Andres Morfin Veytia

Andrei Calin

Joost Badea

Jacco Ellerbroek

Niki Hoekstra

Ioannis Patrinopoulou

Vaios Daramouskas

V Lappas

V Kostopoulos

Jacomijn De Vries

Van Ham

Badea

A M Veytia

C A Badea

J Hoekstra

N Patrinopoulou

I Daramouskas

V De Vries

J Van Ham

E Sunil

P Menendez

Ponte Alonso

J Rodrigo

Villazon Terrazas

D Bereziat

A Vidosavljevic

L Sedov

Metropolis II: Benefits of Centralised Separation Management in High-Density Urban Airspace

Keywords: Unmanned Traffic Management, Conflict Detection & Resolution (CD&R), Self-Separation, U-space, UAS, UTM

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

I. INTRODUCTION

Urban air traffic mobility is a subject of high interest across the research community, as the applications of use of aircraft in the Very Low Level (VLL) urban airspace are increasing. The use of autonomous aircraft for delivery transportation has been suggested as a measure to combat congestion in high density ground vehicle traffic in city centres and excessive emissions. Unmanned Traffic Management (UTM) systems need to be designed and tested to investigate methodologies for integrating highly dense air traffic in the urban environment.

Previous research has been conducted with the scope of shaping future operations of small aircraft within VLL airspace. The CORUS project [START_REF] Sedov | U-space Concept of Operations[END_REF] categorised airspace sections, and defined urban airspace as being above the tallest obstacle in a city. However, that might not be feasible for missions such as local small package deliveries, as it would be inefficient for small aircraft to travel at such high altitudes in cities with very tall buildings (e.g., New York).

Furthermore, the question of how to control traffic within U-space operations still remains. Previous research has emphasised centralised, strategic separation management, with tactical separation as a potentially federated or distributed fallback. The question remains, however, to what extent (highdensity) drone traffic will be accurate and predictable enough to enable effective strategic separation, and whether proposed business models for urban drone applications would be compatible in terms of required planning horizons.The Metropolis II project aims to investigate the impact of the degree of centralisation of a separation management method on resulting traffic safety, efficiency, capacity and equity. n The project developed three separate separation management concepts with varying degrees of centralisation; a centralised concept including only centralised strategic deconfliction techniques, a decentralised system including only decentralised tactical deconfliction techniques and a hybrid system including both centralised strategic deconfliction and decentralised tactical deconfliction.

This paper presents an overview of the experimental results of the Metropolis II project, regarding safety, efficiency and other performance metrics of three different concepts in highly dense urban airspace. Section II-A presents the design of the urban airspace, missions and operations used in the project. The three concepts are briefly introduced in Section II-B. The experimental and simulation methodology is described in Section III, while the metrics used for concept comparison and the most relevant results are shown in Section IV. Finally, the discussion and conclusion are presented in sections V and VI, respectively.

II. METHODS

The following section presents the background information of urban airspace and UTM concept design used in simulations within the Metropolis II project. More detailed information about the developed concept of operations can be found in [START_REF] Patrinopoulou | Metropolis II: Investigating the Future Shape of Air Traffic Control in Highly Dense Urban Airspace[END_REF].

A. Urban environment

It is expected that urban air traffic will be fundamentally different from conventional aviation due to the increased presence of physical obstacles, the higher expected traffic densities, the different operational characteristics, and the use of small unmanned aerial vehicles. Globally, several developments are undergoing for creating a concept of operations capable of enabling aircraft operations within cities.

A large volume of these operations is expected to be conducted within Very-Low Level (VLL) airspace, defined by FAA [3] and EU [START_REF] Sedov | U-space Concept of Operations[END_REF] projects as being below an altitude of approximately 500ft. Considering building heights, aircraft flying at such low altitudes might be required to fly above the existing street network for safety, efficiency and privacy reasons. Thus, the Metropolis II project aims to include both constrained (i.e., aircraft are constrained to flying above streets and between buildings) and open (i.e., aircraft can fly freely above obstacles in any direction) airspace within the study. Furthermore, previous research [START_REF] Doole | Constrained urban airspace design for large-scale drone-based delivery traffic[END_REF] performed within constrained airspace has used orthogonal street networks, which provide an idealised traffic situation. Within the Metropolis II project, a broader scope was taken by also considering organically developed street networks, common in cities in Europe.

For this reason, the city of Vienna was chosen as the basis for the simulation environment, due to the presence of a combination of orthogonal and organic street network topology, and its relatively low street alignment [START_REF] Boeing | Urban spatial order: street network orientation, configuration, and entropy[END_REF]. Vienna also has a low disconnectness index [START_REF] Barrington-Leigh | A global assessment of street-network sprawl[END_REF], meaning that all areas of the city are reachable without much detours. Lastly, the city has a great amount of open data available for both the buildings [START_REF] Baukörpermodell | LOD0.4) Wien[END_REF] and the streets [START_REF]Hall of Fame/Streets complete -OpenStreetMap Wiki[END_REF], providing the opportunity to create a more realistic simulation area.

Alongside buildings, urban airspace will have to account for areas in which flights will not be permitted, such as parks or other privacy-sensitive locations. Within the Metropolis II project, we chose to enforce these areas using geofencing. Within the experiment airspace designed for Vienna, several geofenced areas were created within open airspace. Thus, although aircraft may freely fly in open airspace, they are not allowed to fly within these restricted areas.

Vertiports and distribution centres were placed within open and constrained airspace to act as origins and destinations for aircraft, based on the municipal demand expected in the future. These demands are based on parcel delivery data from [START_REF] Doole | Constrained urban airspace design for large-scale drone-based delivery traffic[END_REF], [START_REF] Bowes | Annual report[END_REF]. A linear regression is used that takes into account area, population size, and average annual gross salary per municipality. Fig. 1 shows how the city centre contains more vertiports as it is expected to have more demand. Distribution centres have a higher number of aircraft taking off from the same location, while vertiports have less demand but are more numerous. Within this work, the percentage of traffic originating from vertiports and distribution centres is 40%.

B. Description of concepts

In the Metropolis 2 project, three different separation management concepts were studied, which differed in terms of the degree of centralisation of the separation management task: a centralised, a hybrid, and a decentralised concept. Table I shows an overview of the main differences between the concepts, and the following sections present a summary of the design choices made within each. The open airspace design is different among concepts due to the need for it to be compatible with the concept philosophy: in a decentralised concept, the airspace is less structured, while in a more centralised one, structure is beneficial to the global optimisation of trajectories.

1) The Centralised Concept: The Centralised concept aims to strategically deconflict all aircraft prior to take-off. To do this, the open airspace is structured in hexagonal cells and is connected to the constrained airspace street network to enable use of the A* path-finding algorithm [START_REF] Duchamp | Air traffic deconfliction using sum coloring[END_REF], [START_REF] Duchamp | Density-Adapting Layers towards PBN for UTM[END_REF]. The airspace is divided into 16 vertical layers, where each layer is assigned a heading range, similar to the layers concept created within the Metropolis I project [START_REF] Sunil | Metropolis: Relating Airspace Structure and Capacity for Extreme Traffic Densities[END_REF].

Flight planning and strategic deconfliction is performed in a two-step approach. In the first step, a relaxed graph colouring model is used to assign flight layers such that the number of unsolved conflicts is minimised. In the second step, the remaining conflicts are solved by delaying the departure slot of flights.

In the centralised concept, a single authority is responsible for planning all flights and ensuring that all aircraft maintain separation throughout their mission. Flight plans are therefore 4-dimensional (latitude, longitude, height tracked in time). UAS agents are responsible for following the flight plan as closely as possible considering their performance limitations and uncertainties. The centralised concept does not include a tactical separation component.

2) The Hybrid Concept: The Hybrid concept attempts to leverage the use of a central entity for flight planning and strategic separation management while letting individual aircraft tactically separate if necessary. In the hybrid concept, the strategic separation component relies on using horizontal separation while the tactical separation component uses vertical and speed-based manoeuvres to solve conflicts not predicted by the central entity. In open airspace, airspace is structured as a unidirectional radial grid that connects to the street network in constrained airspace.

The hybrid concept adds an extra cost to travelling in constrained airspace, to reduce traffic density within the centre of the city. The total airspace is represented as a graph and the paths are solved using the Dijkstra algorithm. When an edge is occupied by an aircraft that edge is removed from the graph for a certain amount of time. This means that if an aircraft is not able to find an adequate solution, it gets delayed.

This concept also makes use of 16 layers to vertically separate aircraft. However, there are a reduced number of cruising layers when compared to the Centralised concept in order to provide aircraft with the option of using deconfliction layers for tactical conflict resolution maneuvers.

3) The Decentralised Concept: The decentralised concept only makes use of tactical separation to maintain safe separation between aircraft. The tactical separation algorithm employs both speed and altitude based manoeuvres. In the decentralised concept, open airspace is discretised into polygonal cells in the horizontal plane. In the vertical plane, layers are used to separate aircraft vertically in function of heading range, similar to the centralised concept (16 layers). In constrained airspace, all streets are uni-directional and are assigned a cruise height allocation and direction. The directions are assigned in a way that ensures that the whole airspace is accessible. The directionality allocation aims to minimise the probability of two cruising aircraft intersecting with each other. However, due to the street network containing non-nominal intersections (e.g., 3-way intersections), this will not always be possible. Aircraft that turn can make use of turn layers, located between cruising layers.

The decentralised concept does not use any strategic separation during flight planning. However, there is a central entity that monitors the traffic in constrained airspace and updates the cost of travelling through the airspace. Each aircraft then has the option to replan if the cost has changed due to high traffic, but also has the option to maintain its current route.

III. EXPERIMENT

The Metropolis II simulation experiments were designed to see how the level of centralisation influences safety, efficiency, and other metrics. More detailed information about the experiment set up may be found in the Metropolis II reports [START_REF]Deliverables -Metropolis 2[END_REF] and in [START_REF] Patrinopoulou | Metropolis II: Investigating the Future Shape of Air Traffic Control in Highly Dense Urban Airspace[END_REF].

A. Simulation software

The experiments were simulated through fast-time simulations using BlueSky, an open air traffic simulator [START_REF] Hoekstra | Bluesky ATC simulator project: an open data and open source approach[END_REF]. BlueSky has an extensible plugin system, to which different conflict detection and resolution (CD&R) implementations and other modules can be added, allowing for testing of different concepts under similar conditions.

B. Simulation Area

The Metropolis II simulation airspace is an 8 kilometre wide and 500 feet high cylinder centered in the center of Vienna (Fig. 1). It includes open and constrained airspace. Aircraft are allowed to fly without constraints in open airspace unless there is a geofenced area. In constrained airspace, aircraft must fly above streets.

C. Aircraft models

For the Metropolis II project two aircraft types (MP20 and MP30) based on a simplified DJI Matrice 600 Pro hexacopter drone performance model were used in the simulations. The difference between the two is that MP20 cruises at a speed of 20 kts while MP30 at 30 kts.

D. Missions

In order to achieve a diverse and representative set of traffic patterns, and to provide a means to study the use of mission prioritisation, four mission types were defined:

1) Parcel Deliveries: These missions start from distribution centers and travel to a vertiport, and thus generate converging/diverging traffic patterns; 2) Food Deliveries: These missions travel from vertiport to vertiport, leading to point to point traffic patterns; 3) Loitering missions: Point to point missions that activate a dynamic geofence upon arrival; 4) Emergency missions: These missions are similar to food deliveries but are announced to the UTM system closer to their desired take-off time. There are four levels of priority: low, medium, high, and emergency. The parcel, food and loitering traffic is assigned in equal proportion for the first three levels, with emergency missions having the highest priority. These levels are taken into consideration during tactical and strategic deconfliction decisions.

E. Uncertainties and non-nominal situations

Uncertainties and non-nominal events are introduced in the experiments in order to test the robustness of the separation management concepts. These are in the form wind, and rogue aircraft, respectively. Rogue aircraft are vehicles that do not adhere to the rules developed by the concept. They travel from outside the airspace through the center and then back outside the airspace, ignore traffic rules, and do not respond to the presence and actions of other traffic. The number of rogue aircraft in the air is kept constant during the simulation. Note that while rogue aircraft do not follow the rules of the concept, they still travel through constrained airspace by following the streets. The aircraft also randomly vary altitude at several different locations which are not known to the operators. Rogue aircraft do not make any attempts to tactically solve conflicts, and thus should affect the tactical components of the concepts most.

The experiments also include a limited simulation of the effect of wind on UTM operations. The goal of including wind in the experiments was to induce speed and timing uncertainty, which would impact the strategic planning of concepts with a central component. Therefore, a uniform wind field is defined over the whole airspace but only the component parallel to the direction of the aircraft affects the speed of the aircraft. This means that the aircraft will only speed up or slow down and not experience cross-winds.

F. Independent variables

The following independent variables were defined for the experiments:

1) Degree of centralisation: Decentralised, Hybrid, and Centralised; 2) Traffic density: Very low, low, medium, high, and very high traffic densities are used (see Table II); 3) Wind level: The speed of the uniform vector field (0, 1, 3, and 5 knots); 4) Rogue aircraft: The number of rogue aircraft at one time in the airspace (0, 33, 66, 100). For each scenario characterised by a choice of independent variables, nine repetitions were performed using a different set of randomly-generated traffic.

G. Dependent measures

The dependent measures shown in this work are as follows:

• Number of pairwise conflicts, • Number of intrusions,

• Achieved instantaneous traffic density, • Mission duration efficiency, • Mission delay. A conflict is counted when there is a predicted intrusion with a certain look ahead-time. This work uses state-based detection, meaning that trajectories are linearly extracted with a certain look-ahead time. The horizontal and vertical minimum separation is set to 32 metres and 25 feet, respectively. The look-ahead time is 10 seconds.

H. Simulation time

Dependent variables were measured throughout the entire simulation in all scenarios. Each scenario ran until all aircraft reached their destinations. Although flight departures are only planned within the first hour, the simulation time varied between 90 min and 180 min.

IV. RESULTS

The current section presents the most important results of the Metropolis II project. The calculated metrics results are presented using boxplot graphs, each boxplot is computed based on the value of the metrics across the 9 scenario repetitions.

A. Traffic density

Figs. 2, 3, and4 show the cumulative traffic density heat maps throughout the simulations with high density traffic and 40 percent of traffic coming from distribution centers for the centralised, hybrid, and decentralised concepts respectively. Here, lighter color indicates denser traffic.

A comparison of Figs. 2,3, and 4 reveals several differences in terms of flight planning. In Fig. 2 the hexagonal grid of the Centralised concept is visible in open airspace. Fig. 3 shows the radial grid of the hybrid concept. Since there was a higher cost in travelling through constrained airspace, the traffic tended to form a beltway around the city center. Fig. 4 shows the effects of the absence of strategic planning. As aircraft tend to use the shortest path though the network in constrained airspace, there are certain streets and intersections which are more used than the others. It is also interesting to note that the centralised concept tended to spread the traffic more evenly in constrained airspace.

Fig. 5 shows that the decentralised and centralised concepts have very similar instantaneous traffic densities. The peak of the hybrid density in constrained airspace is about half of that for the decentralised and centralised concept due to the ringroad strategy used by the former. It is also clear that the hybrid concept took longer to taper off, showing that aircraft were in the air for longer as they avoided constrained airspace.

B. Airspace safety

Fig. 6 shows the number of conflicts per flight for each concept for the different traffic demand scenarios. The effect of strategic pre-take-off deconfliction can clearly be seen: the decentralised concept has the highest number of conflicts at all densities, reaching to an order of magnitude difference at the high and very high densities. The hybrid and centralised concepts both have similar number of conflicts with the centralised concept having slightly fewer. Fig. 7 shows the total number of intrusions for each of the concepts in the vertical axis and the scenario density in the horizontal axis. Here it is clear that the hybrid concept has the smallest number of intrusions at all traffic densities. The decentralised and centralised concepts are more comparable in number of intrusions. The levels appear similar at most traffic densities, though at the very high traffic density the decentralised concept has slightly more intrusions than the other concepts.

As the three concepts achieved different densities within constrained airspace, the number of intrusions is not directly comparable. Therefore, the intrusion rate was computed in function of aircraft density in constrained airspace, presented in Fig. 8. The intrusion trend of the centralised and decentralised concepts are nearly identical. Although the hybrid concept achieves the fewest intrusions per minute per number of aircraft in the air, the difference becomes less pronounced when compared to the total number of intrusions presented in Fig. 7.

C. Mission efficiency

The duration efficiency is an estimate of how much longer the simulated route took versus an ideal route, presented in Fig. 9. The centralised and hybrid concepts both have relatively constant efficiencies across densities. The hybrid concept achieved a lower overall efficiency due to the beltway airspace structure, which made missions last longer. The decentralised concept is similar in magnitude to the centralised concept, however the efficiency tends to decrease with the density due to aircraft having to resolve more conflicts tactically.

D. Mission delay

The average demand delay is measured as the mean of the delay of all the flights in a scenario, presented in Fig. 10. The centralised and decentralised concepts present similarly low values of delay, with the decentralised concept showing a slightly increasing pattern with increasing density. The hybrid concept induces notably higher delays for all densities and presents a rapid raise in the delays with increasing density. Three main factors contribute to demand delay. The first is the departure delay applied by the concepts to the aircraft as an effort to increase safety. The second is the route length, as the hybrid concept discourages aircraft from using the constrained airspace and plans longer routes to avoid it. The last one is the flight time with lower speed limit or hovering while resolving conflicts as a part of the tactical separation.

E. Service access and equity

Demand delay dispersion metric is used as an indicator of equity and access to UTM services. This metric gives an indication of how delay is distributed over airspace participants. A low value indicates that delay was evenly distributed, and a higher value means that some aircraft were delayed disproportionately more than others. It is computed as the standard deviation over the arrival delay of all aircraft which arrived at their destination, where the arrival delay is the difference between realized arrival time and ideal expected arrival time.

The demand delay dispersion can be seen in Fig. 11. The decentralised concept had the lowest demand delay dispersion which means that most aircraft were similarly delayed. In the very low, low, and medium densities the hybrid concept had a lower dispersion delay than the centralised concept. However, for the high and very high the effect is reversed and the hybrid concept achieves values above those of the centralised concept, indicating that for those densities the hybrid concept applies significantly longer delay to a subset of aircraft.

F. Effect of uncertainties and non-nominals on safety

Fig. 12 shows the additional number of intrusions due to the presence of wind for all three concepts in a high density scenario. The centralised concept shows a very low number of additional intrusions at 1 and 3 knots. However, it is higher at 5 knots. For the hybrid concept the additional number of intrusions is similar at 1, and 3 knots and then increases at 5 knots. For the decentralised concept, the number of intrusions is similar across all wind speeds. However, the spread of the number of intrusions moves to the positive side of the vertical axis for the 3 and 5 knots case.

Fig. 13 shows the additional number of intrusions due to rogue aircraft for all three concepts in the high density scenarios. Overall, the number of intrusions increases as the number of rogue aircraft increases. The hybrid concept resulted in the lowest number of additional intrusions. The number of intrusions for the decentralised and centralised concepts is similar across rogue levels, although the decentralised does show more spread in its results.

V. DISCUSSION

There are a number of observations that can be made from the results of the Metropolis 2 concept comparison. The first is that the hybrid concept managed to achieve the lowest number of intrusions of the three concepts, even when considering intrusion count as a function of instantaneous traffic density. However, when looking at, e.g., flight efficiency and demand delay it can be seen that the safe performance of the hybrid concept comes, in part, at the cost of flight efficiency and capacity. This shows the potential benefit of the combination of strategic and tactical separation management measures. However, not all of these differences can be attributed to the difference in degree of centralisation between the concepts. The design of the airspace was also a large factor in the observed differences between the three concepts.

Firstly, because the hybrid concept made use of dedicated deconfliction layers, it only has half of the allowable cruising space compared to the centralised concept, which can have an impact on capacity. This means that the strategic planner has half the vertical space available. From Fig. 11 it can be seen that at lower densities the dispersion of the delays of the hybrid concept was below the centralised concept. However, as the traffic density increased the availability of cruising space became denser and the hybrid concept was not able to match the centralised concept.

Secondly, the beltway-like structure of the hybrid concept for unconstrained airspace had aircraft in the air for longer because the strategic algorithm tended to prefer these beltways over flying in constrained airspace. This can be seen in the density heat maps (Fig. 3) and the instantaneous density plot in Fig. 5. This behaviour causes longer flight distances, but because of its high degree of traffic alignment (on any point along a roundabout all vehicles are going in approximately the same direction at the same speed) conflict probability can be reduced by a large amount. This shows that the airspace structure can have a significant impact on safety, especially if traffic is better aligned and spread out (albeit at the expense of route efficiency). The decentralised concept does not globally optimize the paths of the aircraft. It does attempt to spread traffic through flow control but local hotspots can still be seen within the airspace. Moreover, the number of conflicts in the decentralised concept are an order of magnitude higher than both the centralised and hybrid concepts. However, when comparing the number of resulting intrusions to the centralised concept results, it can be seen that the decentralised concept is able to maintain similar levels of intrusions to the centralised concept. The higher number of conflicts, therefore, achieve reactively, what a pre-departure optimisation algorithm does proactively. It is interesting that by only using strategic separation or tactical separation, the number of intrusions remains comparable. The centralised concept was able to reduce network bottlenecks and maintain path efficiency. However, it is not able to plan completely for all circumstances, therefore a tactical element remains indispensable. However, it can be difficult to effectively trade-off between both as it is seen from the performance of the hybrid concept in the duration efficiency (Fig. 9) and the demand delay (Fig. 10).

Another interesting result is the similarity between the centralised and decentralised concept in the safety results within constrained airspace (Fig. 8). This can be attributed to the nature of constrained airspace, where the action and solution space are limited. Thus, when the separation methods are limited (i.e., using either tactical or strategic), the airspace network structure itself has a greater influence on the safety.

The results in terms of robustness to rogue aircraft show that the hybrid concept is able to have the least number of intrusions for all rogue levels. This can be related to the fact that the hybrid concept distributes traffic efficiently in the strategic phase, and that its tactical resolution algorithm proved more effective in solving the type of conflicts that occurred with rogue vehicles.

VI. CONCLUSION

The hybrid concept was able to achieve the highest level of safety by combining centralised strategic and decentralised tactical separation management. The "ring road" routing method helped increase safety by a large factor in the hybrid concept as there were fewer aircraft in constrained airspace. The strategic planning of the hybrid concept distributed traffic and mitigated network bottlenecks. Moreover, the tactical separation strategy further improved the safety by resolving the conflicts that the strategic planning did not detect, due to uncertainties regarding the positions of all aircraft. However, this came at the expense of route efficiency and demand delay.

The decentralised concept provided the most equitable access to the airspace. In terms of safety and demand delay, the decentralised concept was on par with the centralised concept, resulting in a similar number of intrusions for both. However, the centralised concept was able to redistribute traffic more efficiently than the hybrid concept due to allowing more aircraft in constrained airspace.

To answer the question on the degree of centralisation of an urban air traffic management system, we propose an improved hybrid system with more elaborate path planning and less strict strategic planning that allow higher aircraft densities.

Figure 1 .

 1 Figure 1. Urban airspace structure used within the Metropolis II traffic scenario simulations.

Figure 2 .

 2 Figure 2. Cumulative density heat map for high density for the Centralised concept.

Figure 3 .

 3 Figure 3. Cumulative density heat map for high density for the Hybrid concept.

Figure 4 .

 4 Figure 4. Cumulative density heat map for high density for the Decentralised concept.

Figure 5 .

 5 Figure 5. Aircraft density in constrained airspace in function of time for a high-density scenario.

Figure 6 .

 6 Figure 6. Number of conflicts per flight as a function of the scenario density for the three concepts.

Figure 7 .

 7 Figure 7. Number of intrusions as a function of the scenario density for the three concepts.

Figure 8 .

 8 Figure 8. Intrusions per minute as a function of the number of aircraft in constrained airspace for the three concepts.

Figure 9 .

 9 Figure 9. Flight duration efficiency as a function of the traffic density for the three concepts.

Figure 10 .

 10 Figure 10. Average demand delay as a function of the traffic density for the three concepts.

Figure 11 .

 11 Figure 11. Demand dispersion delay as a function of the traffic density for the three concepts.

Figure 12 .

 12 Figure 12. Additional number of intrusions as a function of the wind level for the three concepts (1, 3, and 5 kts wind speed).

Figure 13 .

 13 Figure 13. Additional number of intrusions as a function of the number of rogue aircraft for the three concepts (33, 66 and 100 aircraft).

TABLE I .

 I OVERVIEW OF THE MAIN DIFFERENCES BETWEEN THE THREE CONCEPTS AND THEIR PROPERTIES[START_REF] Patrinopoulou | Metropolis II: Investigating the Future Shape of Air Traffic Control in Highly Dense Urban Airspace[END_REF].

	Features	Centralised	Hybrid	Decentralised
	Open Airspace	Hexagonal cells	Radial grid	Polygonal
				cells
	Global knowl-	Yes	Yes	No
	edge of flights
	Main	Pre-flight	Pre-flight Strate-	In-flight Tac-
	separation	Strategic	gic and In-flight	tical
	management		Tactical
	method

TABLE II

 II

	. NUMBER OF AIRCRAFT CONCURRENTLY IN-FLIGHT FOR EACH
	TRAFFIC DENSITY.
	Density	Average number of aircraft per scenario
	Very Low	1660
	Low	3340
	Medium	4990
	High	6650
	Very High	8290

ACKNOWLEDGMENT

This research has received funding from the SESAR Joint Undertaking under the European Union's Horizon 2020 research and innovation programme under grant agreement No 892928 (Metropolis II).


Yuri Goegebeur

Armelle Guillou

Jing Qin

Estimation of marginal excess moments for Weibull-type distributions

Keywords: Weibull-type distribution, bivariate extreme value statistics, tail copula, empirical process convergence

may come

Introduction

Modeling and prediction of rare events are topics that have given rise to an abundant literature in recent decades, especially to model climate, earthquakes and other environmental phenomena, such as precipitation, temperature and flooding, situations where we have to face significant risks that can have dramatic financial or human consequences. For these applications, the natural toolbox is the extreme value theory, where the pioneering paper is due to [START_REF] Fisher | Limiting forms of the frequency distribution in the largest particle size and smallest member of a sample[END_REF] who formulated the three types of limiting distributions for the maximum of a random sample X 1 , . . . , X n correctly normalised. Later, [START_REF] Gnedenko | Sur la distribution limite du terme maximum d'une série aléatoire[END_REF] proposed a rigorous proof that the three types are the only possible limit distributions for univariate extremes and characterized the domains of attraction of these limit laws. In this paper, we will consider some estimation problems within a rich subclass of the max-domain of attraction, namely the Gumbel class. Although the distributions in the Gumbel class may have different types of tail behaviour, their commonality is that they all have an extreme value index of zero. Therefore, it is impossible to differentiate them on the basis of this parameter alone. To solve this issue, we restrict our 1 study to Weibull-type distributions, which means that the sample at our disposal, X 1 , . . . , X n , will have a common distribution function F X which satisfies

F X pxq " 1 ´e´x 1{θ X pxq , x ą 0, ( 1
)
where θ is a positive parameter and X is a slowly varying function at infinity, i.e., a positive measurable function such that lim tÑ8 X ptxq X ptq " 1 for all x ą 0.

(2)

The parameter θ is called the Weibull-tail coefficient and it governs the tail behaviour, where larger values correspond to a slower decay of the survival distribution function F X :" 1 ´FX towards zero. Different values of this parameter allow the Weibull-type distributions to cover a large part of the Gumbel class, which implies that these distributions constitute a flexible family. Commonly used members of this family include the Weibull, normal, gamma and logistic distributions, to name only a few. As expected, the estimation of θ has led to a huge literature. We refer to [START_REF] Broniatowski | On the estimation of the Weibull tail coefficient[END_REF], [START_REF] Beirlant | The mean residual life function at great age: applications to tail estimation[END_REF], [START_REF] Girard | A Hill type estimator of the Weibull tail coefficient[END_REF], [START_REF] Gardes | Estimating extreme quantiles of Weibull tail distributions[END_REF], 2008, 2016), [START_REF] Diebolt | Bias-reduced estimators of the Weibull tail-coefficient[END_REF], [START_REF] Dierckx | A new estimation method for Weibulltype tails based on the mean excess function[END_REF], Goegebeur et al. (2010), [START_REF] Goegebeur | A weighted mean excess function approach to the estimation of Weibull-type tails[END_REF], [START_REF] Goegebeur | Robust conditional Weibull-type estimation[END_REF], de [START_REF] De Wet | Kernel regression with Weibulltype tails[END_REF] and the references therein.

In this paper we will study the estimation of the following bivariate risk measure

M EM β,p :" E " pX ´UX p1{pqq β `ˇY ą U Y p1{pq ı ,
where y `:" maxp0, yq, β ą 0, pX, Y q is a random vector with X having a continuous distribution function F X of Weibull-type and Y is a random variable with continuous distribution function F Y kept general, and U ' denotes the tail quantile function defined as U ' pxq " inftz : F ' pzq ě 1 ´1{xu, x ą 1, with ' being either X or Y . The role of the power parameter β is to allow the computation of both the mean and the second moment and thus the variance of the excess pX ´UX p1{pqq `given that the related variable Y is extreme. The risk measure M EM β,p will be called the marginal excess moment.

Similar risk measures have been already considered in the literature, but only in the Fréchet max-domain of attraction, i.e., in case where the extreme value index is strictly positive. See, e.g., Das andFasen-Hartmann (2018, 2019) who consider the estimation of a risk measure closely related to M EM β,p with β " 1, called the marginal mean excess, and [START_REF] Goegebeur | Nonparametric estimation of conditional marginal excess moments[END_REF] where the pair of interest pX, Y q is recorded together with a random covariate in R d . The novelty in the present paper lies in the max-domain of attraction considered, and the fact that no assumption on the distribution function of Y is imposed, whereas in, e.g., [START_REF] Goegebeur | Nonparametric estimation of conditional marginal excess moments[END_REF], the two variables X and Y are both in the same max-domain of attraction, namely the Fréchet. In the present paper we study thus the estimation of the marginal excess moment when the distribution function of X is of Weibull-type. These Weibull-type distributions have an upper tail that is lighter than that of distributions in the Fréchet max-domain of attraction, and find important applications in areas like environmental science, climate science, hydrology, engineering and medicine.

The remainder of the paper is organized as follows. In Section 2, we expand M EM β,p in case p Ó 0. This allows us to propose an estimator for M EM β,p under extrapolation, i.e., for p ă 1{n, where n is the size of the sample on which the estimation is based. This estimator also depends on a nonparametric estimator for M EM β,k{n introduced in Section 3, where k is an intermediate sequence, i.e., a sequence such that k " kpnq Ñ 8 with k{n Ñ 0 as n Ñ 8. The weak convergence of this latter estimator properly normalised is also established in Section 3.

A similar result for the estimator for M EM β,p under extrapolation is proved in Section 4. We examine the finite sample properties of the estimator with a simulation experiment in Section 5, while in Section 6 we illustrate how the proposed method can be used for wave height estimation at extreme wind speeds. Some auxiliary results are given in Section 7, whereas the proofs of the main results are postponed to Section 8.

Construction of the estimator

To motivate the construction of our estimator for M EM β,p in case of extrapolation, i.e., when p ă 1{n, we first look at a theoretical expansion of it assuming p Ó 0. Since both the variables X and Y are involved in the conditional expectation of interest, a condition on the joint distribution of pX, Y q is required. This condition describes the right-hand upper tail dependence of pX, Y q and is formulated as follows, after denoting R t px, yq :" tPpF X pXq ď x{t, F Y pY q ď y{tq:

Assumption pRq For all px, yq P r0, 8s 2 ztp8, 8qu we have that lim tÑ8 R t px, yq ": Rpx, yq exists.

Note that the convergence in Assumption pRq is in fact uniform for px, yq P r0, T s 2 for any T ą 0, due to the continuity of R and the monotonicity of R t .

Let RV ψ denote the class of regularly varying functions at infinity with index ψ P R, i.e., positive measurable functions f satisfying f ptxq{f ptq Ñ x ψ , as t Ñ 8, for all x ą 0. Denote Hpxq " ´log F X pxq, and hence F X pxq " e ´Hpxq . In the case where F X is of Weibull-type we have then H P RV 1{θ . Note that if H is differentiable with derivative H 1 then H 1 " h where h is the hazard rate function.

Lemma 2.1 Assume F X satisfies (1), H is differentiable with derivative h P RV 1{θ´1 . If Assumption pRq holds, then, as p Ó 0, we have

M EM β,p " a X ´1 p ¯ıβ ÝÑ ż 8 0 R `e´z , 1 ˘dz β ,
where a X ptq :" 1{hpU X ptqq.

Considering now an intermediate sequence k, Lemma 2.1 yields the approximation

M EM β,p { " a X ´1 p ¯ıβ M EM β,k{n { " a X `n k ˘‰β " 1,
from which we deduce that, with h denoting the slowly varying function associated with h,

M EM β,p " ˜aX p 1 p q a X p n k q ¸β M EM β,k{n " ˜UX p 1 p q U X p n k q ¸βp1´1{θq ˜ h pU X p n k qq h pU X p 1 p qq ¸β M EM β,k{n .
Note that (1) is equivalent to assume that U X satisfies

U X pxq " plog xq θ U plog xq, (3)
where U is also a slowly varying function at infinity. Thus, we deduce the following estimator for M EM β,p { M EM β,p :"

ˆlogp1{pq logpn{kq ˙βp p θ k ´1q M EM β,k{n , (4)
where p θ k is an estimator of θ and M EM β,k{n an intermediate estimator of M EM β,k{n . We refer to Theorem 4.2 for the weak convergence of this estimator, after proper normalisation, which is the main result of this paper.

Estimation of M EM β,k{n

Assume we have at our disposal a sample pX 1 , Y 1 q, . . . , pX n , Y n q of independent copies of the random vector pX, Y q. Let X i,n , i " 1, . . . , n, and Y i,n , i " 1, . . . , n, denote the order statistics pertaining to the X and Y sample, respectively. We define

M EM β,k{n :" 1 k n ÿ i"1 rX i ´Xn´k,n s β `1l tY i ąY n´k,n u ,
and introduce

M EM β,k{n px, yq :" 1 k n ÿ i"1 " X i ´UX ´n k ¯´x a X ´n k ¯ıβ `1l tY i ąU Y p n ky qu .
Then, M EM β,k{n can be rewritten as

M EM β,k{n " M EM β,k{n pp u n,k , p e n,Y q , a.s.,
where p u n,k :" pX n´k,n ´UX pn{kqq{a X pn{kq and p e n,Y :" n k F Y pY n´k,n q.

To study the main asymptotic behaviour of M EM β,k{n , we need to strengthen Assumption pRq by specifying the uniform rate of convergence of R t px, yq towards its limit Rpx, yq as t Ñ 8.

This assumption can be expressed as follows.

Assumption pSq There exist κ ą 0 and τ ă 0 such that

sup xą0,yPr 1 2 ,2s |R t px, yq ´Rpx, yq| x κ ^1 " O pt τ q .
This assumption can be viewed as a second-order condition in the bivariate framework. It excludes the case Rpx, yq " 0, @x, y ě 0, corresponding to a pair pX, Y q asymptotically upper tail independent.

Theorem 3.1 Assume F X satisfies (1), H is differentiable with derivative h which is ultimately monotone, and h is differentiable with derivative h 1 satisfying xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8. Assume also pSq and Rpy, 1q continuously differentiable in y. Then, if k Ñ 8 as n Ñ 8 such that k{n Ñ 0, ? ka X pn{kq{U X pn{kq Ñ 0 and ? kpn{kq τ Ñ 0, we have

? k ˜M EM β,k{n M EM β,k{n ´1¸d ÝÑ Θ where Θ :" 1 ş 8 0 Rpe ´z , 1qdz β "ż 8 0 W R `e´v , 1 ˘dv β ´WR p1, 8qG 1 p1q ´WR p8, 1q " Gp1q ´G1 p1q ‰ * ,
with W R a zero centered Gaussian process with covariance structure

E tW R px 1 , y 1 q W R px 2 , y 2 qu " R px 1 ^x2 , y 1 ^y2 q ,
and Gptq :" ş 8 0 R pe ´v t, 1q dv β . The variance of the limit in Theorem 3.1 is given by

VarpΘq " 1 `ş8 0 Rpe ´z , 1qdz β ˘2 "ż 8 0 Rpe ´v, 1qdv 2β `pG 1 p1qq 2 `pGp1q ´G1 p1qq 2 ´2Gp1q ż 8 0 Rpe ´v, 1qdv β `2G 1 p1qpGp1q ´G1 p1qqRp1, 1q  .
Note that the condition xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8 implies that h P RV 1{θ´1 (see Proposition 2.5 in [START_REF] Resnick | Heavy-tail phenomena, probabilistic and statistical modeling[END_REF].

4 Estimation of M EM β,p

In this section we study the asymptotic properties of the estimator { M EM β,p given in (4). As estimator for the Weibull-tail coefficient θ we use the estimator proposed by [START_REF] Girard | A Hill type estimator of the Weibull tail coefficient[END_REF], given by p θ k :"

1 k ř k i"1 plog X n´i`1,n ´log X n´k,n q 1 k ř k i"1 `log log n i ´log log n k ˘.
As a first step we obtain the limiting distribution of this estimator, properly normalised, in terms of a stochastic process that is closely related to the limiting process in Theorem 3.1.

Theorem 4.1 Assume F X satisfies (1), H is differentiable with derivative h which is ultimately monotone, and h is differentiable with derivative h 1 satisfying xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8. Then, if k Ñ 8 as n Ñ 8 such that ? k a X pn{kq U X pn{kq Ñ 0 and

? k U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq Ñ 0, we have ? k ´p θ k ´θ¯d ÝÑ Γ,
where

Γ :" θ "ż 8 0 W R `e´v , 8 ˘dv ´WR p1, 8q
* .

Note that this convergence result can be restated as ? k ´p θ k ´θ¯d ÝÑ N p0, θ 2 q, as obtained also by [START_REF] Girard | A Hill type estimator of the Weibull tail coefficient[END_REF] under similar conditions on k and n but through arguments based on properties of upper order statistics. In fact the condition ? ka X pn{kq{U X pn{kq Ñ 0 can be replaced by ? k{ logpn{kq Ñ 0, as used in [START_REF] Girard | A Hill type estimator of the Weibull tail coefficient[END_REF], and the condition ? k

U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq
Ñ 0 is closely related to the condition ? kbplog n{kq Ñ 0 in [START_REF] Girard | A Hill type estimator of the Weibull tail coefficient[END_REF]. Our contribution lies in obtaining the weak convergence result for p θ k using arguments based on empirical process convergence. As is clear from the arguments in Sections 7 and 8, the process that underlies p θ k is closely related to the process that underlies M EM β,p , leading to the joint convergence of p θ k and M EM β,p , after normalisation, which is needed for the main theorem given below.

We can now state the main result of the paper, namely the weak convergence of { M EM β,p , properly normalised.

Theorem 4.2 Assume F X satisfies (1), H is differentiable with derivative h which is ultimately monotone, and h is differentiable with derivative h 1 satisfying xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8. Assume also pSq and Rpy, 1q continuously differentiable in y.

Then, if k Ñ 8 as n Ñ 8 such that ? k a X p n k q U X p n k q Ñ 0, ? k `n k ˘τ Ñ 0 and ? k sup xěU X pn{kq x| 1 X pxq|
X pxq Ñ 0, we have for p satisfying p ď k n such that log 1{p log n{k Ñ λ P r1, 8q that

? k ˜{ M EM β,p M EM β,p ´1¸d Ñ β logpλqΓ `Θ.
The covariance between the random variables Γ and Θ is given by

CovpΓ, Θq " θ ş 8 0 Rpe ´z , 1qdz β "ż 8 0 Rpe ´v, 1qdv β`1 ´pGp1q ´G1 p1qq ż 8 0 Rpe ´v, 1qdv ´ż 8 0 Rpe ´v, 1qdv β `pGp1q ´G1 p1qqRp1, 1q  .
Note that in case λ " 1, which corresponds asymptotically with no extrapolation, we have that the limiting distribution for { M EM β,p corresponds with that of M EM β,p , which is what could be expected intuitively.

Simulation experiment

In this section a simulation study is implemented to investigate the finite sample performance of the proposed estimator. Two models are considered:

Model 1. The logistic copula model given by Cpu 1 , u 2 q " e ´tp´log u 1 q ξ `p´log u 2 q ξ u 1{ξ , u 1 , u 2 P r0, 1s, with ξ " 4. This model satisfies pSq with Rpx, yq " x `y ´px ξ `yξ q 1{ξ , τ " ´1 and κ " 1 ´ε for some small ε ą 0. For the marginal distribution of X we consider two settings, namely X " Weibullpθq with θ " 1 and θ " 1.5, while Y " Γpα, λq with pα, λq " p2, 1q.

Model 2. The copula function of p|Z 1 |, |Z 2 |q where pZ 1 , Z 2 q follows the bivariate standard Cauchy distribution with density function f given by

f pz 1 , z 2 q " 1 2π 1 p1 `z2 1 `z2 2 q 3{2 , pz 1 , z 2 q P R 2 .
This model was also considered in [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] and satisfies pSq with Rpx, yq " x `y áx 2 `y2 , τ " ´1 and κ " 2. For the marginal distributions we consider the settings • X " |N p1.2, 1q| and Y " U r0, 10s. The folded normal distribution is of Weibull-type with θ " 1{2.

• X " Extended Weibullpθ, ξq, i.e., X has distribution function satisfying 1 ´FX pxq " rpxqe ´x1{θ , x ą 0, where θ ą 1 and r P RV ξ with ξ P R. We take θ " 1.25 and rpxq " x ´1{5 , so ξ " ´0.2.

As for Y we take Y " Burrp1, 1, 1q, where the distribution function of the Burrpζ, λ, φq distribution is given by

F Y pyq " 1 ´ˆζ ζ `yφ ˙λ , y ą 0,
and ζ, λ, φ ą 0.

In the considered models, the distribution of X is always of Weibull-type, though with different Weibull-tail index, namely θ P t0.5, 1, 1.25, 1.5u, while we consider models with different forms of tail heaviness for Y . The Γpα, λq distribution is in the max-domain of attraction of the Gumbel distribution, the uniform distribution belongs to the max-domain of attraction of the (extreme-value) Weibull distribution and the Burr distribution is in that of the Fréchet distribution.

From each distribution we generate 200 samples with sample sizes n " 500 and n " 2000, and we estimate M EM β,p for p P t1{n, 1{p5nqu, and β P t1, 2u. In Figure 1 we show for Model 1 with X " Weibullp1q the boxplots of { M EM β,p {M EM β,p , computed over the 200 replications, as a function of k for n " 500 (left) and n " 2000 (right), and, from top to bottom, pβ, pq " p1, 1{nq, p1, 1{p5nqq, p2, 1{nq and p2, 1{p5nqq. Figures 2 till 4 are constructed in an analogous way for the other distributions. From the simulations we observe the following:

• For all the distributions considered the estimation is very good for k small compared to n, with boxplots nicely centered at the value one. This is in agreement with the theoretical condition k{n Ñ 0. When k increases (for fixed n) then at some point, which depends on the model, an estimation bias occurs. On the other hand, increasing k leads to less variable estimates.

• The estimation results improve with increasing n, as expected.

• Larger values of θ lead to more variable estimates. This can be expected as θ determines the upper tail heaviness of the Weibull-type model.

• For fixed n the variability of the estimates increases for smaller values of p, as smaller p lead to more extreme extrapolations.

• Also, for fixed n, the variability increases with β, so higher order moments are more difficult to estimate than those of lower order.

Wind speed and wave height data

Accurate information on wave climatology is important for many coastal and ocean engineering applications, in particular for the design of marine and coastal infrastructures like offshore wind farms, drilling platforms, sea walls, harbours and ships. As a measure for the severity of the sea state, the significant wave height is one of the most important wave parameters [START_REF] Vanem | A truncated, translated Weibull distribution for shallow water sea states[END_REF]. Consequently, a lot of attention has been devoted to the statistical modelling of this variable, and commonly used distributions are the Weibull, Rayleigh, normal, gamma and logistic distributions, see, e.g., [START_REF] Ferreira | Modelling the long-term distribution of significant wave height with the Beta and Gamma models[END_REF], [START_REF] Athanassoulis | Probabilistic description of metocean parameters by means of kernel density models 1. Theoretical backgrounds and first results[END_REF], and [START_REF] Shariff | Modelling significant wave height data of north sea: Rayleigh vs Weibull distributions[END_REF], among others. However, there are large model uncertainties in fitting parametric models to significant wave height data [START_REF] Vanem | A truncated, translated Weibull distribution for shallow water sea states[END_REF], and hence semi-parametric models like the Weibull-type model considered in the present paper can be a valuable alternative, as they provide a unifying framework that contains many distributions with similar distribution tails. Moreover, for many applications the joint distribution of several metocean variables, like wave height and wind speed, is of interest, e.g., [START_REF] Mittendorf | Joint description methods of wind and waves for the design of offshore wind turbines[END_REF]. We illustrate our developed method and estimate the excess of the significant wave height when wind speed is above a high quantile of its distribution.

The dataset considered here is collected at a surface buoy of the Ocean Observatories Initiative (OOI) Coastal Endurance Array, Oregon Line, and publicly available at https://dataexplorer.oceanobservatories.org/#ooi/array/CE/subsite/CE04OSSM/node/ CE04OSSM-SB/data. The dataset contains hourly measurements of, among others, air and sea surface temperature, wind speed, significant wave height and wave period from 2015, April 8 until 2022, September 8. For our analysis we consider a dataset of weekly measurements of significant wave height pXq, in meters, and wind speed pY q, in meters per second. This gives n " 210. We use weekly measurements in our analysis in order to reduce the temporal dependence between the observations. In Figure 5 we show the scatterplot of significant wave height versus wind speed, which indicates a positive association between the two variables, as expected. To investigate the assumption of an underlying Weibull-type distribution for X we construct a Weibull quantile-quantile plot, for which the coordinates are given by plogp´logp1 ´i{pn `1qqq, log X i,n q, i " 1, . . . , n, see Figure 6, left panel. This quantile-quantile plot becomes linear in the largest observations, supporting the assumption of a Weibull-type distribution for X; see [START_REF] Goegebeur | Goodness-of-fit testing for Weibull-type behavior[END_REF] for a discussion of Weibull quantile plots. The plot of p θ k as a function of k shown in Figure 6, right panel, is stable for k in the range 50 till 80 and indicates a θ estimate of approximately 0.4. Finally, we show { M EM 1,p and { M EM 2,p with p " 1{n (black line) and p " 1{p5nq (red dashed line) in Figure 7, left and right panel, respectively. As is clear from (4), for p θ k less than one, which corresponds with a 'light-tailed' Weibull-type distribution, { M EM β,p increases with p. The opposite effect will be observed for situations where p θ k exceeds one, so for 'heavy-tailed' Weibull-type distributions. Following the theoretical results, and as also observed in the simulation experiment, k should be taken small compared to n. However, for small k the estimates show quite some variability, especially for β " 2. As estimate for M EM β,p we report the median of { M EM β,p computed for k-values up to 50. This gives 0.369 for pβ, pq " p1, 1{nq, 0.314 for pβ, pq " p1, 1{p5nqq, 0.573 for pβ, pq " p2, 1{nq and 0.417 pβ, pq " p2, 1{p5nqq. Estimation of bivariate risk measures like M EM β,p at extreme levels is practically clearly challenging, especially when the size of the dataset is small, as is the case here. Despite this, the method performs quite well on this dataset of wave heights and wind speeds.

Auxiliary results

Remark that M EM β,k{n px, yq

" β 1 k n ÿ i"1 ż 8 U X p n k q`x a X p n k q " u ´UX ´n k ¯´x a X ´n k ¯ıβ´1 1l tX i ąuu du 1l tX i ąU X p n k q`x a X p n k qu 1l tY i ąU Y p n ky qu " β ż 8 U X p n k q`x a X p n k q 1 k n ÿ i"1 1l tX i ąu,Y i ąU Y p n ky qu " u ´UX ´n k ¯´x a X ´n k ¯ıβ´1 du " β ż 8 0 1 k n ÿ i"1 1l tX i ąz`U X p n k q`x a X p n k q,Y i ąU Y p n ky qu z β´1 dz " " a X ´n k ¯ıβ ż 8 0 1 k n ÿ i"1 1l tX i ąU X p n k q`a X p n k qpv`xq,Y i ąU Y p n ky qu dv β " " a X ´n k ¯ıβ ż 8 0 1 k n ÿ i"1 1l tF X pX i qăF X pU X p n k q`a X p n k qpv`xqq,F Y pY i qă k n yu dv β a.s. " " a X ´n k ¯ıβ ż 8 0 T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯dv β ,
where

T n px, yq :" 1 k n ÿ i"1 1l tF X pX i qă k n x,F Y pY i qă k n yu .
The asymptotic behavior of T n px, yq, after proper normalisation, is described by the aforementioned W R process (see Theorem 3.1). We refer to Lemma 1 of [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF], which we include below for completeness. For convenient representation, the limiting processes in their Lemma 1 are defined on the same probability space via the Skorohod construction, but it should be kept in mind that they are only in distribution equal to the original processes.

Lemma

F X ´t `z hptq F X ptq ÝÑ e ´z , (5)
uniformly for z ě ´δ, for any δ ą 0. 1) and H is differentiable with derivative h P RV 1{θ´1 , then, for a sequence k such that k Ñ 8 with k{n Ñ 0, we have [START_REF] De Haan | Extreme value theory, an introduction[END_REF]. Now, for z ą 0, the convergence ( 5) is equivalent to the convergence in distribution of hptqpX ´tq given that X ą t towards the standard exponential distribution. The latter distribution being continuous, the convergence is in fact uniform for z ą 0. Further, since the limit in ( 5) is continuous and the left-hand side in ( 5) is monotone, the convergence is also uniform for z P r´δ, 0s. This achieves the proof of Lemma 7.2.

! ? k " T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y Rn{k ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯ı ´WR ´e´pv`xq , y ¯) dv β ˇˇP ÝÑ 0. Lemma 7.3 If F X satisfies (
? k X n´k,n ´UX p n k q a X p n k q d ÝÑ W R p1,
Proof of Proposition 7.1. We have

ˇˇˇż 8 0 ! ? k " T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y Rn{k ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯ı ´WR ´e´pv`xq , y ¯) dv β ˇď ˇˇˇż 8 0 ! ? k " T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y Rn{k ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯ı ´WR ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯) dv β ˇż 8 0 ! W R ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯´W R ´e´pv`xq , y ¯) dv β ˇˇ"
: Q 1,n px, yq `Q2,n px, yq.

We will study the two terms separately. Concerning Q 1,n px, yq, according to Lemma 7.2, there exists some T ą 0 such that for n large and x ě ´δ, we have

Q 1,n px, yq ď sup 0ăy 1 ďT,yPr 1 2 ,2s
ˇˇ?krT n py 1 , yq ´Rn{k py 1 , yqs ´WR py 1 , yq ˇy

η 1 ˆż 8 0 ˜F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘¸η dv β .
The supremum term is negligeable a.s. according to Lemma 7.1. We need now to prove that the integral is bounded. We have, by the mean value theorem with ξ " ξpvq a value between zero and one,

ż 8 0 ˜F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘¸η dv β " ż δ 0 e ´η h p U X p n k q `aX p n k q ξpv´δq q h p U X p n k qq pv´δq dv β `ż 8 δ e ´η h p U X p n k q `aX p n k q ξpv´δq q h p U X p n k qq pv´δq dv β ": I 1,n `I2,n .
We start with I 1,n . By the use of the Potter bound (see Proposition B.1.9.5 in de Haan and [START_REF] De Haan | Extreme value theory, an introduction[END_REF], for δ 1 , δ 2 ą 0 and n large we have

I 1,n ď ż δ 0 e ´ηp1`δ 1 q ˜1`a X p n k q U X p n k q ξpv´δq ¸1{θ´1´δ 2 pv´δq dv β .
In case θ ă 1, take 0 ă δ 2 ă 1{θ ´1, and obtain, for n large, the bound

I 1,n ď ż δ 0 e ´ηp1`δ 1 qpv´δq dv β .
In case θ ě 1 we have, for 0 ă ε 1 ă 1{δ and n large

I 1,n ď ż δ 0 e ´ηp1`δ 1 qp1´ε 1 δq 1{θ´1´δ 2 pv´δq dv β .
In both cases the integrals are finite. Now consider I 2,n . Again by the Potter bound, for 0 ă δ 1 ă 1, δ 2 ą 0 and n large

I 2,n ď ż 8 δ e ´ηp1´δ 1 q ˜1`a X p n k q U X p n k q ξpv´δq ¸1{θ´1´δ 2 pv´δq dv β .
In case θ ă 1, take 0 ă δ 2 ă 1{θ ´1, and obtain, for n large, the bound

I 2,n ď ż 8 δ e ´ηp1´δ 1 qpv´δq dv β .
In case θ ě 1, for ε 1 ą 0 and n large

I 2,n ď ż 8 δ e ´ηp1´δ 1 qp1`ε 1 pv´δqq 1{θ´1´δ 2 pv´δq dv β .
Now take v ˚ą δ and 0 ă δ 2 ă 1{θ, and obtain

I 2,n ď ż v δ dv β `β ż 8 v ˚e´ηp1´δ 1 q ´ε1 `1 v ˚´δ ¯1{θ´1´δ 2 pv´δq 1{θ´δ 2 v β´1 dv.
Again, in both cases all integrals are finite.

Overall, we have then sup xPr´δ,δs,yPr 1 2 ,2s

Q 1,n px, yq " o P p1q.

We move now to the study of Q 2,n px, yq for which we use the following decomposition for T ą 0 where Γp., .q denotes the incomplete Gamma function, namely Γps, zq :" ş 8 z v s´1 e ´v dv. Using Lemma 2 in [START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] and by choosing T large enough, we can make the upper bound arbitrary small. Similarly, now using Lemma 7.2 we have for n large,

Q 2,n px, yq ď ż T 0 ˇˇW R ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯´W R ´e´pv`xq , y ¯ˇˇd v β `ż 8 T ˇˇW R ´e´pv`xq , y ¯ˇˇd v β `ż 8 T ˇˇW R ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, y ¯ˇˇd v β ": Q p1q 2,
P ˜sup xPr´δ,δs,yPr1{2,2s Q p3q 2,n px, yq ą ε{4 ḑ P ˜sup 0ăzďe δ `c,0ăyă8 |W R pz, yq| z η ż 8 T " n k F X ´UX ´n k ¯`a X ´n k ¯pv ´δq ¯ıη dv β ą ε{4 ¸.
We study the integral from the above display. Without loss of generality we take already T ě δ`ι for some small ι ą 0. The analysis is similar to that of I 2,n above. We have, for 0 ă δ 1 ă 1, δ 2 ą 0 and n large

ż 8 T « F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘ff η v β´1 dv ď ż 8 T e ´ηp1´δ 1 q ˜1`a X p n k q U X p n k q ξpv´δq ¸1{θ´1´δ 2 pv´δq v β´1 dv.
In case θ ă 1 we take 0 ă δ 2 ă 1{θ ´1, and obtain, for n large,

ż 8 T « F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘ff η v β´1 dv ď e ηp1´δ 1 qδ ż 8 T e ´ηp1´δ 1 qv v β´1 dv.
In case θ ě 1 take 0 ă δ 2 ă 1{θ. For ε 1 ą 0 and n large

ż 8 T « F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘ff η v β´1 dv ď ż 8 T e ´ηp1´δ 1 qp1`ε 1 pv´δqq 1{θ´1´δ 2 pv´δq v β´1 dv ď ż 8 T e ´ηp1´δ 1 qpε 1 `1 ι q 1{θ´1´δ 2 pv´δq 1{θ´δ 2 v β´1 dv.
In both cases, for n large the integrals can be made arbitrary small by taking T large.

So, for ε ą 0 there exists a n 1 such that for n ą n 1 we have We now turn to the first probability in the right-hand side of ( 7). With the T obtained above we have

P ˜sup xPr´δ,δs,yPr 1 2 ,2s Q p1q 2,n px, yq ą ε{2 ḑ P ˜sup xPr´δ,δ`T s,yPr 1 2 ,2s ˇˇW R ´n k F X ´UX ´n k ¯`a X ´n k ¯x¯, y ¯´W R `e´x , y ˘ˇˇą ε 2T β ¸.
Using Lemma 7.2 combined with the uniform continuity of W R on compact sets, we have for

n ą n 2 P ˜sup xPr´δ,δs,yPr 1 2 ,2s Q p1q 2,n px, yq ą ε{2 ¸ă ε{2.
Combined, for ε ą 0 we have for n ą maxpn 1 , n 2 q that P ˜sup xPr´δ,δs,yPr 1 2 ,2s

Q 2,n px, yq ą ε ¸ă ε, and hence Proposition 7.1 follows.

Proof of Lemma 7.3. First remark that

P ˆ?k X n´k,n ´UX p n k q a X p n k q ď x " P ˆXn´k,n ď U X ´n k ¯`x ? k a X ´n k ¯" 1 ´P ˆ?k " n k F X pX n´k,n q ´1ı ď ? k " n k F X ˆUX ´n k ¯`x ? k a X ´n k ¯˙´1 ˙. (8)
Now, under (1) and by using two applications of the mean value theorem, with ξ and ∆ being values between zero and one,

? k » - F X ´UX `n k ˘`x ? k a X `n k ˘F X `UX `n k ˘˘´1 fi fl " ? k » - -e ´hˆU X p n k q `ξx ? k a X p n k q ḣp U X p n k qq x ? k ´1fi ffi fl " ´e´∆ h ˆUX p n k q `ξx ? k a X p n k q ḣp U X p n k qq x ? k h ´UX `n k ˘`ξx ? k a X `n k ˘h `UX `n k ˘˘x Ñ ´x, (9)
since h P RV 1{θ´1 and by using the uniformity of the convergence in regular variation. Now, recall that F X pX n´k,n q d " U k`1,n where U k`1,n is the pk `1q´th smallest value in a sample of size n from a Uniform distribution on r0, 1s. Hence n k F X pX n´k,n q P ÝÑ 1. Additionally, we have

? k " n k F X pX n´k,n q ´1ı " ´?k " T n ´n k F X pX n´k,n q , 8 ¯´n k F X pX n´k,n q ı a.s.,
which yields by Lemma 7.1

? k " n k F X pX n´k,n q ´1ı d ÝÑ ´WR p1, 8q. (10)
Combining ( 8), ( 9) and ( 10), Lemma 7.3 follows.

Proof of Lemma 7.4. First we use the Lipschitz property of the function R to obtain

? k ˇˇˇż 8 0 " R ´n k F X ´UX ´n k ¯`a X ´n k ¯px `vq ¯, y ¯´R ´e´px`vq , y ¯ı dv β ˇˇď ? k ż 8 0 ˇˇn k F X ´UX ´n k ¯`a X ´n k ¯px `vq ¯´e ´px`vq ˇˇdv β .
Then, by the differentiability of H and two applications of the mean value theorem

? k ż 8 0 ˇˇˇˇF X `UX `n k ˘`a X `n k ˘px `vq F X `UX `n k ˘˘´e ´px`vq ˇˇˇˇd v β " ? k ż 8 0 ˇˇˇˇˇe ´hp U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq px`vq ´e´px`vq ˇˇˇˇˇd v β " ? k ż 8 0 e ´px`vq ˇˇˇˇˇˇe ´« h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff px`vq ´1ˇˇˇˇˇˇˇd v β " ? k ż 8 0 e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇ| x `v|dv β " ? k ż 8 0 e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇ| x `v|dv β 1l txě0u `» - - ? k ż ´x 0 e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇ| x `v|dv β `?k ż 8 ´x e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇ| x `v|dv β fi ffi fl 1l txă0u ": T 1 1l txě0u `rT 2,1 `T2,2 s1l txă0u ,
where ξ " ξpx, vq and ∆ " ∆px, vq are values between zero and one.

We bound the above integrals in turn, by constants that do not depend on x. We start with T 1 .

For the exponential function appearing in the integral we have the following: piq h is ultimately increasing (θ ď 1), then e

´#1`∆

« h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ď e ´px`vq .
piiq h is ultimately decreasing (θ ě 1), then for 0 ă δ 1 ă 1, δ 2 , ε 1 ą 0 and n large we have

e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ď e ´hp U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq px`vq ď e ´p1´δ 1 q ˜1`a X p n k q U X p n k q ξpx`vq ¸1{θ´1´δ 2 px`vq ď e ´p1´δ 1 qp1`ε 1 px`vqq 1{θ´1´δ 2 px`vq ,
where line 2 follows from the use of the Potter bound.

For the absolute difference in the integral we proceed as follows. We use the differentiability of h and obtain, with ξ " ξpx, vq a value between zero and one,

ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇď ˇˇh 1 `UX `n k ˘`a X `n k ˘ξpx `vq ˘ȟ `UX `n k ˘˘a X ´n k ¯px `vq " pU X `n k ˘`a X `n k ˘ξpx `vqq|h 1 `UX `n k ˘`a X `n k ˘ξpx `vq ˘| h `UX `n k ˘`a X `n k ˘ξpx `vq ˘h `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘ȃ X `n k ȖX `n k ˘`a X `n k ˘ξpx `vq px `vq.
Again we distinguish the case where h is ultimately increasing or decreasing. piq h is ultimately increasing. We use the fact that xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8 and the Potter bound to obtain, for δ 3 , ε 2 ą 0, and n large

ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇď C ˜1 `aX `n k ȖX `n k ˘ξpx `vq ¸1{θ´1`δ 3 a X `n k ȖX `n k ˘px `vq ď Cp1 `ε2 px `vqq 1{θ´1`δ 3 a X `n k ȖX `n k ˘px `vq,
where C is a positive constant. piiq h is ultimately decreasing. For n large

ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇď C a X `n k ȖX `n k ˘px `vq,
for some positive constant C.

We now return to T 1 . piq h is ultimately increasing. We have, for n large, and v ˚ą 0,

T 1 ď C ? k a X `n k ȖX `n k ˘ż 8 0 e ´px`vq p1 `ε2 px `vqq 1{θ´1`δ 3 px `vq 2 dv β ď C ? k a X `n k ȖX `n k ˘ż 8 0 e ´v p1 `ε2 pδ `vqq 1{θ´1`δ 3 pδ `vq 2 dv β ď C ? k a X `n k ȖX `n k ˘«ż v 0 e ´v p1 `ε2 pδ `vqq 1{θ´1`δ 3 pδ `vq 2 dv β `ż 8 v ˚e´v p1 `ε2 pδ `vqq 1{θ´1`δ 3 pδ `vq 2 dv β  .
The first integral in the right-hand side can clearly be bounded by a constant K not depending on x. Hence

T 1 ď C ? k a X `n k ȖX `n k ˘«K `β ż 8 v ˚e´v ˆε2 `1 `ε2 δ v ˙1{θ´1`δ 3 ˆ1 `δ v ˙2 v β`1{θ`δ 3 dv ff ď C ? k a X `n k ȖX `n k ˘«K `β ˆε2 `1 `ε2 δ v ˚˙1{θ´1`δ 3 ˆ1 `δ v ˚˙2 ż 8 v ˚e´v v β`1{θ`δ 3 dv ff .
So we can conclude that there is a constant K not depending on x such that for n large

T 1 ď ? k a X `n k ȖX `n k ˘K.
piiq h is ultimately decreasing. Then, for 0 ă δ 2 ă 1{θ, v ˚ą 0, and n large, we have,

T 1 ď C ? k a X `n k ȖX `n k ˘ż 8 0 e ´p1´δ 1 qp1`ε 1 px`vqq 1{θ´1´δ 2 px`vq px `vq 2 dv β ď C ? k a X `n k ȖX `n k ˘«ż v 0 e ´p1´δ 1 qp1`ε 1 pδ`vqq 1{θ´1´δ 2 v pδ `vq 2 dv β `ż 8 v ˚e´p1´δ 1 qp1`ε 1 pδ`vqq 1{θ´1´δ 2 v pδ `vq 2 dv β  ď C ? k a X `n k ȖX `n k ˘«K `β ˆ1 `δ v ˚˙2 ż 8 v ˚e´p1´δ 1 q ´ε1 `1`ε 1 δ v ˚¯1{θ´1´δ 2 v 1{θ´δ 2 v β`1 dv ff .
So also in this case there is a constant K not depending on x such that for n large

T 1 ď ? k a X `n k ȖX `n k ˘K.
We now turn to T 2,2 . In the integral T 2,2 we have that x `v ą 0 and hence the same bounds can be applied as for T 1 making the analysis very similar. For brevity we omit the details here.

Finally we analyse T 2,1 . Note that in this case ´δ ď x ď x `v ď 0. The exponential function in the integral can be bounded as follows: piq h is ultimately increasing, then

e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ď e ´px`vq .
piiq h is ultimately decreasing, then for δ 1 , δ 2 ą 0, 0 ă ε 1 ă 1{δ, and n large we have

e ´#1`∆ « h p U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq ´1ff+ px`vq ď e ´hp U X p n k q `aX p n k q ξpx`vq q h p U X p n k qq px`vq ď e ´p1`δ 1 q ˜1`a X p n k q U X p n k q ξpx`vq ¸1{θ´1´δ 2 px`vq ď e ´p1`δ 1 qp1`ε 1 px`vqq 1{θ´1´δ 2 px`vq ,
where line 2 follows from the use of the Potter bound.

As for the absolute difference in the integral we obtain the following: piq h is ultimately increasing, then for 0 ă ε 2 ă 1{δ and n large

ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇď C 1 1 `ε2 px `vq a X `n k ȖX `n k ˘|x `v|.
piiq h is ultimately decreasing, then for 0 ă ε 2 ă 1{δ, δ 3 ą 0, and n large

ˇˇˇˇh `UX `n k ˘`a X `n k ˘ξpx `vq h `UX `n k ˘˘´1 ˇˇˇˇď C p1 `ε2 px `vqq 1{θ´2´δ 3 a X `n k ȖX `n k ˘|x `v|.
These bounds are now used in the integral T 2,1 . piq h is ultimately increasing. We have, for n large,

T 2,1 ď C ? k a X `n k ȖX `n k ˘ż ´x 0 e ´px`vq 1 1 `ε2 px `vq px `vq 2 dv β ď C ? k a X `n k ȖX `n k ˘δ2 e δ 1 ´ε2 δ ż δ 0 e ´vdv β ď ? k a X `n k ȖX `n k ˘K,
where K is a constant not depending on x. piiq h is ultimately decreasing. Then, for n large,

T 2,1 ď C ? k a X `n k ȖX `n k ˘ż ´x 0 e ´p1`δ 1 qp1`ε 1 px`vqq 1{θ´1´δ 2 px`vq p1 `ε2 px `vqq 1{θ´2´δ 3 px `vq 2 dv β ď C ? k a X `n k ȖX `n k ˘δ2 p1 ´ε2 δq 1{θ´2´δ 3 e p1`δ 1 qp1´ε 1 δq 1{θ´1´δ 2 δ ż δ 0 e ´p1`δ 1 qp1´ε 1 δq 1{θ´1´δ 2 v dv β ď ? k a X `n k ȖX `n k ˘K,
where K does not depend on x.

Since all integrals can be bounded by a constant not depending on x the result of the lemma follows.

Proofs of the main results

Proof of Lemma 2.1. By integration by parts, we have

M EM β,p " E " pX ´UX p1{pqq β `ˇY ą U Y p1{pq ı " β ż 8 U X p1{pq px ´UX p1{pqq β´1 P pX ą x, Y ą U Y p1{pqq p dx " β ra X p1{pqs β ż 8 0 z β´1 P ´F X pXq ă p F X pU X p1{pq`a X p1{pqzq p , F Y pY q ă p p dz.
The aim now is to apply the dominated convergence theorem. First remark that, combining Lemma 7.2 with our Assumption pRq, we have for z ą 0

z β´1 P ´F X pXq ă p F X pU X p1{pq`a X p1{pqzq p , F Y pY q ă p p ÝÑ z β´1 R `e´z , 1 ˘, and
z β´1 P ´F X pXq ă p F X pU X p1{pq`a X p1{pqzq p , F Y pY q ă p p ď z β´1 min " F X pU X p1{pq `aX p1{pqzq p , 1 * ď z β´1 F X pU X p1{pq `aX p1{pqzq F X pU X p1{pqq " z β´1 e ´hpU X p1{pq`a X p1{pqξzq hpU X p1{pqq z ,
where the last step follows from the mean value theorem, and ξ " ξpzq is a value between zero and one. The latter term in the above display can be handled in the same way as the integrand of I 2,n in the proof of Proposition 7.1 with δ " 0, η " 1 and k{n replaced by p. Thus it can be bounded by an integrable function for p small, and the dominated convergence theorem applies, which establishes Lemma 2.1.

Proof of Theorem 3.1. We consider the following decomposition ´?k ´?k

? k ˜M EM β,k{n M EM β,k{n ´1" 1 M EM β,k{n ? k `M EM β,k{n ´M EM β,k{n " ra X pn{kqs β M EM β,k{n ? k "ż 8 0 T n ´n k F X ´Xn´k,n `aX ´n k ¯v¯, p e n,Y ¯dv β ´ż 8 0 n k P ˆF X pXq ă k n " n k F X ´UX ´n k ¯`a X ´n k ¯v¯ı , F Y pY q ă k n ˙dv β * " ra X pn{kqs β M EM β,k{n ? k "ż 8 0 T n ´n k F X ´Xn´k,n `aX ´n k ¯v¯, p e n,Y ¯dv β ´ż 8 0 R n{k ´n k F X ´UX ´n k ¯`a X ´n k ¯v¯, 1 ¯dv β * " ra X pn{kqs β M EM β,k{n "ż 8 0 W R `e´v , 1 ˘dv β `ż 8 0 " ? k ! T n ´n k F X ´Xn´k
ż 8 0 " R n{k ´n k F X ´UX ´n k ¯`a X ´n k ¯v¯, 1 ¯´R ´n k F X ´UX ´n k ¯`a X ´n k ¯v¯, 1 ¯ı dv β `?k ż 8 0 " R ´n k F X ´Xn´k
ż 8 0 " R ´n k F X ´UX ´n k ¯`a X ´n k ¯v¯, 1 ¯´R `e´v , 1 ˘ı dv β `?k ż 8 0 " R ´e´pv`p u n,k q , p e n,Y ¯´R `e´v , p e n,Y ˘ı dv β `?k ż 8 0 " R `e´v , p e n,Y ˘´R `e´v , 1 ˘‰ dv β `ż 8 0 " W R ´e´pv`p u n,k q , p e n,Y ¯´W R ´e´pv`p u n,k q , 1 ¯ı dv β `ż 8 0 " W R ´e´pv`p u n,k q , 1 ¯´W R `e´v , 1 ˘ı dv β * ": ra X pn{kqs β M EM β,k{n # ż 8 0 W R `e´v , 1 ˘dv β `9 ÿ i"1 Q i,n
+ .

We will study each term separately. We start with Q 1,n for which we apply Proposition 7.1 combined with Lemma 7.3 and the fact that p e n,Y " 1 `oP p1q, in order to obtain, with arbitrary large probability, and for n large, We want to apply the mean value theorem, and thus we need to compute the derivative of G. This can be done by applying the dominated convergence theorem and leads to

|Q 1,n | " ˇˇˇż 8 0 " ? k ! T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `p u n,k q ¯, p e n,Y Rn{k ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `p u n,k q ¯,
|R n{k pu, yq ´Rpu, yq| u κ ^1 ż 8 0 « F X `UX `n k ˘`a X `n k ˘pv `p u n,k q F X `UX `n k ˘˘ff κ ^1 dv β ď ? k sup uą0,yPr 1 2 ,2s |R n{k pu, yq ´Rpu, yq| u κ ^1 sup xPr´δ,δs ż 8 0 « F X `UX `n k ˘`a X `n k ˘pv `xq F X `UX `n k ˘˘ff κ ^1 dv β ď ? k sup uą0,yPr 1 2 ,2s |R n{k pu, yq ´Rpu, yq| u κ ^1 ˆsup xPr´δ,δs # ż 8 0 « F X `UX `n k ˘`a X `n k ˘pv `xq F X `UX `n k ˘˘ff κ ^1 dv β 1l txě0u `ż ´x 0 « F X `UX `n k ˘`a X `n k ˘pv `xq F X `UX `n k ˘˘ff κ ^1 dv β 1l txă0u `ż 8 ´x « F X `UX `n k ˘`a X `n k ˘pv `xq F X `UX `n k ˘˘ff κ ^1 dv β 1l txă0u + ď ? k sup uą0,yPr 1 2 ,2s |R n{k pu, yq ´Rpu, yq| u κ ^1 # 2 ż 8 0 « F X `UX `n k ˘`a X `n k ˘pv ´δq F X `
G 1 ptq " ż 8 0 e ´v R 1 `e´v t, 1 ˘dv β ,
with R 1 py, 1q the first derivative of Rpy, 1q with respect to y. Thus

Q 6,n " ? k " e ´p u n,k ´1ı G 1 pξ n q,
where ξ n is a random value between Concerning now Q 8,n , for T ą 0, we have

|Q 8,n | ď ż 8 T ˇˇW R ´e´pv`p u n,k q , p e n,Y ¯ˇˇd v β `ż 8 T ˇˇW R ´e´pv`p u n,k q , 1 ¯ˇˇd v β `ż T 0 ˇˇW R ´e´pv`p u n,k q , p e n,Y ¯´W R ´e´pv`p u n,k q , 1 ¯ˇˇd v β ": Q p1q 8,n `Qp2q 8,n `Qp3q 8,n .
Take ε ą 0 arbitrary. We have Finally, a similar treatment can be done on Q 9,n and shows that Q 9,n " o P p1q. Combining all these results with Lemma 2.1 achieves the proof of Theorem 3.1.

Proof of Theorem 4.1. Remember

p θ k " 1 k ř k i"1 plog X n´i`1,n ´log X n´k,n q 1 k ř k i"1 `log log n i ´log log n k ˘": N D .
Concerning the denominator D, we clearly have

D " 1 k k ÿ i"1 log ˆ1 `log k{i log n{k ˙" 1 k ř k i"1 log k{i log n{k `O ˆ1 plog n{kq 2 ˙,
since for all x ą 0 we have ´x2 2 ď logp1 `xq ´x ď x and 1 k ř k i"1 plog k{iq 2 Ñ 2 as k Ñ 8. Then, for all k ě 1, we can show that

ż k 1 log t dt ď k ÿ i"1 log i ď ż k 1 log t dt `log k, from which we deduce that 1 k k ÿ i"1 log k i " 1 `O ˆlog k k ˙.
This yields

D " 1 log n{k ˆ1 `O ˆlog k k ˙`O ˆ1 log n{k ˙˙. (11)
Define now

q θ k,n pxq " 1 k n ÿ i"1 " log X i ´log ´UX ´n k ¯`x a X ´n k ¯¯ı 1l tXiąUXp n k q`x a X p n k qu . Then N " q θ k,n pp u n,k q .
We have

q θ k,n pxq " 1 k n ÿ i"1 ż 8 U X p n k q`x a X p n k q 1 u 1l tX i ąuu du " ż 8 U X p n k q`x a X p n k q 1 k n ÿ i"1 1l tX i ąuu 1 u du " a X ´n k ¯ż 8 0 1 k n ÿ i"1 1l tX i ąU X p n k q`aXp n k q pv`xqu 1 U X `n k ˘`a X `n k ˘pv `xq dv " a X ´n k ¯ż 8 0 1 k n ÿ i"1 1l tF X pX i qă k n n k F X pUXp n k q`aXp n k q pv`xqqu 1 U X `n k ˘`a X `n k ˘pv `xq dv, a.s. " a X ´n k ¯ż 8 0 T n ´n k F X ´UX ´n k ¯`a X ´n k ¯pv `xq ¯, 8 ¯1 U X `n k ˘`a X `n k ˘pv `xq dv " a X `n k ȖX `n k ˘ż 8 0 T n `n k F X `UX `n k ˘`a X `n k ˘pv `xq ˘, 8 1
`aXp n k q U X p n k q pv `xq dv.

This implies that

? k ´p θ k ´θ" ? k ˜q θ k,n pp u n,k q D ´θ" ? k ¨aXp n k q U X p n k q D ż 8 0 T n `n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘, 8 1 `aXp n k q U X p n k q pv `p u n,k q dv ´θ‹ ' " a X p n k q U X p n k q D ? k ¨ż 8 0 T n `n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘, 8 1 `aXp n k q U X p n k q pv `p u n,k q dv ´1‹ ' `?k ¨aXp n k q U X p n k q D ´θ‹ ' ": a X p n k q U X p n k q D Q 1 `Q2 .
Concerning Q 2 , using (11), we have

Q 2 " ? k ˜aX `n k ȖX `n k ˘log n k ´θ¸`O ˜?k a X `n k ȖX `n k ˘log n{k log k k ¸`O ˜?k a X `n k ȖX `n k ˘¸.
Since H is differentiable, thus also X , which implies that

hpxq " 1 θ x 1 θ ´1 X pxq " 1 `θ x 1 X pxq X pxq  . ( 12
)
Consequently

a X `n k ȖX `n k ˘log n k " HpU X pn{kqq U X pn{kqhpU X pn{kqq " θ 1 `θ U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq , (13)
from which we deduce that

Q 2 " ´θ2 1 `θ U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq ? k U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq `O ¨1 1 `θ U X pn{kq 1 X pU X pn{kqq X pU X pn{kqq log k ? k '`O ˜?k a X `n k ȖX `n k ˘" op1q.
Now, concerning Q 1 , we have the decomposition

Q 1 " ż 8 0 W R
`e´v , 8 ˘dv `?k ´e´p u n,k ´1¯`ż 8 0 " W R ´e´pv`p u n,k q , 8 ¯´W R `e´v , 8 ˘ı dv

´?k a X `n k ȖX `n k ˘ż 8 0 v `p u n,k 1 `aXp n k q U X p n k q pv `p u n,k q e ´pv`p u n,k q dv ´aX `n k ȖX `n k ˘ż 8 0 v `p u n,k 1 `aXp n k q U X p n k q pv `p u n,k q W R ´e´pv`p u n,k q , 8 ¯dv `ż 8 0 W R `n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘, 8 ˘´W R `e´pv`p u n,k q , 8 1 `aXp n k q U X p n k q pv `p u n,k q dv `?k ż 8 0 n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘´e ´pv`p u n,k q `aXp n k q U X p n k q pv `p u n,k q dv `ż 8 0 ? k " T n `n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘, 8 ˘´n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘‰ 1 `aXp n k q U X p n k q pv `p u n,k q ´WR `n k F X `UX `n k ˘`a X `n k ˘pv `p u n,k q ˘, 8 1 `aXp n k q U X p n k q pv `p u n,k q dv ": Similarly to the study of Q 8,n in the proof of Theorem 3.1, we can show that Q p1q 1 " o P p1q.

ż 8 0 W R `e´v
Concerning Q p2q 1
The integral is clearly bounded in probability, and we assume that ? k a X p n k q U X p n k q Ñ 0. Thus

Q p2q 1 " o P p1q. Concerning Q p3q 1
With arbitrary large probability, for n large and 0 ă ε ă 1{δ, we have The supremum term is negligeable a.s. according to Lemma 7.1. To prove that the integral is bounded, we can follow the lines of proof of Proposition 7.1.

|Q p3q 1 | ď 1 1 ´ε δ a X `
Combining all these results, Theorem 4.1 follows.

Figure 1 :

 1 Figure 1: Model 1, X " Weibull(1), Y " Γp2, 1q. Boxplots of { M EM β,p {M EM β,p as a function of k for n " 500 (left) and n " 2000 (right) with pβ, pq " p1, 1{nq, p1, 1{p5nqq, p2, 1{nq and p2, 1{p5nqq (from top to bottom).

Figure 2 :

 2 Figure 2: Model 1, X " Weibull(1.5), Y " Γp2, 1q. Boxplots of { M EM β,p {M EM β,p as a function of k for n " 500 (left) and n " 2000 (right) with pβ, pq " p1, 1{nq, p1, 1{p5nqq, p2, 1{nq and p2, 1{p5nqq (from top to bottom).

Figure 3 :

 3 Figure 3: Model 2, X " |N p1.2, 1q|, Y " U r0, 10s. Boxplots of { M EM β,p {M EM β,p as a function of k for n " 500 (left) and n " 2000 (right) with pβ, pq " p1, 1{nq, p1, 1{p5nqq, p2, 1{nq and p2, 1{p5nqq (from top to bottom).

Figure 4 :

 4 Figure 4: Model 2, X " Extended Weibull(1.25,-0.2), Y " Burrp1, 1, 1q. Boxplots of { M EM β,p {M EM β,p as a function of k for n " 500 (left) and n " 2000 (right) with pβ, pq " p1, 1{nq, p1, 1{p5nqq, p2, 1{nq and p2, 1{p5nqq (from top to bottom).

Figure 5 :

 5 Figure 5: Wave height data. Scatterplot of significant wave height versus wind speed.

Figure 6 :

 6 Figure 6: Wave height data. Weibull quantile plot of significant wave height (left) and p θ k as a function of k for significant wave height (right).

Figure 7 :

 7 Figure 7: Wave height data. { M EM 1,p (left) and { M EM 2,p (right) with p " 1{n (black line) and p " 1{p5nq (red dashed line).

8 0

 8 Now, according to Lemma 7.2, we have for an arbitrary c ą 0 and n large, sup 0ăyďe δ `c ˇˇ?krT n py, 8q ´ys ´WR py, 8q ˇyη ż ˜F X `UX `n k ˘`a X `n k ˘pv ´δq F X `UX `n k ˘˘¸η dv.

  7.1 (Lemma 1 in[START_REF] Cai | Estimation of the marginal expected shortfall: the mean when a related variable is extreme[END_REF] Assume pRq holds and let k Ñ 8 as n Ñ 8 such that k{n Ñ 0. For any η P r0, 1{2q and T ą 0 we have, for n Ñ 8,

	sup x,yPp0,T s	ˇˇˇˇ? krT n px, yq ´Rn{k px, yqs ´WR px, yq x η	ˇˇˇˇÑ 0 a.s. ,
		sup xPp0,T s	ˇˇˇˇ? krT n px, 8q ´xs ´WR px, 8q x η	ˇˇˇˇÑ 0 a.s. ,
		sup
		yPp0,T s

ˇˇˇˇ? krT n p8, yq ´ys ´WR p8, yq y η ˇˇˇˇÑ 0 a.s. . Lemma 7.2 Let F X satisfy (1), H be differentiable with derivative h P RV 1{θ´1 , then, as t Ñ 8

  8q. Assume F X satisfies (1), H is differentiable with derivative h which is ultimately monotone, and h is differentiable with derivative h 1 satisfying xh 1 pxq{hpxq Ñ 1{θ ´1 as x Ñ 8. Assume also pRq. Then, if k Ñ 8 as n Ñ 8 such that k{n Ñ 0 and ? ka X pn{kq{U X pn{kq Ñ 0, we have, for any δ ą 0, ¯´R ´e´px`vq , y ¯ı dv β ˇˇˇÑ 0.Proof of Lemma 7.2. By the mean value theorem, with ξ a value between zero and one, pointwise, as t Ñ 8, since h P RV 1{θ´1 , and by the uniformity of the convergence in regular variation (see Theorem B.1.4 in de Haan and

	Lemma 7.4 sup xPr´δ,δs,yPr 1 2 ,2s	?	k ˇˇˇż 0 8	" R	´n k	F X ´UX ¯px `vq ¯, y F X ´t `z hptq ´n k ¯`a X ´n k F X ptq " e ´hˆt `ξz hptq ḣptq z ÝÑ e ´z ,

  This yields Q 1,n " o P p1q. Now, concerning Q 2,n , we have, with arbitrary large probability, and for n large,

													¯)
													p e n,Y
				´WR	´e´pv`p u n,k q , p e n,Y ¯ı dv β ˇď
				sup xPr´δ,δs,yPr 1 2 ,2s	ˇˇˇż 0 8	" ?	k	! T n	´n k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯, y	Rn{k
					´n k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯, y ¯) ´WR	´e´pv`xq , y ¯ı dv β ˇˇ.
	|Q 2,n | ď	?	k	sup
				uą0,yPr 1 2 ,2s

  by Assumption pSq, the integrals being bounded similarly as the study of Q 1,n px, yq in the proof of Proposition 7.1. A similar treatment forQ 3,n yields Q 3,n " O ´?k `n k ˘τ ¯.The two next terms, Q 4,n and Q 5,n , can be handled by Lemma 7.4 since for i " 4 or 5, we have, with arbitrary large probability, and for n large,

	Now, from the homogeneity of the R´function, we have
		Q 6,n "	?	k	ż 8 0	« e n,Y R p	˜e´pv`p u n,k q p e n,Y	, 1 ¸´p e n,Y R	ˆe´v p e n,Y	, 1 ˙ff dv β
			" p e n,Y	? k	"	G	ˆe´p u n,k p e n,Y	˙´G	ˆ1 p e n,Y	˙ .
												UX `n k	˘˘ff κ	dv β `δβ	+
	" O P ´?k	´n k	¯τ ¯,
	|Q i,n | ď	sup xPr´δ,δs,yPr 1 2 ,2s	?	k ˇˇˇż 0 8	! R	´n k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯, y ¯´R ´e´pv`xq , y ¯) dv β ˇˇ"
		o P p1q.

  Hence, for n ą maxpn 1 , n 2 q P p|Q 8,n | ą εq ď 4r ε, and thus Q 8,n " o P p1q.

	We have for η P p0, 1{2q
		P	´Qp1q 8,n ą	ε 4	¯ď P ˜sup 0ăy 1 ďe δ´T , 1 2 ďy 2 ď2	|W R py 1 , y 2 q| y η 1	sup |x|ďδ	T ż 8	e ´ηpv`xq dv β ą	4 ε	P
							pp e n,Y R r1{2, 2sq `P p|p u n,k | ą δq
							ď P ˜sup 0ăy 1 ďe δ , 1 2 ďy 2 ď2	|W R py 1 , y 2 q| y η 1	ą	4 ε	β η β	Γpβ, η T q e ´ηδ	P
							pp e n,Y R r1{2, 2sq `P p|p u n,k | ą δq
	and the same bound can be also used for P	´Qp2q 8,n ą ε 4 ¯. Choose T such that the first probability
	in the right-hand side is at most r ε{2. Then, for n ą n 1
							P	´Qp1q 8,n ą	ε 4	¯`P	´Qp2q 8,n ą	ε 4	¯ď 3r ε.
	Now, concerning Q p3q 8,n we have
	P	´Qp3q 8,n ą	ε 2	¯ď P ˜sup
	By the uniform continuity of W R on compact sets and Lemma 7.3 combined with the fact that ? kpp e n,Y ´1q ´Qp3q 8,n ą ε 2 ¯ď r ε.
				P p|Q 8,n | ą εq ď P	´Qp1q 8,n ą	ε 4	¯`P	´Qp2q 8,n ą	ε 4	¯`P	´Qp3q 8,n ą	2 ε	¯.
											29

e ´pT `δq ďy 1 ďe δ ,|y 2 ´1|ďk ´1{4 |W R py 1 , y 2 q ´WR py 1 , 1q|

ż T 0 dv β ą ε 2 P ´|p e n,Y ´1| ą k ´1{4 ¯`P p|p u n,k | ą

δq ď P ˜sup e ´pT `δq ďy 1 ďe δ ,|y 2 ´1|ďk ´1{4 |W R py 1 , y 2 q ´WR py 1 , 1q| ą ε 2 T ´β P ´|p e n,Y ´1| ą k ´1{4 ¯`P p|p u n,k | ą δq . d ÝÑ ´WR p8, 1q, we have for n ą n 2 that P

  , 8 ˘dv ´WR p1, 8q `oP p1q `6 ÿ

	Q piq 1 ,
	i"1
	by Lemma 7.3.
	p1q Concerning Q 1

  Using Lemma 7.2, we can again follow the lines of proof of the term Q 2,n px, yq in the proof of Proposition 7.1 in order to show that Q p4q 1 " o P p1q.With arbitrary large probability, for n large and 0 ă ε ă 1{δ, we have With arbitrary large probability, for n large and 0 ă ε ă 1{δ, we have

	p5q Concerning Q 1
	|Q p5q 1 | ď	1 1 ´ε δ	sup xPr´δ,δs	? k	ż 8 0	ˇˇn k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯´e ´pv`xq ˇˇdv
			" o P p1q,
	by the proof of Lemma 7.4.
	p6q Concerning Q 1
	|Q p6q 1 |
	ď	1 1 ´ε δ	sup xPr´δ,δs	ż 8 0	! ?	k	" T n	´n k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯, 8 n
		k	F X ´UX	´n k	¯`a X	´n k	¯pv `xq ¯ı ´WR	´n k	F X ´UX	´n k	¯`a
														n k `n k	ȖX ˘sup 0ăyďe δ	|W R py, 8q| y η	ż 8 0	pv `δq e ´ηpv´δq dv
					" o P p1q.
	p4q Concerning Q 1

X ´n k ¯pv `xq ¯, 8 ¯) dv.

  Proof of Theorem 4.2. Consider the following decomposition by Lemma 7.4 and the study of Q 3,n in the proof of Theorem 3.1. A similar result can be obtained for M EM β,p {ra X p1{pqs β . Thus, we haveM EM β, k n {ra X pn{kqs β M EM β,p {ra X p1{pqs β ´1 " o ˆ1 ? k ˙.

	{ M EM β,p M EM β,p	´1 "	# loooooooooooomoooooooooooon ˆlog 1{p log n{k ˙βr p θ k ´θs +	# looooooomooooooon M EM β, k n M EM β, k n +	loooooooooooooooooomoooooooooooooooooon $ & % , . plog n{kq βp1´θq M EM β, k n -plog 1{pq βp1´θq M EM β,p	´1
							T 1							T 2	T 3
			"	# ˆlog 1{p log n{k	˙βr p θ k ´θs	´1+	T 2 T 3	`# M EM β, k n M EM β, k n	´1+	T 3	`$ & %	plog n{kq βp1´θq M EM β, k n plog 1{pq βp1´θq M EM β,p	. ´1, -
			"	" βr p θ k ´θs log	log 1{p log n{k	*	p1 `oP p1qq T 2 T 3	`# M EM β, k n n M EM β, k	´1+	T 3
				`# M EM β, k n M EM β,p {ra X p1{pqs β {ra X pn{kqs β	a X p1{pq ´1+ ˆaX pn{kq	log n{k ˙β ˆlog 1{p	˙βpθ´1q
				`ˆa X pn{kq a X p1{pq	˙β ˆlog 1{p log n{k	˙βpθ´1q	´1.
	We have
	M EM β, k n ra X pn{kqs β "	ż 8 0	n k	P ˆF X pXq ă	k n	" n k	F X ´UX	´n k	¯`a X	´n k	¯z¯ı	, F Y pY q ă	n k	˙dz β
		"	ż 8	R `e´z , 1 ˘dz β
			0
			`ż 8 0	" R n{k ´n k	F X ´UX	´n k	¯`a X	´n k	¯z¯,	1 ¯´R	´n k	F X ´UX	´n k	¯`a X	´n k	¯z¯,	1 ¯ı dz β
			`ż 8 0	" R	´n k	F X ´UX	´n k	¯`a X	´n k	¯z¯,	1 ¯´R `e´z , 1 ˘ı dz β
		"	ż 8 0	R `e´z , 1 ˘dz β `O ´´n k	k ¯τ ¯`o ˆ1 ?	"
			ż 8 0	k R `e´z , 1 ˘dz β `o ˆ1 ?	˙,

Acknowledgement

The research of Armelle Guillou was supported by the French National Research Agency under the grant ANR-19-CE40-0013-01/ExtremReg project and by the CNRS International Research Network MaDeF.

Now, using (12) and the fact that U plog xq " r X pU X pxqqs ´θ, we have ˆaX pn{kq a X p1{pq

Concerning T 4 , using the mean value theorem with u n P rU X pn{kq; U X p1{pqs, we have, for δ 1 , δ 2 ą 0 and n large,

Concerning T 5 , using again the mean value theorem, we have

where ω n is an intermediate value between

´1 and zero.

Combining Theorem 3.1 with Theorem 4.1, Theorem 4.2 follows.