Keywords: Hall thruster, low-frequency oscillations, resistive instabilities, ion and plasma propulsion, electrostatic waves and oscillations

correct qualitative and quantitative errors.

I. INTRODUCTION

H ALL THRUSTERS are promising devices for satellite station keeping and propulsion because of their high thrust density, and they are undergoing significant development. Understanding and controlling their instabilities is a major research topic.

In this context, low-frequency instabilities have been investigated in particular by Singh and Malik [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]. We refer to Wei et al. [START_REF] Li-Qiu | Low-frequency oscillations in Hall thrusters[END_REF] and Smolyakov et al. [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E× B discharges[END_REF] for recent reviews.

II. ASSUMPTIONS AND DISPERSION RELATION FOR ELECTROSTATIC MODES

It is well-known that "[the] full picture of instabilities in Hall plasmas is complex and, for typical experimental conditions, likely involves a number of interacting modes which require numerical simulations" [START_REF] Smolyakov | Fluid theory and simulations of instabilities, turbulent transport and coherent structures in partially-magnetized plasmas of E× B discharges[END_REF]. However, analytical approaches with simplified models help to grasp processes intuitively. This motivates the interest for classical stability analyses near simple references states of the plasma. Unfortunately, Hall plasmas are often out-of-equilibrium, nonuniform, unsteady systems, challenging elementary descriptions.

Singh and Malik's model describes the plasma as two fluids in a uniform magnetic field B 0 = B 0 e z , with ions flowing along the x-axis and electrons along the y-axis, respectively with velocities v i0 = v i0 e x (denoted v 0 in [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]) and v e0 = v e0 e y (denoted u 0 in [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]). This geometry assimilates the cylindrical chamber to a slab, with z radial, x parallel to the chamber axis, and y azimuthal. Due to the axial electric field in the chamber, electrons have a significant azimuthal drift, while ions move essentially axially, and, due to their large mass ratio, ions can be considered unmagnetized while electrons are sensitive to the magnetic force. Both species have finite temperatures, T i and T e , and the small pressure oscillations should be understood to be adiabatic1 (δ(p s n -Ys s ) = 0 with Y s = 2 for both species s = e, i treated as two-dimensional monatomic ideal gases). The model also involves electron collisions with neutrals, with a frequency ν.

Dynamics is essentially two-dimensional, in the xy plane (normal to B 0 ). The reference state is taken to be uniform, with density n i0 = n e0 = n 0 . The effect of collisions involving ions is neglected. The effect of elastic electron-neutral collisions is taken into account. The linearized evolution equations read

∂ t n i1 + v i0 ∂ x n i1 + n i0 ∇ • v i1 = 0 (1)
∂ t v i1 + v i0 ∂ x v i1 = e m i E 1 - 1 m i n i0 ∇p i (2)
∂ t n e1 + v e0 ∂ y n e1 + n e0 ∇ • v e1 = 0 (3)
∂ t v e1 + v e0 ∂ y v e1 = - e m e ( E 1 + v e1 ∧ B 0 + v e0 ∧ B 1 ) - 1 m e n e0 ∇p e -ν v e1 (4)
∇ • E 1 = e 0 (n i1 -n e1 ) (5)
Focusing on electrostatic modes, the contribution of B 1 is neglected, while electric field oscillations derive from a potential, E 1 = -∇φ 1 . This system is Fourier-transformed with ∂ t = iω and ∇ = -i k, and the Poisson equation reads

(k 2 x + k 2 y )eφ 1 = e 2 0 (n i1 -n e1 ) = ω 2 pi m i n i1 -n e1 n i0 (6)
where we highlight the ion plasma frequency which will be used to normalize all frequencies and growth rates.

The dispersion relation of system (1)-( 5) is then a fifthdegree polynomial in ω := ω/ω pi . Focusing on low-frequency oscillations leads the authors to a third-degree polynomial, under the assumption that the electron gyrofrequency Ω = |e|B 0 /m e be much larger than max(ω, k y v e0 , ν). Computing the polynomial coefficients is tedious and unilluminating, but straightforward, leading to the lengthy Eq. (18) in Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]. Inspection of the coefficient of the ω 2 term in the polynomial shows that the sum of all three roots has a positive imaginary part (proportional to iν) ; actually, there are generally two roots with positive Im ω, corresponding to damped modes. Therefore, the largest Im (-ω) characterizes either the instability (when it is positive) or the least damped stable mode (when it is negative).

With the authors' hypotheses, we agree on their algebraic Eq. ( 18). However, the numerical values reported in this paper from studying this equation turn out to be erroneous. With the given numerical data, we find much smaller growth rates and qualitatively different dependences on physical parameters.

III. THE LITVAK-FISCH APPROXIMATION

The inconsistency of the graphs (and their interpretation) is manifest when considering the Litvak-Fisch [START_REF] Litvak | Resistive instabilities in Hall current plasma discharges[END_REF] estimate for the growth rate, which reads in dimensionless form [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] recovers this expression analytically as the limit of Eq. ( 18)'s solution for T e = T i = 0 and ω pe max(ω pi , Ω). With Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]'s typical parameter values n 0 = 10 18 m -3 , B 0 = 15 mT, k x = 20 rad/m, k y = 400 rad/m, T e = 10 eV, T i = 1 eV, v e0 = 10 6 m/s, v i0 = 5 • 10 4 m/s, ν = 10 6 s -1 and xenon ions (m i = 131 Da), one finds ω pi ≈ 1.15 • 10 8 rad/s, ω pe /ω pi = m i /m e ≈ 490 and Ω/ω pi ≈ 23, leading to the Litvak-Fisch estimate γ/ω pi ≈ 6 • 10 -5 , three orders of magnitude smaller than Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]'s typical values.

γ/ω pi ≈ ν(2k y v e0 ) -1 (Ω/ω pe )(1 + Ω 2 /ω 2 pe ) -3/2 . (7) Ref.
The relevance of Litvak and Fisch's approximation T e = T i = 0 in estimating coefficients in Eq. (18) of Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] can be assessed by noting that no coefficient of the cubic polynomial involves the temperatures alone. Rather, the electron temperature appears only through the sums ω 2 pe + k 2 Y e T e /m e and Ω 2 + k 2 Y e T e /m e , so that k 2 T e /m e ≈ 7.5 • 10 8 rad/s ≈ 6.5 ω pi brings a minor correction (see also Fig. 5).

The effect of ion temperature in the cubic equation may seem more dramatic at a first reading, as it occurs in the combination k 2

x v 2 i0 -k 2 Y i T i /m i , with k x v i0 = 10 6 rad/s and k 2 T i /m i ≈ 0.5 • 10 6 rad/s. But other terms in the coefficients always overwhelm the ion temperature contribution, as Fig. 6 confirms numerically.

IV. PLOTS

To ease comparison with the original figures, we follow Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] for all data selections on figures 1-6, and use the same figure numbering. Our plots display a single curve and ordinate axis, as we consider only electrostatic waves.

For all figures, except for the abscissa, parameters have the typical values listed above. The ordinate (note factor 10 6 ) is the instability growth rate scaled to the ion plasma angular frequency ω pi = 1.15 • 10 8 rad/s, according to Eq. ( 18) in Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF].

For Figure 7 (which has no counterpart in Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]), we consider another range of electron drift velocities, showing that a smaller v e0 (outside the thruster regular operation regime) results in a significantly stronger instability, except for a tiny velocity interval around k x v i0 /k y (the value for which both ion drift and electron drift generate the same Doppler shift in the system (1)-( 4)) where the growth rate decreases and the instability can even be suppressed : all eigenmodes (with the same wavenumbers) are stable for these small v e0 's.

We leave the analysis of these figures for a later endeavour, as this discussion requires deeper investigation in the physics of the instability process. The present comment aims at providing the correct figures implied by the equations in Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] for electrostatic modes.

It is worth noting that the salient behaviours (linear in Figs 1 and 3, hyperbolic in Figs 2 and4) are already captured by the Litvak-Fisch estimate (7), to which Eq. (18) of Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] brings only moderate corrections, so that the need for computing the heavy coefficients and solving Eq. (18) of Ref. [START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF] is unclear. Besides, the consistency of the linearized equations may be questioned, as Hall thruster plasmas are often neither uniform nor stationary. In particular, the wavelength of the instability is not small compared to the gradient length of the mean density profile, which should in consequence be taken into account. His main interest is in plasma instabilities and turbulent transport through magnetic fields, in the frame of research on Hall thrusters and magnetic confinement fusion.

Fig. 1 .

 1 Fig.1. Growth rate as a function of magnetic field.

Fig. 2 .

 2 Fig.2. Growth rate as a function of azimuthal wavenumber.

Fig. 3 .

 3 Fig. 3. Growth rate as a function of electron-neutral collision frequency.

Fig. 4 .Fig. 5 .

 45 Fig. 4. Growth rate as a function of electron drift velocity.

Fig. 6 .Fig. 7 .

 67 Fig.6. Growth rate as a function of ion temperature.

Rather than obeying Ref.[START_REF] Singh | Growth of low-frequency electrostatic and electromagnetic instabilities in a Hall thruster[END_REF]'s unconventional ps = YsnsTs.

ACKNOWLEDGMENT

The authors are pleased to thank anonymous reviewers for fruitful comments, and H.K. Malik for instructive discussions in June 2018. This work was initiated thanks to the Indo-French Centre for the Promotion of Advanced Research (IFC-PAR/CEFIPRA project 5204-3), New Delhi. Assistance from L. Couëdel and discussions with F. Doveil were helpful to YE.


S Lay
email: sabine.lay@simap.grenoble-inp.fr

Atom scale study of cracks in the carbide phase of WC-Co alloys

Keywords: Ductile crack, dislocation multipoles, cleavage, atom scale, tungsten carbide

In this study, cracks propagating in WC grains are investigated at the atom scale along the [0001] axis. A brittle crack, characterised by cleavage along {1,0,-1,0} prismatic facets and the absence of dislocations is observed. The crack follows a zig-zag path allowing a higher deformation energy to be absorbed by the crystal. A ductile crack is also characterised. The plastic zone is limited and contains mainly multipoles consisting in unextended dislocations. The crack path is tortuous and delineated by dislocation arrangements sometimes involving extended dislocations. Close to the crack tip, the fracture is discontinuous and nanocracks occur at the level of dislocation arrangements. These dislocation configurations could provide crack nucleation sites for the crack propagation.

Introduction

WC-Co composites are structural materials combining high hardness and toughness. They are used in various applications such as rock cutting tools in mining industry, road planning picks in construction or drill tips in microelectronics. They undergo intense abrasive wear and fatigue resulting in the degradation of the tools, mainly due to the initiation and propagation of cracks [START_REF] Bailey | Wear processes exhibited by WC-Co rotary cutters in mining[END_REF][START_REF] Konyashin | Novel industrial hardmetals for mining, construction and wear applications[END_REF]. Studies of the fracture behaviour of cemented carbides have pointed out the role of the binder being the main toughening phase in the material [START_REF] Sigl | Experimental study of the mechanics of fracture in WC-Co alloys[END_REF][START_REF] Schleinkofer | Microstructural processes during subcritical crack growth in hard metals and cermets under cyclic loads[END_REF][START_REF] Kotoul | On the shielding effect of a multiligament zone of a crack in WC-Co[END_REF][START_REF] Tarragó | FIB/FESEM experimental and analytical assessment of R-curve behavior of WC-Co cemented carbides[END_REF]. Quantification of the crack path in the microstructure indicates that the path in the binder is higher than its volume ratio. However, the major part of the crack occurs in the carbide phase across WC grains or along WC grain boundaries [START_REF] Sigl | Experimental study of the mechanics of fracture in WC-Co alloys[END_REF][START_REF] Mingard | In situ observation of crack growth in a WC-Co hardmetal and characterisation of crack growth morphologies by EBSD[END_REF]. WC behaves as a brittle material below about 800 K [START_REF] Mari | Experimental strategy to study the mechanical behaviour of hardmetals for cutting tools[END_REF]. However, a significant plastic deformation occurs in fractured carbide grains at room temperature [START_REF] Takahashi | Determination of the slip systems in single crystals of tungsten monocarbide[END_REF][START_REF] Lewis | Plastic Deformation in Tungsten Carbide[END_REF][START_REF] Luyckx | Slip system of tungsten carbide crystals at room temperature[END_REF][START_REF] Sarin | On the deformation of tungsten carbide-cobalt cemented carbides[END_REF][START_REF] Gee | Application of EBSD to the evaluation of plastic deformation in the mechanical testing of WC/Co hardmetal[END_REF][START_REF] Liu | Mechanisms of WC plastic deformation in cemented carbide[END_REF][START_REF] Csanádi | Deformation and fracture of WC grains and grain boundaries in a WC-Co hardmetal during microcantilever bending tests[END_REF][START_REF] De Luca | Nanomechanical Behaviour of Individual Phases in WC-Co Cemented Carbides, from Ambient to High Temperature[END_REF]. A mechanism based on the reaction between partials of extended dislocations and further coalescence was suggested for the nucleation of cracks in WC grains [START_REF] Hibbs | Defect interactions in deformed WC[END_REF][START_REF] Jayaram | Plastic deformation of tungsten carbide-cobalt at high confining pressure[END_REF]. An effort to study the deformation structure in cemented carbides after severe working conditions including temperature increase up to more than 800°C was also addressed and showed the high level of deformation of WC grains, the breaking of grain boundaries and their infiltration by Co [START_REF] Östberg | Mechanisms of plastic deformation of WC-Co and Ti(C, N)-WC-Co[END_REF]. scale [START_REF] Özden | Mesoscopical finite element simulation of fatigue crack propagation in WC/Co-hardmetal[END_REF], a better understanding of the wear behaviour of the carbide phase would be useful with a view to improving its properties. Especially very little work at the nanoscale was devoted to the analysis of crack propagation in the carbide, so a better knowledge of the failure mechanisms of the WC phase in cemented carbides is still required. More generally, few detailed analyses of cracks in materials were performed up to now at the nano-or atom scale while experimental data are mandatory to understand how they initiate and propagate [START_REF] Robertson | An HVEM study of hydrogen effects on the deformation and fracture of nickel[END_REF][START_REF] Higashida | Dislocation emission from a crack tip in MgO thin crystals[END_REF][START_REF] Xing | Experimental study about nano-deformation field near quasicleavage crack tip[END_REF][START_REF] Ii | Identification of crack path of interand transgranular fractures in sintered silicon nitride by in situ TEM[END_REF][START_REF] Ly | Dynamical observations on the crack tip zone and stress corrosion of two-dimensional MoS 2[END_REF].

This work presents an experimental study of cracks running inside WC grains at the atom scale. Two modes of cracks are analysed.

2-Experimental procedure

2-1 Materials and method

While conducting atom scale characterization of cemented carbides containing grain growth inhibitors, cracks could be imaged in two WC-Co alloys: alloy called A contains 9.5 Co, 0.6 Cr 3 C 2 and 0.3 VC, alloy B contains 9.2 Co, 0.3 Cr 3 C 2 and 0.5 VC (wt. %). They were prepared from WC powder about 0.5 µm in size and sintered at 1400 °C for 2 h. Thin foils for high resolution transmission electron microscopy (HRTEM) investigations were prepared by a standard argon milling method. As the HRTEM specimens have a thickness of about 100 nm in the regions with atomic resolution, some deformation and cracks possibly occur during the handling of the specimens. The origin of cracks studied in this work is probably this one. Observations were carried out using a 4000EX JEOL microscope operated at 400 kV. Image processing was performed using the Digital Micrograph TM software. The contrast of the images was improved by subtracting the signal arising from an amorphous phase by means of the average background subtraction (ABS) filter [START_REF] Mitchell | HRTEM filter[END_REF][START_REF] Kilaas | Optimal and near-optimal filters in high-resolution electron microscopy[END_REF]. The dislocations were studied owing to the HRPACK software that simulates the displacement field around a defect in the framework of the linear elasticity theory [START_REF] Bonnet | HRPACK: a software describing the elastic fields near dislocations and interfaces at atomic scale[END_REF]. HRPACK was used because the dislocations were very close together and it was important to precisely position the dislocation cores. Some dislocation dissociations could be revealed by this way. The elastic constants for WC taken in the calculations are those determined by Lee et al. [START_REF] Lee | Single crystal elastic constants of tungsten monocarbide[END_REF].

2-2 Dislocations in WC and conventions

WC has a hexagonal structure with W atoms at (0,0,0) and C atoms at (1/3,2/3,1/2) positions. Its lattice parameters are a = 0.2906nm and c = 0.2837nm. Plasticity studies in WC have revealed that the dominant slip system at room temperature is 1/3<1,-2,1,3> {1, 0,-1,0} [START_REF] Takahashi | Determination of the slip systems in single crystals of tungsten monocarbide[END_REF][START_REF] Luyckx | Slip system of tungsten carbide crystals at room temperature[END_REF]. In highly deformed regions, it was observed that the 1/3<1,-2,1,3> dislocations were dissociated into two 1/6<1,-2,1,3> partial dislocations bounding a stacking fault [START_REF] Hibbs | Room-temperature deformation mechanisms and the defect structure of tungsten carbide[END_REF]. It was therefore proposed that the deformation of WC at room temperature mainly operates through le nucleation and glide of 1/3<1,-2,1,3> dislocations that dissociate under stress [START_REF] Greenwood | The defect structure of tungsten carbide in deformed tungsten carbide-cobalt composites[END_REF]. However, dislocations with 1/3<1,-2,1,0> , <0001> Burgers vectors or undissociated 1/3<1,-2,1,3> dislocations were also observed in as-sintered, annealed or deformed materials [START_REF] Hibbs | Room-temperature deformation mechanisms and the defect structure of tungsten carbide[END_REF][START_REF] Greenwood | The defect structure of tungsten carbide in deformed tungsten carbide-cobalt composites[END_REF][START_REF] Lay | Deformation Mechanisms of Refractory Carbides -Application to W-Co Carbides Composites[END_REF][START_REF] Jablon | Subgrain structure and dislocations in WC-Co hard metals revealed by electron channelling contrast imaging[END_REF]. Therefore, 1/3<1,-2,1,0> {1, 0,-1,0} and <0001> {1, 0,-1,0} secondary slip systems are also encountered. For the sake of clarity, in the following, vectors will be

expressed as a1 = 1/3[2,-1,-1,0], a2 = 1/3[-1,2,-1,0], a3 = 1/3[-1,-1,2,
0], c = [0,0,0,1] and ai ± c (i=1:3).

The (0,1,-1,0), (1,0,-1,0) and (-1,1,0,0) prismatic glide planes will be called P1, P2 and P3 respectively with ai and ai ± c vectors being parallel to Pi plane (i=1:3) (Fig. 1). The cracked regions studied in this work were observed along [0001] direction of WC. In this orientation, the glide planes are viewed edge on. On the HRTEM images of this work, the dislocation core position is depicted by the symbol "ꓕ". Its orientation is determined by the Burgers vector direction which depends on the dislocation line sense. In this work, the dislocation line is oriented towards the reader to fit with the convention used in HRPACK software. This choice reverses the Burgers vector direction compared to a dislocation line direction pointing into the paper [START_REF] Hirth | Theory of dislocations[END_REF].

3-Results

Brittle crack

The first example of crack is found in material A. A crack propagating along a boundary displays a branch in one adjacent WC crystal at the level of a small step at the boundary (Fig. 2). A misorientation angle of about 0.6° and a lattice shift is depicted between the cracked parts of the WC grain. On the HRTEM image, the crack path is revealed by a blurred contrast. Atom positions are modified along the crack over about three atomic layers with a spacing between atoms larger than in the crystal. The wideness of the crack is very thin, about 1 nm in size. More generally, the crack is not straight, its path follows the three kinds of {1,0,-1,0} prismatic planes of WC as may be seen close to the crack tip where the crack is narrow (Fig. 2d). No dislocation is observed in the vicinity of the crack what indicates that the fracture occurred in a brittle manner.

Ductile crack

Figure 3 is a general view of a crack running inside a WC grain in specimen B. During the ion beam milling of the TEM thin foil, preferential thinning often occurs in the Co phase. In the present case, an elongated hole formed partly in the Co phase and partly in the neighbouring WC grain (Fig. 3a). The hole shape seems to have favoured the initiation of a crack inside the WC grain during the handling of the thin foil. The crack crosses a grain boundary with ∑=1 misorientation (180° <0001>) and stops in the adjacent WC grain. The characterisation focusses on the tip of the crack (Fig. 3b). Several

3.2.a. Dislocations in domain I

The domain around the crack containing dislocations is about 40 nm wide (Fig. 4a). About 25 dislocations were observed inside this domain and no dislocation was detected outside. Most dislocations were characterized. They mainly have a Burgers vector component equal to ±a1 or ±a2.

According to the literature, extended ai±c dislocations were expected, however no dislocation is dissociated in this domain. Moreover, few of them are isolated. They mostly form multipoles (Fig. 45). Figure 4 shows an isolated dislocation and two dislocation dipoles, which are formed by dislocations with opposite Burgers vector components. The displacement field around the isolated dislocation extends over more than 5 nm. Fig. 4c,e illustrates the strong effect of the dipole configuration on the displacement field around the dislocations. As seen, for the dislocation dipole, the deformation is mainly limited to the region between the dislocation cores. The rearrangement of dislocations into dipoles is favoured to decrease the strain energy of the crystal. They form when dislocations gliding on parallel slip planes interact and adopt an equilibrium configuration. When no stress is applied the minimum energy occurs if the dislocations align along a direction 45° from the glide plane [START_REF] Hull | Introduction to dislocations[END_REF]. In the present cases, the angle is found close to 60° and this position highly limits the strain field around the dislocations. If the dislocations have a mixed character, the screw component also affects the force occurring between the dislocations. However, as the exerted force is radial, it has no effect on the equilibrium position of the dislocations which is dictated by their edge components [START_REF] Hull | Introduction to dislocations[END_REF]. Another property of the dipole is that they act as extensional centres: considering the displacement field around dislocation dipoles, they can be contraction or dilatation centres as exampled in Fig. 4f-g depending on the sign of the Burgers vector edge component [START_REF] Gilman | Influence of dislocation dipoles on physical properties[END_REF]. Also multipoles are observed in zone I in the vicinity of the crack. Figure 5 shows HRTEM images of quadrupoles. In each case, dislocations have similar projected Burgers vector magnitude but opposite sign for half of them, leading the whole edge component being equal to zero. A large deformation is encountered in the vicinity of each dislocation. The mutual position of the dislocations is influenced by the presence of each dislocation in the multipole [START_REF] Chen | Dislocation Multipoles and Their Role in Strain-Hardening[END_REF]. It is also to note that the majority of dislocations in domain I have a Burgers vector component parallel to a1, which indicates that mainly the P1 glide plane was activated during the deformation of the WC grain. The occurrence of dislocations with opposite Burgers vector components let assume that dislocation loops were generated during the deformation. The formation of multipoles likely results from the interaction between dislocations during the deformation, as moving dislocations can be trapped by dislocation multipoles [START_REF] Chen | Dislocation Multipoles and Their Role in Strain-Hardening[END_REF].

3.2.b. Dislocations in domain II

Area II refers to the crack path where the crack, although narrow, is well visible on the HRTEM micrograph (Fig. 6a). It roughly runs parallel to the P2 prismatic plane. After subtracting the noise of the image, the crack path is still detected by a brighter contrast. Three arrangements of dislocations called A, B, C are found along the crack and are described in the following (Fig. 6b-d). Defect A corresponds to a quadrupole of dislocations which is likely stable as it is formed by two couples of dislocations (A1-A4 and A2-A3) with opposite Burgers vector components. Several kinds of dislocations are found for defect B. The dislocations (B3-B5) have opposite Burgers vectors and lie in the same prismatic plane P1. They can be interpreted as a dislocation loop. The dislocation couples (B1-B'1) and (B2-B'2) are dislocations dissociated in the P1 and P2 prismatic planes, respectively. A stacking fault is observed between the partial dislocations, the one between B2 and B'2 being along the main crack path. B1 and B2 form in fact a single dislocation with a Burgers vector equal to 1/6[-1,-1,2,0] that is the sum of the Burgers vectors of B1 (1/6[-2,1,1,0]) and B2 (1/6[1,-2,1,0]). The dislocation called B4 lying between B and C defects was added to improve the fit between the experimental and the calculated displacement field of dislocation set B.

Defect C also consists of several dislocations, one being dissociated in the P3 prismatic plane (C3-C'3). Dislocation B4 was taken into account in the calculated displacement field to improve it. For this arrangement, the calculated displacement field fits well with the atomic positions of the upper part of the WC grain. In the part below the crack, a small mismatch occurs unlike what was observed at the level of defect A and B. It is probably due to a larger failure of the grain in the vicinity of defect C.

It shows that different levels of damage occur along the crack path.

3.2.c. Dislocations in domain III

Domain III contains the tip of the crack, whose exact location is difficult to determine. Interrupted blurred contrast and small shifts of atomic columns observed in some places suggest that the crack is not continuous in this domain. The discontinuous failure path is indicated in Fig. 7. It is delineated on the HRTEM image by dislocation arrangements arbitrary separated into three sets called D to F. Arrangement D consists of two successive dislocations. At the level of arrangement D, the calculated displacement field coincides with the experimental image, but the contrast is blurred outside the dislocation cores (Fig. 7a) indicating that some crack occurred in this region. The multipole called F is formed by three different types of dislocations. The calculated displacement field also well coincides with the image except in the right part between dislocations F1 and F2 where a discrepancy is observed, what indicates a crack in this area. Defect E contains dislocations with identical Burgers vector but same or opposite signs and dislocations lying above and below the crack path (Fig. 8). It was necessary to include the strain field of dislocations belonging to defect F to get a good fit on the 10 right hand side. A narrow stacking fault bounded by E1 and E'1 partial dislocations is also attached to the crack path (Fig. 8). On the left hand side of array E, there is a mismatch between the calculated displacement field and the image and no better fit including other dislocations could be found. It reveals a cracked part between D and E. These observations confirm that the crack path is discontinuous in this domain.

Discussion

This experimental study has shown the occurrence of two kinds of crack propagating in WC grains. In the first case, the crack propagates mainly along {1,0,-1,0} prismatic planes and no dislocation is observed in the surrounding of the crack. The fracture occurs in a brittle manner by cleavage along prismatic planes. The existence of the three cleavage planes allows the crack to deviate from one plane to the other what increases the ability of the crystal to absorb the deformation energy and to resist to fracture [START_REF] Smallman | Modern Physical Metallurgy[END_REF]. In other experiments, it was found that cracks could also propagate along {1,1,-2,0} planes, which is not observed in the present study [START_REF] Hibbs | Defect interactions in deformed WC[END_REF]. In the second example, a dislocation activity is observed in the cracked region. The crack path does not preferentially follow prismatic planes and is delineated by dislocation arrangements. In this case, the presence of dislocations denotes a plastic behaviour of the WC crystal although the plastic extension is limited.

The observation of a brittle and a ductile crack in WC is in agreement with knowledge of WC behaviour [START_REF] Sigl | Experimental study of the mechanics of fracture in WC-Co alloys[END_REF]. Their occurrence should depend on the stress level and orientation applied to the crystal, which dictates its ability to deform plastically.

In the present study, it is found that most dislocations in the vicinity of the ductile crack are undissociated. According to the literature related to the deformation in WC, extended ai±c dislocations were expected. The dissociation width can be reduced in dipole configuration when the height between the glide planes of the dislocations is smaller than the dissociation [START_REF] Gilman | Influence of dislocation dipoles on physical properties[END_REF]. However, assuming a contraction value as large as the one calculated in [START_REF] Gilman | Influence of dislocation dipoles on physical properties[END_REF], the separation width would still be detectable by HRTEM. Another assumption to explain the presence of undissociated dislocations would be the occurrence of ai instead of ai±c dislocations close to the crack owing to a larger critical resolved shear stress acting on the (ai, Pi) slip systems.

A few dissociated dislocations were however found attached to the crack path (in defects B, C, and E). Moreover, looking carefully at these faults indicates that they are associated with a blurred contrast, so faults would form in highly deformed region. It suggests that fault formation could be in some favourable cases a preliminary step to accommodate the stress field, before complete atomic bond breaking and separation of the crystal.

A model for the initiation of cracks in WC, based on the reaction of a series of extended dislocations gliding in two different prismatic planes, was proposed [START_REF] Hibbs | Defect interactions in deformed WC[END_REF]. Such a reaction was observed at the level of the crack in arrangement B leading to the formation of the sessile 1/6[-1,-1,2,0] dislocation. However, the model is based on the coalescence of a large number of such sessile dislocations. This mechanism could not be detected in the present work.

It is to note that the interaction of extended dislocations lead to the formation of sessile dislocations, which decrease the mobility of multipoles and harden the material. The character of the dislocations -extended or not-should therefore influence the deformation behaviour of WC. Few is known at the moment on the factors influencing the dislocation dissociation. It was proposed that the carbon level in the material could affect the dissociation width of dislocations [START_REF] Jayaram | Deformation enhanced decarburization of tungsten carbide-cobalt alloy[END_REF]. Furthermore, in the present materials, a small solubility of Cr and V is expected in WC at least in the carbide grains in contact with Co and nothing is known about their effect on the plastic behaviour of WC [START_REF] Weidow | Transition Metal Solubilities in WC in Cemented Carbide Materials[END_REF]. A better knowledge of such effects could help to improve the mechanical properties of WC.

In the case of the ductile crack, dislocations were found around the crack and along the crack. It is usually proposed that dislocations are emitted from the crack tip during crack growth to decrease the stress by exerting a back stress onto the crack tip and that these dislocations accommodate the applied stress field [START_REF] Chiao | Direct observation of dislocation emission from crack tips in silicon at high temperatures[END_REF][START_REF] Zielinski | Crack-tip dislocation emission arrangements for equilibrium -I. In situ TEM observations of Fe2wt%Si[END_REF][START_REF] Higashida | HVEM observations of dislocation structures near a crack tip in MgO crystals[END_REF][START_REF] Zhao | Quantitative analysis of nanoscale deformation fields of a crack-tip in single-crystal silicon[END_REF]. Depending on the relative orientation of the glide systems and of the applied stress, slip systems are activated and dislocation loops can be emitted from sources along the crack or in the vicinity of it [START_REF] Sutton | Interfaces in crystalline materials[END_REF]. The loop observed in defect B could correspond to the emission of dislocations from the crack. Around the crack, multipoles consisting of dislocations with edge character and opposite Burgers vectors have formed to lower their energy.

At the level of the crack path, the dislocation arrangements also show specific features. Summing the edge component of Burgers vectors in each arrangement gives their equivalent Burgers vector (Table 1). For most defects, the equivalent Burgers vector is equal to zero, or to a single dislocation Burgers vector, showing that the dislocations arrangements, as multipoles around the crack, tend to adopt stable configurations. Moreover, equivalent Burgers vectors have different magnitudes and directions. It was previously deduced from the observations that only a part of atomic bonds along the crack path was broken and that the WC lattice was more or less continuous across the crack in crack tip area. Especially no tilt misorientation along the crack was detected. Consequently, no geometrically necessary dislocations are required such as an array of edge dislocations in a form of a low angle boundary to compensate for a misorientation, in agreement with the observation [START_REF] Jagannadham | Dislocations at ductile/plastic crack tips: insitu TEM observations[END_REF].

Dislocation arrangement A B C D E F Equivalent b 0 0 -a1 a1-a2 0 - a3
Table 1. Equivalent Burgers vector of the dislocation arrangements lying along the crack path.

One can question about the exact role of multipoles found along the crack path. These dislocation arrangements induce locally a large deformation of the crystal lattice, with no long-range stress field when their associated Burgers vector is equal to zero. They decrease the strain energy of the crystal compared to isolated dislocations, which can result in a higher resistance of WC to failure. On the other hand, multipoles have a strain hardening effect as they form metastable configurations in the crystal, which reduces their mobility, and they are able to trap moving dislocations [START_REF] Chen | Dislocation Multipoles and Their Role in Strain-Hardening[END_REF]. Moreover, it was proposed that dipoles and therefore multipoles could act as fracture nuclei by acting as centres of dilatation or rotation [START_REF] Gilman | Influence of dislocation dipoles on physical properties[END_REF]. Also, cracks could nucleate at cores of edge dislocations under tensile stress [START_REF] Gilman | Importance of dislocation cores in fatigue fracture[END_REF][START_REF] Gu | Lattice weakening by edge dislocation core under tension[END_REF]. In the present study in WC, dislocation multipoles delineating the discontinuous crack close to the crack tip were highlighted owing to atom scale characterization. Their presence suggests that depending on their position relative to the stress field ahead the crack tip, some multipoles could constitute sites of crack nucleation and facilitate crack propagation. Further experiments and modelling could provide a better insight on the role of dislocation multipoles in the deformation and failure of WC and more generally in carbides.

Conclusion

This study aims to better understand fracture phenomena occurring in WC based materials at room temperature. It reports HRTEM observations of cracks occurring in WC grains. In agreement with the literature both a brittle and a ductile crack were encountered. For the brittle crack, cleavage along successive {1,0,-1,0} prismatic planes propagating in a zig-zag way was found. This characteristic emphasizes the effect of crystal orientation on the resistance of WC to failure. A ductile crack with a limited extension of the plastic zone was also observed. Close to the crack tip, stacking faults are detected locally in highly damaged areas. The fracture is discontinuous and dislocation multipoles nearby and along the crack path are present. The crack path follows dislocation multipoles suggesting that multipoles could act as crack nuclei in the carbide according to a mechanism which would need further investigations.

Figure 1 :

 1 Figure 1: (a) Unit cell of WC where lattice vectors are indicated as well as the prismatic glide plane P1. (b) WC lattice viewed along [0001] showing the position of the three prismatic glide planes, P1, P2 and P3.

Figure 2 :

 2 Figure 2: HRTEM image of the crack running in the WC grain viewed along [0001] direction. (a) General image of the WC grain showing the rough path of the crack. (b) Low magnification image showing the branching of the crack from the grain boundary to the WC grain at the level of a small step denoted A. (c) Magnification of area B drawn in (a) (rotated counterclockwise 60°) showing the P3 prismatic plane as a fracture plane. (d) Magnification of area C in (a) displaying the preferential crack path along P1, P2 and P3 prismatic planes of WC.

  domains are distinguished. Domain I consists in the WC volume around the crack path where dislocations are observed. Domain II is the crack path where the failure is clearly visible and domain III is the crack path where the failure is more hardly detected. Dislocation arrangements are found in each domain. For the sake of clarity, they are presented according to their location. As the observation direction is along [0001] in WC crystal, only dislocations with a Burger vector component in the (0001) plane can be detected and therefore only the component of the Burgers vector parallel to the basal plane can be defined.

Figure 3 .

 3 Figure 3. TEM image of a crack in the WC-Co carbide. (a) Low magnification of the crack propagating in WC grains viewed along [0001]. (b) Focus on the crack tip in WC (dotted rectangle in (a)) showing the failure path (II and III) and the area where dislocations are observed (I).

Figure 4 .

 4 Figure 4. HRTEM images of dislocations viewed along [0001] in zone I close to the crack path. (a) General image, (b) Isolated dislocation denoted a in (a) with superimposed calculated displacement field, and indication of the a1 edge component of its Burgers vector. (c) Same displacement field compared with the perfect lattice. (d) Dislocation dipole denoted b in (a) with superimposed calculated displacement field. (e) Same displacement field compared to the perfect lattice. (f) Dislocation dipole denoted c in (a) with superimposed calculated displacement field. (g) Magnification of the displacement field showing the curvature of the atomic planes and the dilatation character associated with the centre of dipole c.

Figure 5 .

 5 Figure 5. HRTEM filtered images of dislocation quadrupoles lying in zone I close to the crack path viewed along [0001] direction. (a) Quadrupole denoted d in Fig. 4a, (b) Quadrupole denoted e in Fig. 4a. The calculated displacement field associated with each quadrupole is superimposed on the image.

Figure 6 .

 6 Figure 6. Dislocation arrangements along the crack path in domain II viewed along [0001]. (a) HRTEM general view of the crack with arrangements called A, B and C. (b-d) HRTEM filtered images of the dislocation arrangements with superimposition of the calculated displacement field. The dotted line indicates the crack path and the continuous segments the stacking fault plane between dissociated dislocations. For sake of clarity, Burgers vectors indications were not added on the images. The same convention as the one applied in Fig. 4 and 5 was used.

Figure 7 .

 7 Figure 7. Domain III of the crack corresponding to the crack tip. (a) general HRTEM image of WC grain viewed along [0001] showing the discontinuous crack path passing through defects denoted D, E and F and (b) same image with the crack path drawn. (c-d) Filtered HRTEM images of D and F defects with superimposed dislocation displacement field. The same convention as in Fig. 4 and 5 is used to indicate the Burgers vector of the dislocations.

Figure 8 .

 8 Figure 8. Defect E in domain III along the crack. (a) HRTEM filtered image viewed along [0001] with superimposed calculated displacement field of the dislocations. The thin dotted white line indicates the discontinuous crack path and the thick dotted line depicts a crack revealed by the discontinuity of the WC lattice at the level of this crack. (b) Magnification of the lower left region in (a) showing the cracked part and the stacking fault bounded by E1 and E'1 partial dislocations attached to the discontinuous crack.

Acknowledgments

The author thanks A. Lackner (Plansee Ceratizit) for former support of this study.


Fatma Ben Khadher
email: fatma.b.khadher@gmail.com

Yousri Slaoui
email: yousri.slaoui@math.univ-poitiers.fr

The stochastic approximation method for recursive kernel estimation of the conditional extreme value index

Keywords: Stochastic approximation algorithm, tail index, extreme value, Pareto distribution

In this research paper, our central focus is upon applying the stochastic approximation method to define a class of recursive kernel estimator of the conditional extreme value index. We investigate the properties of the proposed recursive estimator and compare them to those pertaining to Hill's non-recursive kernel estimator. We attempt to demonstrate that using some optimal parameters, the proposed recursive estimator defined by the stochastic approximation algorithm proves to be very competitive to Hill's non-recursive kernel estimator. Finally, the theoretical results are explored through simulation experiments and illustrated using real dataset about Malaria in Senegalese children.

Introduction

The extreme value theory is a branch of statistics which studies the asymptotic distributions of extreme values. It can be a maximum or a minimum of a set of random variables. This theory was developed by Emil Julius [START_REF] Gumbel | Statistics of Extremes[END_REF]. It is widely applied in many research areas like climate changes, environmental risks, insurance and financial banking (see [START_REF] Beirlant | Statistics of extremes-Theory and applications[END_REF] for a list of interesting examples). Estimation of the tail index, associated with a random variable Y , is one of the main problems in the area of extreme value theory. Therefore, a lot of research, aiming to estimate this parameter, has been performed during last decades (see for example [START_REF] Embrechts | Modelling Extremal Events[END_REF], [START_REF] Beirlant | Statistics of extremes-Theory and applications[END_REF], De [START_REF] De Haan | Extreme Value Theory-An Introduction[END_REF], [START_REF] Reiss | Statistical Analysis of Extreme Values[END_REF], [START_REF] Gardes | A moving window approach for nonparametric estimation of the conditional tail index[END_REF], [START_REF] Gardes | Functional nonparametric estimation of conditional extreme quantiles[END_REF] and [START_REF] Stupfler | A moment estimator for the conditional extreme-value index[END_REF]). We denote by γ the tail index which characterizes the distribution tail heaviness of Y . It can be estimated parametrically using the [START_REF] Hill | A simple general approach to inference about the tail of a distribution[END_REF] estimator and nonparametrically using a kernel version of the Hill's estimator proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Improved approaches have recently appeared in literature. Among them, we state [START_REF] Brilhante | A simple generalisation of the Hill estimator[END_REF] to defined a moment of order p; [START_REF] Beran | The harmonic moment tail index estimator: asymptotic distribution and robustness[END_REF] who proposed a harmonic moment tail index estimator; Paulauskas andVaičiulis (2013, 2017) who elaborated parametric families of functions of the order statistics.

Recently, recursive estimation has drawn the attention and whetted the interest of multiple researchers. Recursivity means that the estimator calculated from the first n observations, say θ n , is a function of θ n-1 . More precisely, we can easily update the estimator value with each additional observation specialy in large sample sizes. The basic objective of the present work lies in introducing a recursive kernel estimator of the conditional extreme value index defined by applying a stochastic approximation algorithm. To the best of our knowledge, this tail index estimator construction was not previously considered in literature and it aims to improve the estimation accuracy.

The application of the stochastic approximation algorithm was introduced first by [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the non-parametric estimation of a regression function[END_REF] and then extended by Mokkadem et al. (2009b) to estimate a regression function. Using this algorithm, [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF] approximated the mode of a probability density, Mokkadem et al. (2009a) estimated a multivariate probability density and recently [START_REF] Slaoui | On the choice of smoothing parameters for semi-recursive nonparametric hazard estimators[END_REF] estimated a hazard function. It turns out that this estimator depends on two important parameters, which are the bandwidth and the stepsize of the stochastic algorithm. By making an adequate choice of the two parameters, the proposed recursive estimator can be very competitive to Hill's non-recursive kernel estimator in terms of estimation error and much better in terms of computational costs.

The structure of this paper is organized as follows. In Section 2, we identify our estimator and we set forward its asymptotic properties. Simulation experiments and investigation of real dataset are depided in Section 3. Finally, the last section wraps up the conclusion and provides new perspectives for future works.

2 Construction of the estimator and asymptotic properties 2.1 The proposed estimator Let (X i , Y i ), i = 1, • • • , n, be independent realizations of the random vectors (X, Y ) ∈ R d × R + 0 , where X is a d-dimensional covariate with joint density function g, d 1. The probability density function of Y given X = x is defined as f (y|x) = P(Y = y|X = x) and the conditional survival function of Y given X = x is denoted by F (y|x) = P(Y > y|X = x). In this paper, we are basically interested in heavy tails. More precisely, we assume that the conditional survival function of Y given X = x satisfies (C1):

F (y|x) = y -1 γ(x) l(y|x
F (y|x) -υ -1 γ(x) b(y|x) = υ -1 γ(x) υ ρ(x) γ(x) -1 ρ(x)γ(x)
.

Additional conditions are needed for ensuring the asymptotic properties of the estimators. Let d(x, y) denote the Euclidean distance between x and y, for all x, y ∈ R d . (C4): There exists c g ∈ R such that for all x, y ∈ R d ,

g (x) -g(y) = c g d(x, y). (C5): There exists c F ∈ R such that for all x, y, z ∈ R d , ln F (y|x) ln F (y|z) -1 = c F d(x, z)
and γ(x)c F < 0. Moreover, we impose a condition on the kernel function K. (C6): K is a bounded density function on R d , with support Ω included in the unit hypersphere of R d and satisfying

R d K(z)dz = 1.
Our idea rests upon to construct a recursive estimator for the conditional tail index γ(x). It will be presented as a ratio of two estimators a n (x) and b n (x). The denominator b n (x) is an estimator of the function b(x) = g (x) F (t n |x), where (t n ) is a nonrandom threshold sequence tending to ∞ as n -→ ∞. The nominator a n (x) is an estimator of the function a

(x) = γ(x)F (t n |x)C x g (x)
, where

C x = 1 + b(t n |x) γ(x)ρ(x) 1 1 -ρ(x) -1 + r n,x
and (r n,x ) is a non-random sequence, tending to 0 as n -→ ∞, defined as

r n,x = ρ(x) γ 2 (x) ∞ 1 z -1 γ(x) -1   γ 2 (x) z 1 γ(x) F (tnz|x) F (tn|x) -1 b(t n |x) - z ρ(x) γ(x) -1 ρ(x)γ(x)   dz.
Construction of a recursive estimator of the function a (x): Let us introduce the stochastic algorithm to estimate the function a(•) at a point x. It is based on searching the zero of the function f 1 : y -→ a(x)y. Following Robbins-Monro's procedure, this algorithm is defined as below

(i) a 0 (x) ∈ R, (ii) ∀n 1, we set a n (x) = a n-1 (x) + γ n Z n (x)
, where the stepsize (γ n ) is a sequence of positive real numbers that goes to zero and Z n (x) is an observation of the function f 1 at the point a n-1 (x).

To construct Z n (x), we follow the approach of [START_REF] Révész | Robbins-Monro procedure in a Hilbert space and its application in the theory of learning processes I[END_REF][START_REF] Révész | How to apply the method of stochastic approximation in the non-parametric estimation of a regression function[END_REF], [START_REF] Tsybakov | Recurrent estimation of the mode of a multidimensional distribution[END_REF] and [START_REF] Slaoui | Large and moderate principles for recursive kernel density estimators defined by stochastic approximation method[END_REF]Slaoui ( , 2014aSlaoui ( ,b, 2018) ) which are based on the classical property of stochastic algorithms (which is E [Z n (x)|F n-1 ] = 0, where F n-1 stands for the σ-algebra of the events occurring at the time n -1). In addition, we introduce a kernel K (which is a function satisfying

R d K(z)dz = 1
), and a bandwidth (h n ) (which is a sequence of positive real numbers that goes to zero when n -→ ∞), and set

Z n (x) = K hn (x -X n ) [ln Y n -ln t n ] 1 {Yn>tn} -a n-1 (x), with K h (x) := h -d K h -1 x .
The stochastic approximation algorithm introduced in Mokkadem et al. (2009a) which estimates recursively the function a at the point x is defined as follows:

a n (x) = (1 -γ n )a n-1 (x) + γ n K hn (x -X n ) [ln Y n -ln t n ] 1 {Yn>tn} . (2.1)
Considering a 0 (x) = 0, the estimator a n defined in (2.1) can be rewritten as

a n (x) = Π n n k=1 Π -1 k γ k K h k (x -X k ) [ln Y k -ln t n ] 1 {Y k >tn} , (2.2) with Π n = n k=1 (1 -γ k ). (2.3)
Construction of a recursive estimator of the function b (x):

We apply the stochastic algorithm to estimate the function b(•) at a point x. It is based on searching the zero of the function f 2 : y -→ b(x)y. Following Robbins-Monro's procedure, this algorithm is defined as below

(i) b 0 (x) ∈ R, (ii) ∀n 1, we set b n (x) = b n-1 (x) + β n T n (x)
, where the stepsize (β n ) is a sequence of positive real numbers that goes to zero and T n (x) is an observation of the function f 2 at the point b n-1 (x).

Based on the same previously used approach, we consider

T n (x) = K hn (x -X n ) 1 {Yn>tn} -b n-1 (x),
with the same bandwidth (h n ) and kernel function K h previously defined. Then, the stochastic approximation algorithm to estimate recursively the function b at the point x is defined as follows:

b n (x) = (1 -β n )b n-1 (x) + β n K hn (x -X n ) 1 {Yn>tn} .
(2.4) Considering b 0 (x) = 0, the estimator b n defined by (2.4) can be rewritten as

b n (x) = Q n n k=1 Q -1 k β k K h k (x -X k ) 1 {Y k >tn} , (2.5)
with

Q n = n k=1
(1β k ).

(2.6) Then, our proposed recursive estimator for the conditional tail index γ(x) is defined as:

γ H n (x) := a n (x) b n (x) = Π n n k=1 Π -1 k γ k K h k (x -X k ) [ln Y k -ln t n ] 1 {Y k >tn} Q n n k=1 Q -1 k β k K h k (x -X k ) 1 {Y k >tn} .
(2.7)

The second objective of our paper is to study the properties of the recursive estimator defined by (2.7) and to compare them with the kernel version of Hill's estimator of the conditional extreme value index proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], which is defined as

γH n (x) = 1 n n i=1 K h i (x -X i ) [ln Y i -ln t n ] 1 {Y i >tn} 1 n n i=1 K h i (x -X i ) 1 {Y i >tn}
.

(2.8)

The asymptotic properties of γ H n are investigated in the next subsection.

Asymptotic results

In order to obtain the bias and the variance of the recursive estimator γ H n defined by (2.7), we first calculate those of the recursive estimator a n defined by (2.2). Then, we calculate those of the recursive estimator b n defined by (2.5).

Throughout this paper, stepsizes and bandwidths are considered to belong to the following regularly varying sequences class. Definition 1 Let u ∈ R and (u n ) n 1 be a nonrandom positive sequence. We say that u n ∈ GS(u) if

lim n→∞ n 1 - u n-1 u n = u.
This condition was introduced by [START_REF] Galambos | Regularly Varying Sequences[END_REF]. The acronym GS stands for (Galambos and Seneta). Typical sequences in GS(u) are, for b ∈ R, n u (log n) b , n u (log log n) b and so on. Finally, we impose the following additional conditions: (C7):

i) γ n ∈ GS(-α) with α ∈ (1/2, 1]. ii) h n ∈ GS(-p) with p ∈ (0, α/d). iii) lim n→∞ nγ n ∈ (min (p, (α -pd) /2) , ∞]. iv) β n ∈ GS(-b) with b ∈ (1/2, 1]. v) lim n→∞ nβ n ∈ (min (p, (b -pd) /2) , ∞]. vi) nh d+2 n ln 2 t n -→ n→∞ ∞.
The following notations will be often used in this paper:

ε = lim n→∞ (nγ n ) -1 .
(2.9)

ε 1 = lim n→∞ (nβ n ) -1 .
(2.10)

c ′ F = c F z 2 such as z ∈ B * R d (0, 1) = {x ∈ R d ; 0 < x 2 ≤ 1}. C = - 1 γ(x) + o(1) c F u 2 for all u ∈ Ω. m n (x) = E K hn (x -X n )(ln Y n -ln t n )1 {Yn>tn} . m n (x) = E (ln Y n -ln t n )1 {Yn>tn} |X n = x .
Since we are interested in the asymptotic behavior of the estimator γ H n , we shall start by giving the asymptotic behavior of the estimator a n . Theorem 1 (Bias and variance of the estimator

a n ) Let Assumptions (C1) -(C7) hold. 1. If p ∈ (0, α/ (d + 2)], then E(a n (x)) = a(x) - C (1 -pε)C x a(x)h n ln t n + o(h n ln t n ).
(2.11)

If p ∈ (α/ (d + 2) , 1/d), then E(a n (x)) = o( γ n h -d n ).
(2.12)

2. If p ∈ (0, α/ (d + 2)), then Var(a n (x)) = o(h 2 n ln 2 t n ).
(2.13)

If p ∈ [α/ (d + 2) , 1/d), then Var(a n (x)) = 1 C x 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ(x)γ n h -d n + o(γ n h -d n ). (2.14)
Departing from the above, we infer that the bias and the variance of the estimator a n heavily depend on the choice of the stepsize (γ n ). We consider two classical choices of (γ n ). The first one rests on the minimization of the variance, while the second one relies on the Mean Integrated Squared Error (MISE).

Choices of (γ n ) minimizing the variance of the estimator a n : As mentioned in Mokkadem et al. (2009a), by considering the point of view of estimation by confidence intervals, it is recommended to minimize the variance of the proposed estimator for confidence interval estimation (see also [START_REF] Hall | Effect of bias estimation on coverage accuracy of bootstrap confidence intervals for a probability density[END_REF]).

Corollary 1 Let the assumptions of Theorem 1 hold. To minimize the asymptotic variance of the estimator a n , α must be chosen equal to 1, (γ n ) must satisfy lim n→∞ nγ n = 1pd, and we then have

Var(a n (x)) = 6 C x (1 -pd) nh d n K 2 1 m n (x)g (x) + o 1 nh d n .
The proof of Corollary 1 follows immediately from (2.14).

Choices of (γ n ) minimizing the MISE of the estimator a n : First, we have

MISE(a n (x)) = R d MSE(a n (x))dx = R d (E(a n (x)) -a(x)) 2 + Var(a n (x)) dx.
The following proposition provides the MISE of the estimator a n . Proposition 1 Let Assumptions (C1)-(C7) hold.

1. If p ∈ (0, α/ (d + 2)), MISE(a n (x)) = C 2 (1 -pε) 2 h 2 n ln 2 t n R d F 2 (t n |x)g 2 (x)γ 2 (x)dx + o(h 2 n ln 2 t n ). 2. If p = α/ (d + 2), MISE(a n (x)) = C 2 (1 -pε) 2 h 2 n ln 2 t n R d F 2 (t n |x)g 2 (x)γ 2 (x)dx + 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ(x) C x γ n h -d n + o h 2 n ln 2 t n + γ n h -d n . 3. If p ∈ (α/ (d + 2) , 1/d), MISE(a n (x)) = 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ(x) C x γ n h -d n + o(γ n h -d n ).
The following corollary is an immediate consequence of the second part of Proposition 1.

Corollary 2 Let Assumptions (C1)-(C7) hold. To minimize the MISE of the estimator a n , the stepsize (γ n ) must be chosen in GS(-1) such that lim n→∞ nγ n = 1 and the bandwidth (h n ) must equal

h n =   3d(d + 1) 2(d + 2) K 2 1 C 2 ln 2 (t n ) R d F (t n |x)g (x) γ 2 (x)dx R d F 2 (t n |x)g 2 (x)γ 2 (x)dx 1 d+2 γ 1 d+2 n   .
Now, we treat the asymptotic behavior of the estimator b n , in order to deduce the one of the estimator γ H n .

Theorem 2 (Bias and variance of the estimator b

n ) Let Assumptions (C1)-(C7) hold. 1. If p ∈ (0, b/ (d + 2)], then E(b n (x)) = b(x) + C 1 -pε 1 b(x)h n ln t n + o(h n ln t n ).
(2.15)

If p ∈ (b/ (d + 2) , 1/d), then E(b n (x)) = b(x) -o( β n h -d n ). (2.16) 2. If p ∈ (0, b/ (d + 2)), then Var(b n (x)) = o(h 2 n ln 2 t n ).
(2.17)

If p ∈ [b/ (d + 2) , 1/d), then Var(b n (x)) = β n h d n 1 2 -(b -pd)ε 1 K 2 2 g (x) F (t n |x) + o β n h d n .
(2.18)

The bias and the variance of the estimator b n defined by the stochastic approximation algorithm (2.4), then heavily depend on the choice of the stepsize (β n ). For an adequate choice, we consider two possible choices of (β n ). The first one is based on the minimization of the variance, while the second one is grounded on the minimization of MISE.

Choices of (β n ) minimizing the variance of the estimator b n : As mentioned in Mokkadem et al. (2009a), it is recommended to minimize the variance of the proposed estimator for confidence interval estimation. Corollary 3 Let the assumptions of Theorem 2 hold. To minimize the asymptotic variance of the estimator b n , b must be chosen equal to 1, (β n ) must satisfy lim n→∞ nβ n = 1pd, and we then have

Var(b n (x)) = (1 -pd) K 2 2 g (x) F (t n |x) 1 nh d n + o 1 nh d n .
The proof of Corollary 3 follows immediately from (2.18).

Choices of (β n ) minimizing the MISE of the estimator b n :

The following proposition provides the MISE of the estimator b n .

Proposition 2 Let Assumptions (C1)-(C7) hold.

1. If p ∈ (0, b/ (d + 2)), MISE(b n (x)) = C 2 (1 -pε 1 ) 2 h 2 n ln 2 t n R d F 2 (t n |x)g 2 (x)dx + o(h 2 n ln 2 t n ). 2. If p = b/ (d + 2), MISE(b n (x)) = C 2 (1 -pε 1 ) 2 h 2 n ln 2 t n R d F 2 (t n |x)g 2 (x)dx + 1 2 -(b -pd)ε 1 K 2 2 β n h d n R d g (x) F (t n |x)dx + o h 2 n ln 2 t n + β n h d n . 3. If p ∈ (b/ (d + 2) , 1/d), MISE(b n (x)) = 1 2 -(b -pd)ε 1 K 2 2 β n h d n R d g (x) F (t n |x)dx + o β n h d n .
The following corollary is an immediate consequence of the second part of Proposition 2.

Corollary 4 Let Assumptions (C1)-(C7) hold. To minimize the MISE of the estimator b n , the stepsize (β n ) must be chosen in GS(-1) such that lim n→∞ nβ n = 1 and the bandwidth (h n ) must equal

h n =   d 4 (d + 1) (d + 2) K 2 2 C 2 ln 2 (t n ) R d F (t n |x)g (x) dx R d F 2 (t n |x)g 2 (x)dx 1 d+2 β 1 d+2 n   .
Now we present the bias and the variance of γ H n . Theorem 3 (Bias and variance of γ H n ) Let Assumptions (C1)-(C7) hold, and suppose that the stepsize

(β n ) = n -1 . 1. If p ∈ (0, α/ (d + 2)], then E γ H n (x) -γ (x) = - C 1 -pε + C 1 -p γ (x) h n ln t n +o (h n ln t n ) . (2.19) If p ∈ (α/ (d + 2)), 1/d), then E γ H n (x) -γ (x) = o γ n h d n .
(2.20)

2. If p ∈ (0, α/ (d + 2)), then Var γ H n (x) = o h 2 n ln 2 t n .
(2.21)

If p ∈ [α/ (d + 2) , 1/d), then Var γ H n (x) = 1 b 2 (x) 1 C x 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ(x) γ n h d n + o γ n h d n .
(2.22)

Clearly, the bias and the variance of the estimator γ H n depend on the choice of the two stepsizes (γ n ) and (β n ).

Let us state now the following Theorem, which gives the weak convergence rate of the proposed recursive estimator γ H n defined in 2.7 in the special case of (

β n ) = n -1 . Theorem 4 Let Assumptions (C1)-(C7) hold, and suppose that (β n ) = n -1 . 1. If there exists r > 0 such that γ -1 n h d+2 n ln 2 t n -→ n→∞ r then γ -1 n h d n γ H n (x) -γ(x) D -→ N √ rB(x), Var(x) ,
where We can consider the case when the stepsize (β n ) is chosen to minimise the variance of the estimator b n . Similarly, we can obtain the weak convergence rate of the estimator γ H n defined in 2.7.

B(x) = - C 1 -pε + C 1 -p γ (x) , Var(x) = 1 b 2 (x) 1 C x 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ(x). 2. If γ -1 n h d+2 n ln 2 t n -→ n→∞ ∞, then 1 h n ln t n γ H n (x) -γ(x) P -→ B(x),

Simulation study

The target of our applications is to compare the performance of the proposed recursive kernel estimator of the conditional extreme value index given in (2.7) to that of Hill's non-recursive estimator defined in (2.8) using the "Leave One Out" cross-validation bandwidth selection.

The study design

We use the following simulation design: we consider the unidimensional case d = 1 and we simulate N = 500 samples of size n (n = 50, 250) of independent replicates (X i , Y i ) where X is uniformly distributed on [0, 1] and the conditional distribution of Y i given X i = x is Pareto with parameter γ(x) = 0.5(0.1 + sin(πx) × (1.1 -0.5 exp(-64(x -0.5) 2 ))) (this function was proposed by [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF], it was also used in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and in [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. The pattern of γ is given in Figure 1. For each of the N simulated samples, we estimate γ(•) at x = (0.1, 0.2, 0.3, • • • , 0.8, 0.9) using the estimator (2.7) with a biquadratic kernel

K(x) = 15 16 (1 -x 2 ) 2 1 [-1,1] .
As mentioned in previous papers (see Slaoui (2014a,b)), there is no big influence on the choice of the kernel K in our setup when the observations are not contamined. However, investigating deconvolution problem of such estimator is out of the scope of our area of research (see [START_REF] Slaoui | Data-driven deconvolution recursive kernel density estimators defined by stochastic approximation method[END_REF]).

In order to calculate our estimator, we need to choose the bandwidth h and the threshold t n . We take t n to be the (nk) th order statistic Y (n-k) as is usual in extreme value statistics. Moreover, we propose an algorithm for choosing (h, k). This algorithm adapted from [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], was considered recently by [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. The purpose is then to select the bandwidth h using the following cross-validation criterion

h cv = arg min h∈H n i=1 n j=1 1 {Y i Y j } -F n,-i (Y j |X i ) 2 , where H= h n = cn -1 ; n 1 and (c, v) ∈ {0.1, 0.2, • • • , 0.9} is a grid of values for h and F n,-i (y|x) := n j=1,j =i K h (x -X j )1 {Y j y} n j=1,j =i K h (x -X j )
. This criterion was introduced in Yao (1999), implemented by [START_REF] Gannoun | Reference ranges based on nonparametric quantile regression[END_REF], and established in an extreme value context by [START_REF] Daouia | Kernel estimators of extreme level curves[END_REF][START_REF] Daouia | On kernel smoothing for extremal quantile regression[END_REF], [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and [START_REF] Ndao | Nonparametric estimation of the conditional extreme value index with random covariates and censoring[END_REF]. Using this bandwidth selection, we consider the following procedure to determine the number of threshold excesses k. This procedure rests on considering for each point x, the following steps:

Step 1: we compute the estimates for

γ H n,Y (n-k) (x) with k = 1, • • • , n -1.
Step 2: we constitute several successive "blocks" of the estimates

γ H n,Y (n-k) (x) (one block for k ∈ {1, • • • , 15}, a second block for k ∈ {16, • • • , 30} and so on).
Step 3: we calculate the standard deviation of the estimate in each block.

Step 4: we determine the k-value (denoted by k 1 ) from the block with minimal standard deviation (in particular, we take the median of the k-values in that block).

Finally, we estimate γ(x) by using the estimator γ H n (2.7) by taking (h, k) = (h cv , k 1 )

Results

For each configuration of the simulation design parameters (sample size n, stepsize parameters (γ n , β n ) and covariate value x), we calculate the average IAE (Integrated Absolute Error), the average ISE (Integrated Squared Error) and L ∞ of the estimators over N = 500 trials;

IAE = 1 N N i=1 R g [i] (x) -γ(x) dx, ISE = 1 N N i=1 R g [k] (x) -γ(x) 2 dx and L ∞ = max i=1,••• ,N R g [i] (x) -γ(x) dx,
where g [i] corresponds to the estimator computed from the ith sample. In order to investigate the comparison estimators, we consider the stepsizes (γ n , β n ) equal to n -1 , n -1 , (2/3) n -1 , n -1 , n -1 , (2/3) n -1 and (2/3) n -1 , (2/3) n -1 respectively. These four choices of parameters of the recursive estimator are referred to as R1, R2, R3 and R4 respectively. Results are highlighted in Table 1.

We point out that the major merit of our proposed estimator lies in its update aspect. Indeed, when new sample points are available, it requires less computational cost than non-recursive estimator. Moreover, Table 1 reveals that our proposed recursive estimator can provide better results in some specific situations that are very close in general to the reference values, which proves the effectiveness of our proposed recursive estimator in terms of the estimation error. Figure 2 discloses that all the considered estimators yield good results since the values of γ at each point x ∈ {0.1, 0.2, . . . 0.9} are very close to the median. 1: Simulation results for γ(x). N = 500 trials are considered. For each configuration of the simulated parameters (n, γ n , β n , x), the first line indicates the value of each estimator in different points x. Lines two, three and four provide the averages IAEs, ISEs and L ∞ respectively of the five different kernel density estimators. NR corresponds to the non-recursive estimator and R1, R2, R3 and R4 correspond to the proposed recursive estimators when (γ n , β n ) takes n -1 , n -1 , (2/3) n -1 , n -1 , n -1 , (2/3) n -1 and

(2/3) n -1 , (2/3) n -1 respectively. Table 2: The comparaison between errors of the non-recursive estimator (2.8) and the proposed recursive estimator (2.7).

Real dataset

We considered a Malaria dataset of 176 families in Senegal, totalizing 505 children between 2 and 19 years old, living in two villages of Niakhar (Toucar and Diohine). The number of observations was 6986. We measured Plasmodium falciparum Parasite Load (PL) from thick blood smears obtained by finger-prick during two different seasons and regularly over a three-year observation period (2001)(2002)(2003). The number of measurements per child ranged from 1 to 15. We refer readers to consult [START_REF] Milet | Genome wide linkage study, using a 250K SNP map, of Plasmodium falciparum infection and mild malaria attack in a Senegalese population[END_REF] for more details about data. These data were used also in Slaoui and Nuel (2014c) in a parametric context and more recently in Slaoui (2019a) in a non-parametric context. In the real example, the proposed recursive estimator (2.7) was compared to Hill's non-recursive kernel estimator (2.8) proposed by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Therefore, for any considered estimator γ of the index function γ, we propose to compute IAE and ISE defined as:

IAE( γ) = R | γ(x) -γ(x)| dx and ISE( γ) = R ( γ(x) -γ(x)) 2 dx.
Departing from Table 2 and Figure 3, we infer that the IAE and the ISE of the proposed recursive estimator are smaller than those of the non-recursive estimator set forward by [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Thus, demonstrating the effectiveness of our considered estimator.

Conclusion

In this research paper, we tackled the estimation of the conditional extreme value index γ(x) of a heavy-tailed distribution when some random covariate information is available. We elaborate recursive kernel estimator of the extreme value index function based on the stochastic approximation algorithm. The proposed estimator asymptotically follows normal distribution. We subsequently compared the proposed estimator to Hill's non-recursive extreme value index estimator. We demonstrated that using some particular stepsizes and a specific bandwidth selection through a cross-validation procedure, the proposed recursive estimator could be very competitive to the non-recursive version. Moreover, we highlighted that the proposed estimator is much better in terms of computational costs. Numerical results illustrate the effectiveness of our recursive approach. To this extent, we would state that although our work is an extension to a wealthy historical background, it may be taken further, extended and built upon since it offers different perspectives and opens new horizons for future research. We can extend our recursive extreme value index estimator to the case of censored data. We can also propose a new estimator of the conditional extreme quantile using our recursive estimator defined by (2.7) and compare it to the classical Weissman estimator. Another direction is to investigate the almost sure convergence and the large and moderate deviation principles of the proposed estimator, which requires non trivial mathematics. This would go well beyond the scope of the present paper.

for t n -→ n→∞ ∞ and h n -→ n→∞ 0 with h n ln t n -→ n→∞ 0 m n (x) = m n (x)g (x) 1 + C C x h n ln t n + o(h n ln t n ) . (A.1) Proof of Lemma 2 m n (x) = E K hn (x -X n )(ln Y n -ln t n )1 {Y >tn} = E K hn (x -X n )E((ln Y n -ln t n )1 {Y >tn} |X n ) = E [K hn (x -X n )m n (X n )] = R d K(z)m n (x -h n z)g(x -h n z)dz.
An integration by parts then yields

m n (x) = Ω K(z) ∞ tn F (y|x -h n z) y dy g(x -h n z)dz,
and consequently,

m n (x) -m n (x)g (x) = Ω K(z) ∞ tn F (y|x -h n z) y dyg(x -h n z)dz -g (x) ∞ tn F (y|x) y dy (C6) = Ω K(z) ∞ tn F (y|x) y F (y|x -h n z) F (y|x) dy(g(x -h n z) -g (x) + g (x))dz - Ω K(z) ∞ tn F (y|x) y dy(g (x) -g(x -h n z) + g(x -h n z))dz = Ω K(z) ∞ tn F (y|x) y F (y|x -h n z) F (y|x) -1 dyg (x) dz + Ω K(z) ∞ tn F (y|x) y dy (g(x -h n z) -g (x)) dz + Ω K(z) ∞ tn F (y|x) y F (y|x -h n z) F (y|x) -1 dy (g(x -h n z) -g (x)) dz =: I 4 + I 5 + I 6 . (A.2)
The expression of I 4 can be written as follows

I 4 = Ω K(z) Ĩ4 g (x) dz, with Ĩ4 = ∞ tn F (y|x) y F (y|x -h n z) F (y|x) -1 dy,
and grounded on the fact that

F (y|x -h n z) F (y|x) = exp ln F (y|x) ln F (y|x -h n z) ln F (y|x) -1 , (A.3) under (C5), we readily obtain ln F (y|x -h n z) ln F (y|x) -1 = c F d i=1 (h n z i ) 2 = c ′ F h n .
Moreover, using the following property: ln l(y|x) ln y -→ 0 as y -→ ∞, ensures that ln

F (y|x) ln F (y|x -h n z) ln F (y|x) -1 = ln F (y|x)c ′ F h n = - 1 γ(x) ln y + ln l(y|x) c ′ F h n = - 1 γ(x) + o(1) c ′ F h n ln y = Ch n ln y, for some positive constant C = - 1 γ(x) + o(1) c ′ F .
Using the following Taylor's formula, exp(x)- =: Ĩ4,1 + Ĩ4,2 + Ĩ4,3 .

1 = x + x 2 2 + o(
Concerning Ĩ4,1 , by integration by parts and using (C1) and (C2), we get

Ĩ4,1 = -Ch n F (t n |x)(ln t n -1) -Ch n ∞ tn ∂F (y|x) ∂y (ln y -1)dy + Ch n ∞ tn F (y|x) y (ln y -1)dy = -Ch n F (t n |x)(ln t n -1) + 1 + 1 γ(x) + o(1) Ch n ∞ tn F (y|x) y ln ydy -Ch n 1 + 1 γ(x) + o(h n ) ∞ tn F (y|x) y dy = -Ch n F (t n |x)(ln t n -1) + 1 + 1 γ(x) + o(1) Ĩ4,1 -Ch n 1 + 1 γ(x) + o(h n ) m n (x).
Hence, it follows that Ĩ4,1 = Ch n F (t n |x)(ln t n -1)

1 γ(x) + o(1) + Ch n 1 + 1 γ(x) + o(1) m n (x) 1 γ(x) + o(1) = Ch n F (t n |x)γ(x) ln t n C x (1 + o(1)) C x - Ch n F (t n |x)γ(x)C x (1 + o(1)) C x + Ch n (1 + γ(x)) (1 + o(1)) m n (x) 1 + o(1) , with C x = 1 + b(t n |x) γ(x)ρ(x) 1 1 -ρ(x) -1 (1 + o(1));
which can be written as

Ĩ4,1 = C C x h n ln t n m n (x) (1 + o(1)) - C C x h n m n (x) (1 + o(1)) +C (1 + γ(x)) h n m n (x) (1 + o(1)) = C C x h n ln t n m n (x) (1 + o(1)) . (A.4)
Concerning Ĩ4,2 , integration by parts and using (C1) and (C2), we get

Ĩ4,2 = - C 2 h 2 n 2 F (t n |x) ln 2 t n -2 ln t n + 2 - C 2 h 2 n 2 ∞ tn ∂F (y|x) ∂y ln 2 y -2 ln y + 2 dy + C 2 h 2 n 2 ∞ tn F (y|x) y ln 2 y -2 ln y + 2 dy = - C 2 h 2 n 2 F (t n |x) ln 2 t n -2 ln t n + 2 + C 2 h 2 n 2 1 γ(x) + o(1) ∞ tn F (y|x) y ln 2 t n -2 ln t n + 2 dy + C 2 h 2 n 2 ∞ tn F (y|x) y ln 2 y dy -C 2 h 2 n ∞ tn F (y|x) y ln y dy + C 2 h 2 n ∞ tn F (y|x) y dy = - C 2 h 2 n 2 F (t n |x) ln 2 t n + C 2 h 2 n F (y|x) ln t n -C 2 h 2 F (y|x) + 1 + 1 γ(x) + o(1) Ĩ4,2 -Ch n 1 + 1 γ(x) + o(1) Ĩ4,1 + C 2 h 2 n 1 + 1 γ(x) + o(1) m n (x).
Then, it comes that

Ĩ4,2 = C 2 2C x h 2 n γ(x)F (t n |x)C x ln 2 t n (1 + o(1)) - C 2 C x h 2 n γ(x)F (t n |x)C x ln t n (1 + o(1)) +C 2 h 2 n γ(x)F (t n |x) (1 + o(1)) + Ch n (1 + γ(x)) (1 + o(1)) Ĩ4,1 -C 2 h 2 n (1 + γ(x)) (1 + o(1)) m n (x) = C 2 2C x h 2 n ln 2 t n m n (x) (1 + o(1)) . (A.5) Moreover, note that Ĩ4,3 = o(1). (A.6)
Now, the combination of (A.4), (A.5) (A.6), ensures that

Ĩ4 = C C x h n ln t n m n (x) (1 + o(1)) .
Furthermore, we have

I 4 = Ω K(z) Ĩ4 g (x) dz = C C x h n ln t n m n (x)g (x) (1 + o(1)) . (A.7)
Concerning I 5 , we have under the assumption (C4) and (C6)

I 5 = Ω K(z) ∞ tn F (y|x) y dy (g(x -h n z) -g (x)) dz = c g m n (x) Ω K(z)d(x -h n z, x) dz = o(1). (A.8)
Moreover, under the assumption (C4)-(C6), we have, for t n sufficiently large,

I 6 = Ω K(z) ∞ tn F (y|x) y F (y|x -h n z) F (y|x) -1 dy (g(x -h n z) -g (x)) dz = o(1). (A.9)
The combination of (A.2), (A.7), (A.8) and (A.9), ensures that

m n (x) -m n (x)g (x) = C C x h n ln t n m n (x)g (x) (1 + o(1)) = m n (x)g (x) C C x h n ln t n (1 + o(1)) ,
which gives (A.1). Lemma 3 Let assumptions (C1) and (C4)-(C6) hold. Then, for all x ∈ R d such that g (x) > 0, we have for t

n -→ n→∞ ∞ and h n -→ n→∞ 0 with h n ln t n -→ n→∞ 0 E K hn (x -X n ) 1 {Yn>tn} = g (x) F (t n |x) (1 + Ch n ln t n + o(h n ln t n )) .
Proof of Lemma 3 Since (X i , Y i ), i = 1, • • • , n are independent and identically distributed, we have under the assumption (C6)

E K hn (x -X n ) 1 {Yn>tn} = R d R 1 h d n K x -t h n 1 {y>tn} f (y|t)g(t)dtdy = R d 1 h d n K x -t h n F (t n |t)g(t)dt = Ω K(u)F (t n |x -uh n )g(x -uh n )du. Now, we consider E K hn (x -X n ) 1 {Yn>tn} -F (t n |x)g (x) = F (t n |x) Ω K(u) F (t n |x -uh n ) F (t n |x) -1 g(x -h n u)du +F (t n |x) Ω K(u)(g(x -h n u) -g (x))du =: J1 + J2 .
Concerning J1 , under the assumption (C5) and using the equation (A.3), we have

F (t n |x -uh n ) F (t n |x) = exp ln F (t n |x) c F u 2 h n = exp ln t n - 1 γ(x) + o(1) c F u 2 h n = exp [ln t n Ch n ] .
Moreover, since g (x) > 0, the application of Taylor's formula, ensures that

J1 = g (x) F (t n |x) Ω K(u) [Ch n ln t n + o(h n ln t n )] g(x -h n u) g (x) du = Cg (x) F (t n |x)h n ln t n + o(F (t n |x)h n ln t n ).
Under (C5), we have

J2 = F (t n |x)c g Ω u 2 h n K(u)du = o(g (x) F (t n |x)h n ln t n ).
Then, we get

E K hn (x -X n ) 1 {Yn>tn} = g (x) F (t n |x) + Cg (x) F (t n |x)h n ln t n + o(F (t n |x)h n ln t n ).
We state now the following technical lemma, which is proved in Mokkadem et al. (2009a), and which will be used throughout the demonstrations. Lemma 4 Let (v n ) ∈ GS (v * ), (γ n ) ∈ GS (-α) and m > 0 such that mv * ε > 0 where ε is defined in (2.9), and Π n in (2.3). Then,

lim n→∞ v n Π m n n k=1 Π -m k γ k v k = 1 m -v * ε .
Moreover, for all positive sequences (α n ) such that lim n→∞ α n = 0, and all C ∈ R,

lim n→∞ v n Π m n n k=1 Π -m k γ k v k α k + C = 0.
Proof of Theorem 1

1. The application of Lemma 2, ensures that

E(a n (x)) = Π n n k=1 Π -1 k γ k mk (x) = Π n n k=1 Π -1 k γ k m n (x)g (x) 1 - C C x h k ln t n + o(h k ln t n ) .
In the case p ∈ (0, α/ (d + 2)], we have lim n→∞ nγ n > p; the application of lemma 4 ensures that

E(a n (x)) = a(x) - C (1 -pε)C x a(x)h n ln t n + o(h n ln t n ),
and (2.11) follows. In the case p ∈ (α/ (d + 2) , 1/d), we have

h n ln t n = o γ n h -d n , Lemma 4 ensures E(a n (x)) -a(x) = o γ n h -d n , which gives (2.12). = Q 2 n n k=1 Q -2 k β 2 k K 2 2 h d k E h -d k H x -X k h k 1 {Y k >tn} -E 2 h -d k K x -X k h k 1 {Y k >tn} , with H(•) =: K 2 (•) K 2 2
also satisfying assumption (C6). Using Lemma 3, we get

Var(b n (x)) = Q 2 n n k=1 Q -2 k β 2 k K 2 2 h d k g (x) F (t n |x) (1 + Ch k ln t n + o(h k ln t n )) -g 2 (x)F 2 (t n |x) (1 + 2Ch k ln t n + o(h k ln t n )) ,
then, we have

Var(b n (x)) = K 2 2 g (x) F (t n |x)Q 2 n n k=1 Q -2 k β 2 k h d k + C ln t n K 2 2 g (x) F (t n |x)Q 2 n n k=1 Q -2 k β 2 k h d-1 k + K 2 2 g (x) F (t n |x)Q 2 n n k=1 Q -2 k β 2 k o ln t n h d-1 k -g 2 (x)F 2 (t n |x)Q 2 n n k=1 Q -2 k β 2 k -2C ln t n g 2 (x)F 2 (t n |x)Q 2 n n k=1 Q -2 k β 2 k h k -g 2 (x)F 2 (t n |x)Q 2 n n k=1 Q -2 k β 2 k o(h k ln t n ).
In the case when p ∈ [b/ (d + 2) , 1/d), we have lim By using Lemma 4 and choosing the stepsize (γ n ) = (n -1 ), computations provide .13) with

Cov(a n (x), b n (x)) = 1 n Q n n k=1 Q -1 k β k h -2d k (J 1 -J 2 J 3 ) , (A
J 1 = E K 2 x -X k h k [ln Y k -ln t n ]1 {Y k >tn} , J 2 = m n (x) and J 3 = E K h k (x -X k ) 1 {Y k >tn} .
Following similar steps as Lemma 2 in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF] and Lemma 2, we infer that

J 1 = m n (x)g (x) K 2 2 h d k 1 - C C x h k ln t n + o(h k ln t n ) ,
J 2 and J 3 are already calculated in Lemmas 1 and 2. Then, the combination of (A.11), (A.12), (2.14), (2.18) and (A.13), gives (2.22), and the combination of (A.11), (A.12), (2.13), (2.17 where

Y k (x) = Π -1 k γ k Z k (x) -γ(x)η n η -1 k β k W k (x) , with Z n (x) = K hn (x -X n ) [ln Y n -ln t n ]1 {Yn>tn} , W n (x) = K hn (x -X n ) 1 {Yn>tn} and η n = Π -1
n Q n . Now, in the case when (β n ) = n -1 , we have η n = (nΠ n ) -1 and η -1 In addition, classical computations and applications of Lemma 4 ensure that Γ 1 = Π -2 n γ(x)

k β k = Π k . Then, Y k (x) = Π -1 k γ k Z k (x) -γ(x)(nΠ n ) -1 W k (x).
1 C x 6 2 -(α -pd)ε K 2 1 m n (x)g (x) γ n h d n + o γ n h d n , Γ 2 = Π -2 n γ(x) 1 C x 1 1 + pd K 2 2 m n (x)g (x) 1 nh d n + o 1 nh d n , Γ 3 = Π -2 n γ(x) 2 1 + pdε K 2 2 m n (x)g (x) 1 nh d n + o 1 nh d n .
As a matter of fact, we infer that

s 2 n = b 2 (x) Π 2 n γ n h d n [Var(x) + o(1)] .
On the other side, we have, for all q > 0,

E | Y k (x) | 2+q = O 1 h (1+q)d k ,
and, since lim n→∞ (nγ n ) > (αpd) /2, there exists q > 0 such that lim n→∞ nγ n > 1 + q 2 + q (αpd).

Applying Lemma 4, we get The convergence in (A.14) then follows from the application of Lyapounov's Theorem.

n k=1 E | T k (x) | 2+q = O n k=1 Π -2-q k γ 2+q k E | Y k (x) | 2+q = O γ 1+q n Π 2+q n h (

  ), where γ(•) is an unknown positive function of the covariate x called the tail function and for a fixed x, l(•|x) is a function that varies slowly at infinity, i.e for all λ > 0means that the conditional distribution of Y given X = x is in the Frechet maximum domain of attraction. The tail function γ(x) is the conditional extreme value index function which needs to be adequately estimated from the available data.(C2): l(•|x) is normalized. The Karamata representation (Theorem 1.3.1 given in Bingham et al. (1987)) of the slowly-varying function, l(•|x), can be written as l(y|x) = c(x) exp y 1 ε(z|x) z dz , where c(•) is a positive function and ε(z|x) -→ 0 as z -→ ∞. Thus, l(•|x) is differentiable and the function ε(•|x) is given by ε(z|x) = z l ′ (z|x) l(z|x) . (C3): There exists a strictly negative function ρ(•), a strictly positive function γ(•) and a rate function b(•|x), b(y|x) -→ 0 as y -→ ∞, of constant sign for large values of y such that for all υ

  the convergence in distribution, N the gaussian-distribution and P -→ the convergence in probability.

Figure 1 :

 1 Figure 1: Pattern of γ(•) on [0, 1]

Figure 2 :

 2 Figure2: Boxplots of the N = 500 estimates of our five considered estimators (NR, R1, R2, R3, R4) in points x = 0.1, 0.2, 0.3 (1st line), x = 0.4, 0.5, 0.6 (2nd line) and x = 0.7, 0.8, 0.9 (3rd line), and dashed lines represent the values of reference estimator γ(•) in each point x as mentioned above.

Figure 3 :

 3 Figure 3: Qualitative comparaison between the non-recursive estimator (2.8) and the proposed recursive one (2.7).

F

  (y|x) y ln 2 y dy + o(1)

  bp(d -1))ε 1 K 2 2 g (x) ln t n F (t n |x)

  ) and (A.13), gives (2.21). Proof of Theorem 4 Let us at first assume that, if p α/(d + 2when p > α/(d + 2), Part 1 of the theorem follows from the combination of (2.20) and (A.14). In the case when p = α/(d + 2), Parts 1 and 2 of the Theorem follow from the combination of (2.19) and (A.14). In the case p < b/(d + 2), (2.21) implies that1 h n ln t n γ H n (x) -E γ H n (x) P -→ 0,and the application of (2.19) gives Part 2 of Theorem. Now (A.14) is proved. Relying on (A.11), we haveD n (x) -E[D n (x)(x) -E[Y k (x)]) ,

SetT

  k (x) = Y k (x) -E [Y k (x)] . Var (Z k (x)) + γ 2 (x)(nΠ n ) Cov (Z k (x), W k (x)) := Γ 1 + Γ 2 + Γ 3 .

Acknowledgments

This work was performed while the first author was at the Laboratory of Mathematics and Applications of Poitiers as a visiting scientist, partly funded by the University of Monastir. The authors would like to thank the Editor-in-Chief Prof. Narayanaswamy Balakrishnan of Journal of Communications in Statistics -Theory and Methods and the reviewers for their helpful comments, which helped us focus on improving the original version of the paper.

Funding

This work benefited from the financial support of the GDR 3477 GeoSto. It is supported also by LR18E16: Analyse, Géométrie et Applications, University of Monastir (Tunisia).

A Proofs

We introduce the following Lemmas that will enable us to obtain the asymptotic expansion of a n .

Lemma 1 Let assumption (C3) holds. Then, for t n -→ ∞ as n → ∞ we have

The proof of Lemma 1 is presented in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF]. Lemma 2 Let assumptions (C1)-(C6) hold. Then, for all x ∈ R d such that g (x) > 0 we have 2. Now, we have

The application of Theorem 1 in [START_REF] Goegebeur | Nonparametric regression estimation of conditional tails: the random covariate case[END_REF], ensures that

In the case when p ∈ [α/ (d + 2) , 1/d), we have lim

, and the application of Lemma 4 ensures that

which proves (2.14). In the case when p ∈ (0, α/ (d + 2)), we have

n ln 2 t n ), which yields (2.13).

Proof of Theorem 2

1. First, the application of Lemma 3 provides

Now, in the case when p ∈ (0, b/ (d + 2)], we have lim n→∞ nβ n > p; the application of Lemma 4 ensures that

and (2.15) follows. In the case when p ∈ (b/ (d + 2) , 1/d), we have

, which gives (2.16).

Now, we have

which proves (2.18). In the case when p ∈ (0, b/ (d + 2)), we have

n ln 2 t n ), which gives (2.17).

Proof of Theorem 3 Let us first note that, for x such that b n (x) = 0, we have .11) It follows from (A.10), that the asymptotic behavior of γ H n (x)γ(x) can be deduced from the one of D n (x). Then, (2.19) follows from (2.11), (2.15) and (A.10) whereas (2.20) follows from (2.12), (2.16) and (A.10). Now it follows from (A.11) that