
	ρ = -1/2	0.3744	0.1586	0.1971	0.5915
			θ = 9/4
	ρ = -2	0.3710	0.8783	0.4365	-
	ρ = -3/2	0.3274	0.6915	0.3847	0.6088
	ρ = -1	0.3401	0.3303	0.2301	0.4946
	ρ = -1/2	0.4359	0.1818	0.2262	0.7415
			θ = 5/2
	ρ = -5/2	0.8958	0.9493	0.3713	-
	ρ = -2	0.4301	0.8754	0.4420	-
	ρ = -3/2	0.3824	0.6891	0.3874	0.4998
	ρ = -1	0.4050	0.3309	0.2526	0.7070
	ρ = -1/2	0.5086	0.2101	0.2629	0.8675

Table 5 :

 5 MAREs associated with the four estimators of the extreme quantile q(α n = 1/n 4 ) computed on simulated data from the D(ζ = 1/θ, η = -ρ, a = 10)-distribution. The best result is emphasized in bold. MAREs larger than 1 are not reported.

		RWT	RSH	LSE	MEF
			N (µ, σ = 1)
	µ = 1	0.1396	0.2804	0.0798	0.1852
	µ = 3	0.0776	0.0564	0.0471	0.2943
	µ = 5	0.0553	0.0882	0.0440	0.2682
	µ = 7	0.0334	0.0789	0.0366	0.1852
	µ = 9	0.0361	0.0822	0.0381	0.1987
			G(υ, λ = 1)
	υ = 4	0.1417	0.2240	0.1089	0.4930
	υ = 6	0.1219	0.2270	0.1071	0.5040
	υ = 8	0.0943	0.2095	0.1086	0.4581
	υ = 10	0.1002	0.2158	0.1017	0.4758
			W(υ, λ)
	υ = λ = 1/2	0.4932	0.2100	0.2593	0.8528
	υ = λ = 2	0.1217	0.0510	0.0646	0.3324

Table 6 :

 6 MAREs associated with the four estimators of the extreme quantile q(α n = 1/n 2 ) computed on simulated data from classical Weibull tail-distributions. The best result is emphasized in bold.

		RWT	RSH	LSE	MEF
			N (µ, σ = 1)
	µ = 1	0.1914	0.4430	0.1212	0.3009
	µ = 3	0.1162	0.1001	0.0797	0.3354
	µ = 5	0.0837	0.1584	0.0897	0.2507
	µ = 7	0.0780	0.1565	0.0892	0.2270
	µ = 9	0.0702	0.1477	0.0856	0.1895
			G(υ, λ = 1)
	υ = 4	0.2206	0.3616	0.2024	0.4721
	υ = 6	0.2008	0.3709	0.2126	0.4677
	υ = 8	0.1906	0.3669	0.2148	0.4492
	υ = 10	0.1825	0.3596	0.2137	0.4275
			W(υ, λ)
	υ = λ = 1/2	0.6775	0.2729	0.3886	0.9070
	υ = λ = 2	0.1750	0.0664	0.0938	0.4333

Table 7 :

 7 MAREs associated with the four estimators of the extreme quantile q(α n = 1/n 4 ) computed on simulated data from classical Weibull tail-distributions. The best result is emphasized in bold.

This work is supported by the French National Research Agency (ANR) in the framework of the Investissements d'Avenir Program (ANR-15-IDEX-02). S. Girard also acknowledges the support of the Chair Stress Test, Risk Management and Financial Steering, led by the French Ecole Polytechnique and its Foundation and sponsored by BNP Paribas.

The code is available from the corresponding author upon request.

A APPENDIX: PROOFS

Proofs of main results are collected in Subsection A.1. Some auxiliary results are provided in Subsection A.2 and proved in Subsection A.3.

A.1 Proofs of main results

Proof of Theorem 1. Clearly, the following expansion holds k n log qn (α n , k n , k n ) q(α n ) = T (1) n + T (2) n + T (3) n , with

Let us consider the three terms separately. First, [16, Lemma 1] shows that, under (A.1), k n → ∞, k n /n → 0 and condition (iii):

where ξ n d -→ N (0, 1). Second, [17, Proposition 2.1] entails that, under assumptions (A.1), (A.2), k n → ∞ and k n /n → 0, the following expansion holds:

where ξ n d -→ N (0, 1) and ξ n d -→ N (0, 1). Third, using (A.2) and condition (iii), one obtains

Collecting ( 16), ( 17) and ( 18), one has

Proof of Lemma 3. The result is straightforwardly true for β = 0. Let us then focus on the case where β ∈ (0, 1/2] and consider u ∈ (0, 1]. Differentiating three times, one gets

be the unique point in u ∈ (0, 1] such that g β (u) = 0 when β

The sign of m 0 (β)

for all β ∈ (0, 1/2]. As a consequence, there exists a unique β 0 ∈ (0, 1/2] such that m 0 (β) = 0 and

Two cases appear:

and gβ (1) = 0. This proves that gβ (u) > 0 for all u ∈ (0, 1).

As a conclusion, in both cases, gβ (u) > 0 and g β (u) > 0 for all u ∈ (0, 1) which implies that g β (•) is an increasing function on (0, 1). Since g β (1) = 0 then g β (u) < 0 for all u ∈ (0, 1) and

Proof of Lemma 4. (i) A first-order Taylor expansion shows that h y (•) can be extended by continuity letting h y (1) = 1 for all y ≤ 0.

(ii) Differentiating twice, one gets for all y < 0 and τ > 1:

(iii) f y (τ ) < 1/2 for all y < 0 and τ > 1 is a direct consequence of Lemma 5(i,ii) while f y (τ ) > 0 follows from Lemma 4(iii).

Proof of Proposition 1. First, note that for all τ > 1, ρ ≤ 0 and y ≤ 0, one has

Clearly, B(1, τ, 0) -B(β (τ, y), τ, 0) = 0 and B(1, τ, 0) = 0 in view of [START_REF] Beirlant | Semiparametric lower bounds for tail index estimation[END_REF]. Let us thus focus on the case where τ > 1, ρ < 0 and y ≤ 0. Combining log(τ ) > 0, ρ/y > 0 and Lemma 4(iii) yields

Second,

Collecting ( 21) and ( 22) concludes the proof.