Xeniya Pystina
email: xeniya.pystina@insa-lyon.fr

Lilia Gzara
email: lilia.gzara@insa-lyon.fr

Vincent Cheutet
email: vincent.cheutet@insa-lyon.fr

Aicha Sekhari
email: aicha.sekhari@univ-lyon2.fr

A conceptual framework elements for Digital Twin deployment in production systems domain

The Digital Twin (DT) as a virtual representation of physical assets aims to support transition to smart manufacturing and Industry 4.0. Today, DT remains challenging to implement on production systems because of its original technical complexity, various functions and purposes. Recent researches are focusing on Asset Administration Shell, a promising concept allowing a systemically defined and semantically described components to form distributed interoperable systems according to Industry 4.0 principles. The paper explores benefits of Asset Administration Shell and Model-Based Systems Engineering approaches to provide a conceptual framework of DT implementation in production systems.

I.

INTRODUCTION

Most companies are interested in the transition to Industry 4.0 (I4.0) by rationally distributing existing capacities and implementing new solutions in full compliance with established requirements [START_REF] Alcácer | Scanning the Industry 4.0: A Literature Review on Technologies for Manufacturing Systems[END_REF]. With the development of Internet of Things and advanced analytic tools, businesses may now use new optimized economic models based on easily accessible data from the field in order to gain in various parameters and in productivity in general. Concerning manufacturing sector and manufacturing systems in particular, the Digital Twin (DT) as an I4.0 representative is of big interest since 2003 both for researchers and practitioners [START_REF] Kritzinger | Digital Twin in manufacturing: A categorical literature review and classification[END_REF]. Defined by ISO 23247 series as a virtual representation of a physical entity, DT is characterized by level of convergence and real-time synchronization between virtual and physical systems. Constantly growing level of complexity of manufacturing systems requires more human capacities for control, optimization and decision-making [START_REF] Negri | A Review of the Roles of Digital Twin in CPS-based Production Systems[END_REF]. One of important DTs roles depending on its architecture is to accurately and reliably assist human in these processes, adding flexibility compared to conventional static manufacturing execution systems [START_REF] Semeraro | Digital twin paradigm: A systematic literature review[END_REF]. Given that, there is no unique DT for any physical system. The typical applications for manufacturing show the scope of DT's deployment. In this research, we focus on DT for production systems. We are addressing DT implementation methodology while considering the above-mentioned specificities.

The theoretical base, i.e. conceptualization of the term [START_REF] Semeraro | Digital twin paradigm: A systematic literature review[END_REF]- [START_REF] Lu | Digital Twin-driven smart manufacturing: Connotation, reference model, applications and research issues[END_REF], formed to date allows to delve into the detailed development of the implementation methodology and practical deployment of DTs on different levels of complexity [START_REF] Villalonga | A decision-making framework for dynamic scheduling of cyber-physical production systems based on digital twins[END_REF]. Following that, the arising generalized question is how to build DT according to physical system characteristics and digitalization objectives. To answer this question, a conceptual framework for DT implementation, based on various criteria, is needed.

The recent proposition made by [START_REF] Barth | Systematization of Digital Twins: Ontology and Conceptual Framework[END_REF] is focused on systematization of existing DT applications. The current gaps in DT, well-defined in [START_REF] Villalonga | A decision-making framework for dynamic scheduling of cyber-physical production systems based on digital twins[END_REF], cover theoretical and practical aspects of the DT deployment on the component, system and system of system levels. Among them, the question of -how to deploy complex DT for existing production system to assure control, optimization, planning and scheduling, virtual commissioning, predictive maintenance and related decision-making processes with the possibility for easy upgrade -remains partially unsolved despite numerous attempts. The DT abilities to accumulate and operate the time dependent knowledge alongside with failure resistance during operation are under development for every specific application. The similar issue highlighting the lack in genericity of the implementation approach has been already brought up in [START_REF] Da Cunha | Designing the Digital Twins of Reconfigurable Manufacturing Systems: application on a smart factory[END_REF] in application to reconfigurable manufacturing systems (RMS). Based on RAMI 4.0 reference architecture, which ensures interoperable distributed networks creating flexible machines, services and systems [START_REF] Schweichhart | RAMI 4.0 reference architectural model for Industrie 4.0[END_REF] and aims to establish a framework particularly for manufacturing domain, DT fits in currently on-going second digitalization phase of Industry 4.0 (I4.0) paradigm and can be further developed to become exhaustive virtual representation for production systems thanks to its similar to real world physical rules and properties [START_REF] Zhang | A reconfigurable modeling approach for digital twin-based manufacturing system[END_REF]. Despite the fact that multiple studies tend to understand DT as a multi-tool software platform, the DT's characteristics and properties, defined in [START_REF] Jones | Characterising the Digital Twin: A systematic literature review[END_REF]- [START_REF] Barricelli | A survey on digital twin: Definitions, characteristics, applications, and design implications[END_REF], reflect its modular polyvalent nature as a technology and a scientific concept. According to various reviews [START_REF] Kritzinger | Digital Twin in manufacturing: A categorical literature review and classification[END_REF], [START_REF] Negri | A Review of the Roles of Digital Twin in CPS-based Production Systems[END_REF], [START_REF] Barricelli | A survey on digital twin: Definitions, characteristics, applications, and design implications[END_REF], the use cases for DT in smart manufacturing and production systems aimed to improve one or several specific functions or attributes of production systems already constructed and operated. However, in order to explicitly fulfill the requirements of I4.0, it is necessary to represent a consistent whole for production system DTs (i.e. their components and functions), that is to say how pieces are fit together. For this, it is important to define a conceptual framework for DT implementation taking into account system dimension and model-based system representation including, for example, analytic and verification models.

The paper is organized as follows: Section 2 gives an overview on the problem statement and related works introducing main concepts to form the framework. The proposition of framework components based on previous part is given in Section 3. Section 4 introduces the application of the proposed approach to define a DT for an

II. PROBLEM STATEMENT

The goal of this section is to provide a state-of-the-art on DT for production systems and define relevant research areas for conceptual framework elements.

The following query "digital twin and (manufacturing system* OR production system*) and (conceptual framework OR framework)"and (*_manufacturing), applied to the search in Web of Science for English written articles dated from 2015 to 2022, resulted in 175 sources. Focusing on the connection between DT, cyberphysical production systems and model-based systems engineering, the following network of clusters was identified (cf. Fig. 1): Red -DT and Industry 4.0 generalized reviews; Blue -Industry 4.0 implementation case studies; Violet -Manufacturing case studies focusing on DT models for product design; Orange -DT Data aspects related studies; Brown -Decision-making processes for DT; Green -Industrial use cases for CPPS and smart manufacturing; Grey -DT related applications based on advanced analytics technologies; Pink -Simulation for DT related studies.

Then 43 articles including DT frameworks, models, architectures were investigated forming the dependencies between terms (cf. Fig. 2).

The results characterise DT from different sides: physical interrelations between production systems components, IIoT data exchange, data and information evaluation, digital modelling for production system virtual representation, simulation for production planning optimisation use cases and decision-making services using data analysis technologies. Then, it is possible to distinguish the following approaches for production systems:

1. RAMI 4.0 and ISA-95 based multi-layered frameworks including 5C architecture for cyberphysical production systems (CPPS) [START_REF] Zhang | Construction of intelligent integrated model framework for the workshop manufacturing system via digital twin[END_REF], [START_REF] Sundaram | Smart prognostics and health management (SPHM) in smart manufacturing: An interoperable framework[END_REF]; 2. ISO 23247 based conceptual architectures e.g. [START_REF] Eirinakis | Cognitive Digital Twins for Resilience in Production : A Conceptual Framework[END_REF]; 3. Modelling and simulation based frameworks [START_REF] Touckia | Digital twin framework for reconfigurable manufacturing systems (RMSs): design and simulation[END_REF]; 4. Multi-agent systems and service systems architectures [START_REF] Nie | A multi-agent and internet of things framework of digital twin for optimized manufacturing control[END_REF]; 5. MBSE for shop floor DT [START_REF] Liu | Construction method of shop-floor digital twin based on MBSE[END_REF], [START_REF] Mhenni | Heterogeneous Models Integration for Safety Critical Mechatronic Systems and Related Digital Twin Definition: Application to a Collaborative Workplace for Aircraft Assembly[END_REF]; 6. Conceptual frameworks with asset administration shell based architectures and interoperable frameworks [START_REF] Ye | An Industry 4.0 Asset Administration Shell-Enabled Digital Solution for Robot-Based Manufacturing Systems[END_REF]. Certain propositions [START_REF] Wu | Conceptual digital twin modeling based on an integrated five-dimensional framework and TRIZ function model[END_REF], [START_REF] Zhuang | The connotation of digital twin, and the construction and application method of shop-floor digital twin[END_REF] use bottom-up systemic approach to explain how DT can be implemented identifying the following dimensions: the physical entity, the virtual entity, the services module, the digital twin data module, and connection module. In [START_REF] Ponomarev | Main principals and issues of digital twin development for complex technological processes[END_REF] the DT is represented in bottom-up multi-layered architecture in correlation with an automated process control systems. The global understanding of DT as an information system also finds a basis in multi-layered architectures consisting for example: in [START_REF] Zhang | Application framework of digital twin-driven product smart manufacturing system: A case study of aeroengine blade manufacturing[END_REF] of system layer -information processing layer -physical layer and model layer; in [START_REF] Redelinghuys | A sixlayer architecture for the digital twin: a manufacturing case study implementation[END_REF] of six layers where the realization of the solution is technology oriented. The top-down methodology of building DT presented in [START_REF] Qamsane | A Methodology to Develop and Implement Digital Twin Solutions for Manufacturing Systems[END_REF] focuses on decisionmaking and the main roles of DT rather than technological aspects which provides a broad view on the specificities of DT requirements and industry context. Based on the multi-agent system framework for DT, in [START_REF] Nie | A multi-agent and internet of things framework of digital twin for optimized manufacturing control[END_REF], the example on system of system level implementation is provided. Authors defend the transformation of rigid architecture for production systems to multi-agent nets governed by DT service system. Currently, the structure level of shop floor prevails in scientific sources in manufacturing domain. The variety of approaches include systems engineering which indirectly relate the DT into an information system with corresponding technology [START_REF] Sundaram | Smart prognostics and health management (SPHM) in smart manufacturing: An interoperable framework[END_REF], infrastructure [START_REF] Stair | Principles of Information Systems: 13th Edition[END_REF], RAMI 4.0 that explains the relationship between users and assets on different functional levels [START_REF] Schweichhart | RAMI 4.0 reference architectural model for Industrie 4.0[END_REF]. In [START_REF] Liu | Construction method of shop-floor digital twin based on MBSE[END_REF] the approach resemble to top-down system engineering development methodology. This methodology is widely implemented for shop floor DT. However, it does not fully answer to the interoperability requirement for DT and define the global objective for the system. The DT system design based on DT metamodel is provided in [START_REF] Mhenni | Heterogeneous Models Integration for Safety Critical Mechatronic Systems and Related Digital Twin Definition: Application to a Collaborative Workplace for Aircraft Assembly[END_REF], where authors use MBSE for top-down physical system design and highlight the difficulty to represent all physical system knowledge in DT metamodel. Authors in [START_REF] Contreras | Developing of industry 4.0 applications[END_REF] conclude that different intelligent manufacturing applications can be developed using RAMI 4.0 Finally, a wide range of DT design methods form a rich theoretical base for system design and drive production system domain to a new stage of development. The bottom-up implementation approach has its limits when it comes to a complex asset such as production systems due to increasing amount of data from asset components. At the same time starting to define the asset from use case objective can be beneficial in cost for new development and renovation. The reasons for renovations are of great concern especially for the existing solutions. The cases related to modernizations cannot fully answer to DT requirements and therefore cannot be referred to DT. The cases that harmonize two approaches are limited and refer to asset and system level, whereas the use cases on system of system level are missing. The main contribution is that for DT application the proposed approach associates the MBSE methods bidirectionally from business to physical levels of RAMI 4.0.

III. PROPOSITION

The necessity to consider DT comprehensively from behavioral, informational and technical aspects can be supported by MBSE formalized methodology, digital models, as well as asset administration shell standardized specification and software. The proposition for implementation procedure is the following sequence: definition of the main objective (in the framework of RAMI 4.0 and domain standards), metamodel definition (relationship between main instances), definition of related services to support objective (including simulation, synchronization and visualization), then connection to the external DT. At the same time, the asset characteristics and its DT's functionalities will pass the verification and validation through the development methodology. Each component of the proposed approach is explained below.

A. Model-based system engineering approach

Systems engineering and its derivative MBSE is able to involve every aspect of the production system and pay attention to interfaces when two or many systems or their elements are working together using representing models. The vision to understand DT as a system of systems provides large opportunities in cross-domain and specific applications in manufacturing. The INCOSE System Engineering Handbook defines the "Systems of Systems" as "an interoperating collection of component systems that produce results unachievable by the individual systems alone". According to ISO/IEC 15288 "the perception and definition of a particular system, its architecture and its system elements depend on an observer's interests and responsibilities". Therefore, the DT inherits different functions and characteristics reflecting those from the systems it constitutes. The interoperability issue is at the core of I4.0 needs. The stated role of interoperability is to synthesise, connect and communicate software components, business processes and application solutions through a diversified heterogeneous and autonomous procedure [START_REF] Burns | A review of interoperability standards for industry 4.0[END_REF]. DT through MBSE ensures this interaction between systems on different functional and hierarchical levels of RAMI 4.0 using models as referential. Consequently, the standardized models and specifications help to overcome growing complexity of models and transit to interoperable solutions and continuously facilitate data flows as an example of Asset Administration Shell. To analyze functional and non-functional requirements of DT, the SysML language is used while descending on RAMI 4.0 hierarchical levels [START_REF] Mhenni | A SysML-based methodology for mechatronic systems architectural design[END_REF]. In addition, it is intended to complement existing domain-specific tools by providing a system view with which domain-specific views must remain consistent. Following the methodology from [START_REF] Incose | Incose Systems Engineering Handbook Version 3.2.2 -A Guide For Life Cycle Processes and Activities[END_REF], the requirements definition for future DT system should be identified. The diagram (Fig. 3) represents the requirement analysis process that could be applied to identify and specify DT in order to propose the conceptual framework and DT architecture for production systems.

It defines functional and performance requirements as well as architectural constraints and verification criteria for the system. The objective is to characterize the DT system context describing the environment in which it operates as well as different entities for interaction, i.e. users and other systems. By identifying different interactions of the system, the context diagram will help identifying the input/output flows that the system exchanges and also the interfaces required to achieve these interactions.

B. RAMI 4.0 for production system applications

The reference architecture unites three different dimensions represented by three axes: "The life cycle and value stream" for product lifecycle management supported by ISO 62890; the layers axis to represent functionalities of the product; the hierarchy levels retrieved from IEC 62264/IEC 61512. RAMI 4.0 defines system architecture with respect to aforementioned dimensions and represent the framework for I4.0 applications in manufacturing domain. The asset characteristics can fully be described by RAMI 4.0 on different levels from business to physical.

C. Asset Administration Shell

The AAS is a well-used application in automation domain forming an effective solution for data and information management and interoperability capabilities thanks to its semantic identification support. Defined as a digital representation for assets, the AAS structures the data from the physical world using system engineering approach. The standardized meta model of AAS is aimed to create interoperable DTs with respect to the structure and reduce the design efforts for data organization. In addition, AASs of existing products provide a base for product-as-a-service business models, where information can be stored during the product lifecycle and used by different parts.

However, the use cases applying AAS to production system DTs are still few, due to the complexity of data flows, the rights' policies and security issues of different components vendors. The possible benefits can be achieved not only in interconnectivity and interoperability capacities in xml file exchange but also as an information aggregation source and a buffer between enterprise resource planning systems, SCADA and control systems. The developers for AAS present its 3 possible roles or interaction patterns: as a metadata specification, as an API for DT services and for active interconnected DT [START_REF] Bader | Details of the Asset Administration Shell Part 2 -Interoperability at Runtime -Exchanging Information via Application[END_REF]. These options maintain a standardized approach and transform systems to I4.0 components. [START_REF] Rasor | Towards collaborative life cycle specification of digital twins in manufacturing value chains[END_REF] claims that AAS provide limited support in DT content entirely up to its developers. Nevertheless, the proposed approach eliminates this inconsistency.

D. DT development process

The development process we propose for DT systems implementation can be organized from system and operation architectures design and system evaluation to technical coordination and system integration. The important stages (Fig. 5) in system logic development start with objectives and constraints identification (Step 1) and dependent from the key events in production process.

Based on that, main scenarios can be developed defining working tasks and schedule packages (Step 2). DT system is in charge of decision, configuration and information management processes. RAMI 4.0 does not restrict the complexity level of the information systems moreover organizes the structure for implementation process. Following the hierarchy level axe of RAMI 4.0 from use cases referring to a connected world it is possible to identify critical asset from other potential assets among the components of production system for future DT (Step 3). The question is where a DT for the assets that are not yet manufactured can be referenced to a definition of the digital mock-up therefore for existing production systems the DT works on the operation/production lifecycle stage until the end of the life. The collected information totally depends on the use case.

Therefore, the DT development process includes the following steps: 1) The objective definition and related performance indicators; 2) The information source -assetidentification; 3) DT system requirements identification in connection with an asset; 4) DT modelling (functional, logical architecture and structure) and analysis; 5) Evaluation and verification. The possibility to use standardized specification and AAS model to organize knowledge about physical asset and at the same time structured MBSE approach to identify corresponding functional and non-functional DT system characteristics will provide more stable solution easy to upgrade with new elements and related properties.

IV. APPLICATION

In this section the first steps of the systems engineering approach that enables to define an architecture for production system DT is presented. The application case concerns the flexible production line on the AIP-PRIMéca Academic Technological Platform -Industry 4.0 S.mart RAO (Fig. 5). The seven working posts of the line ensure the continuous production process of pallets filled with jetons according to the requested production order. The assembly and disassembly posts are equipped with industrial robots which are of main interest.

The procedure (Fig. 6) starts from the "work centers" hierarchical layer for production line instead of omitted "enterprise" layer. For the production line on the "business" layer, the global objective is defined with assigned KPIs for value estimation corresponding to various missions or scenarios. For example, "rescheduling" mission for a specific station of the line. Text="For each mission the system shall assemble requested number of pallets" ID= "3.1" Text="The system shall assemble the requested number of pallets with 1,5 min interval between pallets" ID= "4" Text="The system shall be autonomous in energy during the mission" ID= "5" Text="The system shall be composed of 3 collaborating entities: the pallet, the assembly robot, the production line" ID= "2" Text="The system shall be compliant to the ISO 23247 standard related to digital twin" ID= "1" Text="The system shall autonomously produce the requested number of products" ID= "1.3 p1" Text="The system shall autonomously count the jetons, if the stock is depleted operator should be informed" Consequently, various decisions could be made at functional level depending on thetasks that are assigned to the station, e.g. "detection".

In order to define the information from the data, the product is identified with unique identifier for each product component. On integration and communication layers the raw data transfer is assured by control devices forming control system, OPC-UA communication protocol and RFID reader. Meanwhile, the production order previously generated and recorded on the pallet RFID tag can be rescheduled pointing the specific asset. The asset layer concerns the product itself, the jetons and all components of the system involved in process.

In order to specify components on functional layer of RAMI 4.0, the starting point to understand the DT system can be illustrated by DT requirements for the assembly post of the production line (Fig. 7). The global objective defines the assembly process requirements controlled by DT of the post. The same logic can be applied to other posts equipped with industrial robots. The assembly scenario is controlled by the DT of the assembly post. The system DT will federate DT of the stock and DT of the posts by data exchange. The requirements mirror the physical assets which are the industrial robot, the production line and the product consisting of pallet and jetons. Assuming the fact that the DT system has its own life cycle the diagram may be developed further.

V. CONCLUSION

The proposed approach considers the DT as a system and correlates the existing I4.0 reference architecture with systems engineering elements, to constitute the conceptual framework. This latter is based on information that describes production system, production process, assembly orders, product and resources. Further, the system can be enriched by additional information concerning maintenance. The framework aims to provide a view on an operational and logical problem description. To represent DT functionalities on RAMI 4.0 layers with the help of models, the important aspects of modelling should be taken into account. For example, for decisionmaking -the cross-domain and domain specific ontology, for functions organisation -metamodel entities and relationships and for time scheduled system supplysemantically identified data. The next step is establishing the functional architecture of DT where the system functions and their interconnection are identified. For this step, the inputs and outputs of the system as well as its functions are needed. Moreover, the functional architecture represented by activity diagram should include DT decision-support module.

Figure 1 .

 1 Figure 1. Clusters in DT research domain for production systems

Figure 3 .

 3 Figure 3. Context diagram for requirement analysis process[START_REF] Incose | Incose Systems Engineering Handbook Version 3.2.2 -A Guide For Life Cycle Processes and Activities[END_REF]

Figure 4 .Figure 5 .

 45 Figure 4. RAMI 4.0 application procedure

  system shall autonomously bring the pallet, accurately position it with respect to robot, hold it during operation and fill it with exact jetons" ID= "1.1" Text="The system shall autonomously bring the pallet, from post to post" ID= "1.2" Text="The system shall accurately postion the pallet with respect to robot and hold it during the assembly phase" ID= "1.3" Text="The robot shall assemble the pallet with exact jetons" ID= "1.2 p1" Text="The system shall autonomously postion the pallet, if the postioninng fails operator should be informed" ID= "3"

Figure 7 .Figure 6 .

 76 Figure 7. Initial requirements for DT of the working unit


Israel Jonathan
email: jonathan.israel@onera.fr

Physical-statistical and wideband model of the land mobile satellite propagation channel

Keywords:

This paper presents an extension of the SCHUN (Simplified CHannel for Urban Navigation, [1]) simulator and its validation framework in a wideband context. The hybrid physical-statistical model on which this simulator relies on extends the scope of applicability of standard channel characterization models thanks to its physical modeling of the interactions between the environment and a receiver. It also allows the generation of frequency and polarization diversities, and it improves the representativeness of the urban environment (type of materials, roughness...) while keeping a reasonable computational payload. This model has been validated in a narrowband and wideband context with respect to data acquired in a LMS (Land Mobile Satellite) configuration.

I. INTRODUCTION

The physical interactions between an incident electromagnetic wave and the environment close to a mobile receiver in an urban environment have a primordial effect on the characteristics of the received signal and therefore on the performance of the associated systems, especially in the field of satellite communications and GNSS (Global Navigation Satellite System). The modeling of these interactions is thus necessary for the simulation of the performances of such systems in a constrained environment. Three main families of methods are used to represent these interactions: deterministic, physical-statistical and statistical. Deterministic approaches are based on a realistic modeling of the 3D environment and its electromagnetic properties (roughness, dielectric parameters...) and on a deterministic calculation (exact or approximate) of the interactions to simulate the received signal. Such simulators, based for instance on asymptotic methods like [START_REF] Israel | A wideband and multifrequency propagation channel simulation[END_REF], [START_REF] Mametsa | Improving RCS and ISAR image prediction of terrestrial targets using random surface texture[END_REF], [START_REF] Mametsa | FERMAT: a high frequency em scattering code from complex scenes including objects and environment[END_REF], [START_REF] Israel | A propagation model for trees based on multiple scattering theory[END_REF], allow to obtain accurate results but often require a good knowledge of the environment and an important computation time depending on the complexity of the environment and the frequency of the emitted wave. On the other hand, statistical approaches use mainly existing data and little or no physics to simulate the received signal, often from regression of parametric models on measured databases. This is the case for example for the propagation model provided in the ITU-R (International Telecommunication Union -Recommendation) P681 in its paragraph 6, [START_REF]ITU-R P681-11, propagation data required for the design systems in the land mobile-satellite service[END_REF]. A two-state, narrowband Markov model is regressed on S and C band data and the resulting parameters are used to simulate narrowband time series. This type of approaches offers the advantage of a reduced computational payload and ensures by construction the correspondence between the statistics of simulated signals and the measurements, at least at the frequency and in the type of environment where the regression has been performed. On the other hand, since it introduces little or no physics into the modeling, no applicability of such methods when changing frequency or type of environment can be guaranteed. Between these two domains, hybrid or physico-statistical methods often allow to achieve a compromise in representativeness of the scene and the physical rendering and simplicity in terms of scene modeling and computational cost. The model of paragraph 7 of the ITU-R P681 recommendation falls into this category although the physical parameters introduced in the modeling of scatterers are essentially based on regressions and not on physical modeling. The SCHUN simulator used in this study, [START_REF] Ait-Ighil | Schun -a hybrid land mobile satellite channel simulator enhanced for multipath modelling applied to satellite navigation systems[END_REF], is based on a simplified representation of the urban environment allowing an approximate but physical calculation of the electromagnetic interactions.

II. PRESENTATION OF THE PHYSICAL-STATISTICAL MODEL

A. Generic overview

The simulation of electromagnetic scattering from buildings is based on the EM (Electro-Magnetic) SCHUN simulator initially developed for GNSS simulations in urban areas. The fundamental principles governing the SCHUN simulator are perfectly extensible to a model applied in another frequency band (especially between X and Ka). The concept of the SCHUN EM interaction simulator is to generate the environment using a virtual city approach and model the interactions using simplified EM models. This trade-off allows for fast computations relying on physical models.

The macro architecture of the SCHUN channel simulator is shown in Fig. 1. The statistical component of this approach is represented by the virtual city generation. All other modeling elements are based on ray techniques and physical EM models. The simplified building interaction models are based on physical EM models of models such as GO (Geometric Optics) and PO (Physical Optics). The direct signal shadowing model is based on UTD (Uniform Theory of Diffraction). The SCHUN channel simulator is capable of providing a wide range of results such as CIRs (Channel Impulse Responses) in delay/Doppler space, DoA (Directions of Arrival) information, complex narrowband signals, power time series, Doppler spectrums or masking time series.

B. Approach of the Three Component Model (3CM)

The SCHUN building scattering model is based on EM models with a minimum use of empirical approaches or measurement campaigns and satisfy the trade-off between computation time and versatility.

The scattering phenomenon taking place around complex facades can be divided into three propagation mechanisms, or components, each one being linked to a particular propagation scheme. The three components on which relies the 3CM are presented in Fig. 2. This scattering decomposition principle has the advantage of being potentially applicable to existent raytracing tools without major modifications. scattering phenomenon coming from all small features present on complex fac ¸ades. This component allows a smoother transition between the specular zone and the backscattering zone. It also characterizes the influence zone of the fac ¸ade. Since small features are not supposed to be modelled using asymptotic methods such as PO, the implementation of this empirical component is based on analytical models extracted from [START_REF] Ruck | [END_REF]. The main advantage of the approach chosen for the 3CM is the reduced computational payload: the total scattering pattern of any complex fac ¸ade can be divided into three components, each one being easy to model. Another advantage of the 3CM is its physical EM base which leads to realism and versatility. Contrary to empirical models, the 3CM is able to generate wideband CIRs with realistic delay, Doppler and DoA information. Since it is based on the PO algorithm, the 3CM can also deal with dielectric materials, handle elliptic polarizations and is scalable in frequency from 1 to 5GHz. Finally, the 3CM is suitable for MIMO studies with transmitting/receiving diversity, polarization diversity or frequency diversity.

C. Modelling of the trees and poles

The tree scattering model is based on the multiple scattering theory, [START_REF] Israel | Fading and scattering due to trees in l to ka band propagation simulations[END_REF]. It separates the effects due to the trunk and to the tree canopy. The trunk is modelled by a cylinder which allows taking into account the blocking and the diffraction effect on the edges of the cylinder. When going through the canopy, the incoming wave will be attenuated (the linear specific attenuation being fixed for a given type of tree) and a multipath diffuse component is also computed. This latter effect is based on the random positioning of point scatterers in the canopy which will diffuse in all directions. The pole interaction model is also a diffraction model based on UTD.

III. VALIDATION FRAMEWORK A. Experimental data

On the transmitter side, the experimental setup consists of a high phase stability frequency generator (Agilent E8267D PSG) and a 15W RF amplifier. The PROMOLAB recorder is mounted on a trailer towed by a passenger car. The two antennas for each band are wired simultaneously on the roof of the trailer. The core of the post-processing algorithm is based on the correlation between a known reference PRN (Pseudo-Random Noise) code in the data samples and the locally generated code following a method similar to GNSS signal processing. This method was chosen because, like the GNSS signal, the received signal power is below the noise floor of the digitization chain.

At the receiving chain, the locally generated PRN code is used to perform the acquisition process by estimating the correlation peak. In addition, the various correlation peaks allow multipath to be highlighted and thus provide a broadband characterization of the signal. When the amplitude of the main correlation peak is sufficient, the acquisition is successful and a rough estimate of the signal delay and Doppler is made. In the opposite case, for example when the receiver is behind a building or a tree with a dense canopy, a new acquisition is performed until becoming successful. Then, the processing loop enters the tracking phase where the delays and Doppler shifts of the code are estimated in order to lock the processing chain to the direct path. When the tracking fails to follow the direct path, the processing loop re-enters the acquisition stage until being successful. This general scheme is shown in Fig. 3.

B. Validation of the simulation model

The reconstructed street has been used as input in the SCHUN simulator in order to perform a site specific comparison between the simulation and the measurements in S Band on this 380 m trajectory, Fig. 4. and correspond to the same order of magnitude of fading amplitude. This similitude is clearly visible on all narrowband characteristic parameters, Fig. 6. The fading amplitudes and average durations are very close to each other. The order of magnitude of the level crossing rate is also similar even though it is a little bit higher in the measurements. This characteristic has already been noticed during the SCHUN development process, [START_REF] Ait-Ighil | Schun -a hybrid land mobile satellite channel simulator enhanced for multipath modelling applied to satellite navigation systems[END_REF]. This is due to the fact that in the shadowing areas, SCHUN models a vast but still limited number of electromagnetic contributors, leading to a global contribution that does not exhibit as much constructive and destructive interactions as in the reality. Therefore, the SCHUN signal is more continuous with less second order component than the measured signal. Finally, the Ricean factors are also similar in the simulation and the measurements, the main differences appearing for positive values corresponding mostly to LOS (Line Of Sight) situations: the signal is flatter in LOS situation in the simulation than in the measurement, where noise remains due to calibration and tracking residual errors.

C. Wideband parameters

At the end of the signal processing stage, range/time maps are directly available and can be compared to the range/time maps obtained by the SCHUN simulation. In order to be comparable, the raw SCHUN range time maps have been convoluted by a triangle reproducing the signal correlation process as obtained by the GNSS-like signal processing stage applied to the measured data. The corresponding range/time maps are given in Fig. 7. Once again appear phenomena already detected in the narrowband comparison, such as the blocking or fading processes visible around t = 15 s or t = 40 s for instance. Furthermore, some multipaths can also be seen on both maps, even if they may not be located exactly at the same locations, which is due to the complexity of reproducing exactly (up to a 1m and a 1s resolution) the scene and trajectory characteristics. They are shown in ellipses in both figures, the red one corresponding to multipaths due to the fac ¸ade shown in Fig. 8. Globally, two main differences can be noticed: while the range/time map in the measurement is dense (each pixel corresponding to a given value of the GNSS correlation process), the range/time map in the SCHUN simulation is sparse since SCHUN does not compute a dense but a discrete number of contributors which may not fill the entire map. Furthermore, the scattering life time of contributors may be larger in some cases (e.g. the pink section) due to simplifications made in the inter-objects masking computation. From the measured and simulated range/time maps, the mean power range profiles along the trajectory are computed and given in Fig. 9.

Their shapes are close to each other although the measurement curve is more continuous that the simulated one that exhibits a discontinuity around 4 meters due to the fact that in the simulation environment, no multipath occurs with shorter delays since we did not introduce closer objects like other vehicles.

In Fig. 10 are shown the spectrograms obtained from the measurement and the simulation. The main shape is the same, balancing toward the positive Doppler shifts due to the geometric configuration and leading to Doppler frequencies up to 100 Hz in both cases.

IV. CONCLUSION

A validation of the SCHUN LMS model in S band is presented here. Based on GNSS signal processing techniques, the channel sounding methodology allows to retrieve wideband parameters of the propagation channel and to compare them to SCHUN simulations in either site specific (requiring the precise reconstruction of a given street) and site generic (using a virtual city generation) situations. Globally, a good correspondence has been obtained between simulations and measurements for narrowband and wideband parameters. Some limitations have been pointed out in the SCHUN simulator, either on physical or informatics levels. It leads to potential future improvements that could allow the use of this simulator in multi-frequency and multi-polarization simulations in all kinds of urban environments.

Fig. 1 .

 1 Fig. 1. Macro architecture of the SCHUN LMS channel simulator.

Fig. 2 .

 2 Fig. 2. Decomposition principle of the 3CM.

Fig. 3 .

 3 Fig. 3. State machine of the processing chain allowing the acquisition and tracking of the transmitted signal.

Fig. 4 .

 4 Fig. 4. Saint-Lary main street used for measurements and simulation comparison.

Fig. 5 .

 5 Fig. 5. Comparison of measured and simulated original (left) and filtered (right) narrowband time.

Fig. 6 .

 6 Fig. 6. Comparison of measured and simulated level crossing rates (left) and Ricean factors (right).

Fig. 7 .

 7 Fig. 7. Comparison of measured (left) and simulated (right) range/time maps.

Fig. 8 .

 8 Fig. 8. Example of multipaths created by the side of a building.

Fig. 9 .

 9 Fig. 9. Comparison of measured and simulated power range profiles.

Fig. 10 .

 10 Fig. 10. Comparison of measured (left) and simulated (right) spectrograms.

ACKNOWLEDGMENT

This work has been financially supported by CNES study DSO/RF/ITP2017.0011547.


Marleen Marra
email: marleen.marra@sciencespo.fr

Sample spacings for identification: the case of English auctions with absentee bidding

Keywords: D44, C01, C46, C57) Nonparametric set-identification, English auctions, Order statistics, Shape restrictions, Optimal reserve price

This paper presents new nonparametric identification results for ascending auctions with independent private values. The standard identification approach is infeasible in the motivating setting, because absentee bidding conceals the number of bidders. I exploit insights from the statistics literature about the stochastic ordering of adjacent sample spacings. I show how to use such sample spacings to set-identify structural features, using an incomplete model and without knowing all highest bids or the number of bidders. Applying the sample spacing method to a small sample of wine auctions, I show that it identifies informative bounds on policy-relevant counterfactuals. It turns out that Sotheby's restricts full exploitation of the exclusion principle of optimal reserve prices. As a result, sellers set sub-optimally low reserve prices. They benefit up to 13% from adopting a common reserve price rule equal to 120% of the Wine Department's pre-auction value estimate.

Introduction

English auctions can be notoriously secretive.1 Motivated by the limited information observable in English auctions with absentee bidding, I show how so-called sample spacings [START_REF] Pyke | [END_REF][START_REF] Pyke | Spacings revisited[END_REF]) deliver nonparametric set-identification of structural features of interest. The structural analysis of IPV English auction data typically relies on knowing the number of bidders and at least one bid order statistic [START_REF] Athey | Identification of standard auction models[END_REF]). When bid data does not contain the number of bidders, the known mapping of the distribution of an order statistic from an i.i.d. sample of known size and its parent distribution cannot be applied. In this paper, I propose a new method that relies on the stochastic difference between adjacent order statistics, which contain previously unexplored identifying information.

The new set-identification method can be summarized in one paragraph. First, dropout values of absentee bidders are shown to bound the third-highest valuation. It is already widely known that the second-highest valuation is never more than the winning bid plus bidding increment [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF]). Even without knowing the number of bidders, the bid vector is therefore informative about the stochastic spacing between the second-and third-highest valuation (e.g. the secondto-last spacing). The crucial final step relies on facts from the statistics literature, that properly normalized spacings from distribution functions with increasing failure rates are stochastically decreasing [START_REF] Pyke | [END_REF]). This bounds the last spacing, and also set-identifies (counterfactual) surplus and revenue. This last step is crucial because of the well-known issue in English auctions that the auction stops when the secondhighest value bidder drops out.

The case of English auctions with absentee bidding is a fitting example of a setting where information revealed by sample spacings can benefit structural analysis. Absentee bidders report their maximum willingness to pay to the auctioneer who then bids on their behalf during the live auction. 2 Highest bids and number of bidders cannot be discerned from the bid vector when identities are not known, since bidders may not all place just one bid at their maximum valuation. This limited information content diverges from what is assumed known in previous English auction studies, in-cluding [START_REF] Paarsch | Deriving an estimate of the optimal reserve price: An application to British Columbian timber sales[END_REF] using all bidders' drop-out values and the number of bidders, [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF] and Chesher andRosen (2015, 2017) using a vector of highest bids and the number of bidders, [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF] using a vector of bids that includes the second and third-highest drop-out values in some auctions, and [START_REF] Aradillas-López | Identification and inference in ascending auctions with correlated private values[END_REF] using the second highest drop-out value and number of bidders (relaxing IPV). These papers are all groundbreaking in their econometric use of bid data from English auctions, but their identification strategies cannot be applied to the limited data central to this paper. For the analysis developed here, one needs to observe only: 1) a vector of bids and 2) which bids are submitted by absentee bidders. The results extend to auctions where bids from at least three different bidders can be identified.

To underscore its practical use in overcoming data limitations, I apply the method to an original dataset of fine wine auctions with absentee bidding collected at Sotheby's. Leveraging information from drop-out values of absentee bidders and the secondhighest bidder, expected bidder surplus is estimated without knowing the number of bidders. Empirical results show that expected winning bidder surplus is higher for high-end wines, and lies between 75 to 125 percent of the average highest bid in the full sample. More fundamentally, this empirical application shows that the sample spacings method delivers informative bounds even in small samples with large bidding increments. This qualification matters: even when the number of bidders would be known, large bid increments would result in bounds on outcomes of interest (applying [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF]) unless they are assumed away.

This setting is also interesting from a policy angle as Sotheby's allows sellers to set a secret reserve price, but restricts it to be less than the low bound of the Wine Department's estimated value bracket. My estimates show that this results in sub-optimally low reserves, which precludes sellers from leveraging the exclusion principle of optimal reserve prices (see [START_REF] Krishna | Auction theory[END_REF], based on [START_REF] Myerson | Optimal auction design[END_REF] and [START_REF] Riley | Optimal Auctions[END_REF]). In other words, they place too much weight on the sale probability and too little on revenues conditional on a sale. I consider a simple counterfactual policy where the reserve price is automatically set at α times the pre-auction value estimate, and show that expected seller profit increases by up to 13 percent when setting α = 1.2. This upper bound corresponds to roughly 25.000 pounds of additional seller surplus for a single day of wine auctions, of which the auction house holds about 20 per year in its London branch alone. Also the lower bound is non-negative for any α ∈ [0.75, 1.2], indicating that it is not only the binding value bracket constraint that delivers low sale revenues. Even automatic reserve prices at a value of α ≤ 1 increase expected seller profit compared to the current policy where they set reserves individually.

As such, the results in this paper contribute to the structural analysis of bid data with incomplete auction models. Within that literature, [START_REF] Tang | Bounds on revenue distributions in counterfactual auctions with reserve prices[END_REF][START_REF] Aradillas-López | Identification and inference in ascending auctions with correlated private values[END_REF][START_REF] Coey | Ascending auctions with bidder asymmetries[END_REF] are among the most related, focusing on directly (bounding) structural features of interest rather than latent value distribution. [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF] provides results to point-identify the value distribution using a pair of adjacent order statistics in ascending auctions. Similar to how absentee bidding reveals additional information in my motivating example, [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF] exploits a feature of eBay auctions to extract the third-highest valuation from the bid vector. While other papers rely on multiple order statistics (e.g. [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF], [START_REF] Mbakop | Identification of auctions with incomplete bid data in the presence of unobserved heterogeneity[END_REF], and Luo and Xiao ( 2019)) or non-ordered measurements (e.g. [START_REF] Li | Nonparametric estimation of the measurement error model using multiple indicators[END_REF] and [START_REF] Krasnokutskaya | Bid preference programs and participation in highway procurement auctions[END_REF]) to support identification in (auction) models, the innovation of my approach is that it relies on shape restrictions and their relation to the way adjacent order statistics are spaced out. This adds previously unexploited identifying information that helps facilitate structural analysis. Also related is recent work that uses shape properties to aid identification and estimation in auctions, notably [START_REF] Larsen | A mechanism design approach to identification and estimation[END_REF] and [START_REF] Pinkse | Estimation of auction models with shape restrictions[END_REF].

The paper proceeds as follows. After introducing the model in section 2, section 3 sets out the main results. An application in section 4 illustrates the method's relevance for structural analysis of incomplete bid data. Section 5 concludes.

Auction model

The mechanism is a standard English auction with a flexible closing rule, fixed bidding increments, and a secret reserve price. Bidders have the option to place a sealed bid ahead of the auction. Ex-ante symmetric, risk neutral bidders have unit demands and face negligible entry and bidding cost. I follow the convention to denote random variables in upper case and their realizations in lower case. All results in this paper are conditional on a vector of observed auction covariates, Z. To economize on notation, I therefore restrict attention to conditional values. Formally, bidder i draws a valuation V i ∼ f V i , and V i = ψ(Z, U i ), with ψ(.) some function and U i ⊥ Z. Key assumptions of the auction model are:

Assumption 1. All n bidders symmetrically and independently draw values from a common conditional value distribution, such that: i) F U i (.) = F U (.), ∀i = {1, .., n} (exchangeability) ii) F U (.) = F U (.) n (independence) iii) F U (.) is absolutely continuous and is defined on bounded support [0, ū] (regularity conditions) This is the symmetric (conditional) IPV assumption that is the main tenet in the structural analysis of English auctions [START_REF] Paarsch | An introduction to the structural econometrics of auction data[END_REF]). Exchangeability of U i for all bidders in the auction also guarantees that bidders' preference for live or absentee bidding is independent of their valuation. The following shape restrictions are imposed:

Assumption 2. F U (.) satsifies: i) f U (u) 1-F U (u) weakly increases in u (increasing failure rate, IFR) ii) U ≤ disp Y , where Y ∼ i.i.d. U nif [0, ū] (less dispersed than uniform, LDTU) 3
These are mild conditions useful for identification with sample spacings. From a reliability theory perspective, the assumption rules out distribution functions that age slower than the exponential distribution (i) and that age faster than the uniform distribution on the same support (ii). As extreme dispersion as generated by the uniform distribution is never fitting (to my knowledge) to describe latent values in auction data. But also IFR is minimally restrictive, and many seminal (auction) papers previously relied on this shape restriction to generate insights without imposing a specific parametric distribution function (e.g. [START_REF] Myerson | Optimal auction design[END_REF] and [START_REF] Riley | Optimal Auctions[END_REF] to derive a unique optimal reserve price). 4Furthermore, the following assumption restricts variation of the unobserved number of bidders n. This "exogenous participation" assumption formalizes what is already implied by the combination of IPV and negligible entry cost: the absence of selective entry.

Finally, I impose intuitive behavioral assumptions that define bidding strategies in the incomplete model of [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF]:

Assumption 4. Bidders are rational and attentive.

While these assumptions are satisfied in all symmetric separating equilibria of the button auction model of [START_REF] Milgrom | A theory of auctions and competitive bidding[END_REF], they also allow for alternative behavior including not bidding at all or bidding less than one's valuation.

Absentee bidding

Absentee bidding is a widely adopted practice both in Egnlish auctions, as also documented by [START_REF] Akbarpour | Credible auctions: A trilemma[END_REF]. For the data in my empirical application, Sotheby's provides the following guidance: Absentee bids are to be executed as cheaply as permitted by other bids or reserves and in an amount up to but not exceeding the specified amounts. Bids will be rounded down to the nearest amount consistent with the bidding increment. In the case of identical (rounded) bids, the earliest submitted form will take precedence. 5Auction mechanisms with absentee bidding are designed in a way that does not disadvantage absentee bidders. 6 Hence, if only one absentee bid is submitted for an item, the opening bid must start below that value. The rest of the bidding sequence will vary by institution. The following stylized sequence is based on my empirical observations that: i) the data contains various auctions with multiple absentee bids before live bidding starts and ii) after initial absentee bids the live bidding alternates with absentee bids:

Bidding sequence. With one absentee bid, the opening bid equals a fixed share of the reserve price. With multiple absentee bids, the opening bid equals the minimum of the absentee bids. Subsequent bids follow fixed bidding increments until all but one absentee bidder drops out. Afterwards, live bids are alternated against the last remaining absentee bidder unless he has dropped out.

Crucially, the bidding sequence formalizes that absentee bidders are not (dis)advantaged relative to live bidders, which is maintained throughout. To measure this restriction against structural analysis of bid data in other IPV English auction studies, consider the two benchmark models. [START_REF] Paarsch | Deriving an estimate of the optimal reserve price: An application to British Columbian timber sales[END_REF] assume that data is generated by the button auction model in which case all bids are assumed to be a different bidder's maximum willingness to pay. But even when relaxing the behavioral restrictions of the button auction model and using the incomplete model of [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF] for structural analysis of bid data, it is required to observe a vector of highest bids. Instead of needing to know highest bids and the number of bidders, I assume that:

Informational requirement. The econometrician observes: i) a vector of bids, ii) which ones are absentee bids.

Note that English auctions with absentee bidding are merely a motivating example for using sample spacings for structural analysis of bidding data. This core idea applies to any setting where drop-out values of at least two groups of bidders are observed, and those groups are ex-ante identical. Clearly, this is a relatively weak informational requirement nested in the case of observing bidder identities.

Equilibrium bidding strategies

This section describes the equilibrium bid strategies as a function of a bidder's conditional valuation, β k (u), for k ∈ {abs, live} for respectively absentee and live bidders. I restrict attention to type-symmetric Bayes Nash equilibria in weakly undominated strategies. Recall that bidders are symmetric up to their bidding mode and conditional valuation draw. However, there is a crucial difference in the set of bidding strategies available to them. Live bidders can squat or jump bid, bid only once or bid many times incrementally. 7 The strategy available to absentee bidders is limited to the height of the bid submitted.

Lemma 1. It is optimal for an absentee bidder with U = u to bid: β abs (u) = u.

Proof. For absentee bidders, the auction is strategically equivalent to an IPV secondprice sealed bid auction. It therefore follows directly from [START_REF] Vickrey | Counterspeculation, auctions, and competitive sealed tenders[END_REF] that truthful revelation is a weakly undominated strategy for absentee bidders.

Lemma 2. It is optimal for a live bidder with U = u to bid: β live (u) ≤ u, with the added constraint that a final bid is placed if u ≥ standing price plus bidding increment.

Proof. For live bidders, the auction is strategically equivalent to the English auction setting in [START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF], where bidding up to one's valuation is optimal and where bidders won't let an opponent win at a price they are willing to beat.

Resulting bounds on valuation order statistics

In this section, I leverage additional information revealed by absentee bids to bound an additional order statistic of the valuation distribution. Lemma 1 established that absentee bidders bid truthfully. However, the sequence of observed bids also include intermediate bids as the auctioneer is to determine the lowest price given competing bidders and increments, and bidder identities are unobserved. The stylized bidding sequence is used to determine which bids correspond to bidders' drop-out values and which bids correspond to such intermediate bids.

Let B denote the random variable of bids conditional on observables (Z), b the number of submitted bids in an auction, ∆ ≥ 0 the minimum bidding increment, and subscripts + and -respectively upper and lower bounds on the relevant order statistics. Using order statistic notation,

U n:n ≥ B b:b ≡ U - n:n (1) U n-1:n ≥ B b-1:b ≡ U - n-1:n (2) U n-1:n ≤ B b:b + ∆ ≡ U + n-1:n (3)
I add an additional relationship, using information revealed by absentee bidders' drop-out values, to bound the third-highest valuation:

U n-2:n ≥ B x ≡ U - n-2:n (4)
This equation holds by the definition of B x . It is identified in any auction where a drop-out value is observed that is less than the two highest observed bids. Simply put, the highest two bids must be submitted by different bidders and the highest drop-out value less than the second-highest bid must be by a third bidder, delivering a lower bound on the third-highest valuation.

Characterizing B x : examples. To illustrate, consider hypothetical bid vectors.

Let A i and L j respectively denote the ith lowest overall bid, submitted by an absentee bidder, and the jth lowest overall bid, submitted by a live bidder: Auction

1: A 1 , A 2 , A 3 , L 4 , L 5 Auction 2: A 1 , L 2 , A 3 , L 4 , L 5 Auction 3: A 1 , L 2 , A 3
In Auction 1 and Auction 2, L 4 and L 5 must be placed by different (live) bidders because nobody outbids himself, so both are (lower bounds) on drop-out values of different live bidders. A 3 is the highest other drop-out value identified, from an absentee bidder, and hence equal to B x by definition.8 In Auction 3, A 3 is the lowest identified drop-out value of an absentee bidder and L 2 is another drop-out value, but no lower bound on the third-highest valuation can be established.

The identifying power of sample spacings

The censoring problem in English auctions arises from the fact that the auction stops when the bidder with the second-highest valuation drops out. A challenge to identification of the latent value distribution and related structural features of interest is therefore that the highest valuation is never observed. In this section, I address this issue by exploiting results from the statistics literature on the stochastic spacing of order statistics. All identification results rely on the econometrician observing a large set of independent auctions. Definition: sample spacings. Following [START_REF] Pyke | [END_REF][START_REF] Pyke | Spacings revisited[END_REF], spacings D i:n between two adjacent order statistics are defined as:

D i:n = U i:n -U i-1:n , ∀i = 2, .., n. Nor- malized spacings are defined as: Di:n = (n -i + 1)(U i:n -U i-1:n ), ∀i = 2, .., n.
Both D i and Di are random variables with CDF F D i:n and F Di:n ∀i = 2, ..., n.

To understand the usefulness of the last spacing, consider that it contains same information as the highest two conditional order statistics combined. Specifically, the density function of D i:n based on U ∼ i.i.d. F U (.) [START_REF] Pyke | [END_REF]):

f D i:n (d) = n! (i -2)!(n -i!) ū 0 F U (x) i-2 [1 -F U (x + d)] n-i f U (x)f U (x + d)dx, (5)
which equals the density of U i:n conditional on the realization of U i-1:n in expectation over all such realizations. Seller surplus (π S ) and winning bidder revenue (π B ) at counterfactual reserve prices are of primary interest in structural auction studies. They can be expressed in terms of the last spacing and the marginal distribution of the second-highest conditional valuation:

π S (r) = ū 0 [1 -F Dn:n (r -u n-1 )] max(r, u n-1 )dF U n-1:n (u n-1 ) (6) π B (r) = ū 0 [1 -F Dn:n (r -u n-1 )]{ ū max(r,u n-1 ) u n dF Dn:n (u n -u n-1 )}dF U n-1:n (u n-1 ) (7) [1 -F Dn:n (r -u n-1 )
] is the sale probability when U n-1:n = u n-1 and with reserve price r, which is equal to: 1 -F Un:n|U n-1:n (r|u n -1). For expositional clarity of the novel identification approach, the rest of this section proceeds under the assumption that ∆ = 0, so that F U n-1:n is point-identified by the distribution of the second-highest bid. This is in line with other contributions to the set-identification of auction primitives in ascending auctions (e.g. [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF], [START_REF] Aradillas-López | Identification and inference in ascending auctions with correlated private values[END_REF], [START_REF] Coey | Discounts and deadlines in consumer search[END_REF]).

Let F - Dn:n (.) and F + Dn:n (.) respectively denote the lower and upper bound on the distribution of D n:n , such that:

F - Dn:n (d) ≤ F Dn:n (d) ≤ F + Dn:n (d), ∀d ∈ [0, ū]
. By stochastic dominance, bounds on π B and π S are defined as:

ū 0 [1 -F - Dn:n (r -u n-1 )] max(r, u n-1 )dF U n-1:n (u n-1 ) ≥ π S (r) (8) ≥ ū 0 [1 -F + Dn:n (r -u n-1 )] max(r, u n-1 )dF U n-1:n (u n-1 ) ū 0 [1 -F - Dn:n (r -u n-1 )]{ ū max(r,u n-1 ) u n dF - Dn:n (u n -u n-1 )}dF U n-1:n (u n-1 ) ≥ π B (r) (9) ≥ ū 0 [1 -F + Dn:n (r -u n-1 )]{ ū max(r,u n-1 ) u n dF + Dn:n (u n -u n-1 )}dF U n-1:n (u n-1 )
As such, the upper bound on the sale probability is derived with F - Dn:n ; a stochastically larger last spacing means a larger probability that U n:n exceeds r. The upper bound on the winning bidder surplus conditional on a sale (the inner integral in 7) is also derived from F - Dn:n as it translates directly into a larger U n:n -U n-1:n . If n would be known, one could work with the marginal distribution of F Un:n by taking it out of the (outer) integrals in ( 6)-( 7).9 My identification approach instead exploits the relation between ageing properties of F U (.) and the stochastic ordering of its sample spacings.

Identification of F

- D (n:n) (.) and F + D (n:n) (.)
Due to the censoring problem, F - D (n:n) (.) is the more challenging bound to pin down. To do so, I rely on the result by [START_REF] Barlow | Inequalities for linear combinations of order statistics from restricted families[END_REF] that normalized spacings from distribution functions with increasing failure rates are stochastically decreasing. Intuitively, this is because normalized spacings from the exponential distribution are exchangeable random variables [START_REF] Pyke | [END_REF]), and F U (.) ages faster than the exponential in the reliability theory sense (by the shape assumption 2 ii).

Lemma 3. U n-1:n and U - n-2:n identify F - D (n:n) (.) without knowing n:

F - Dn:n (d) ≡ P [2(U n-1:n -U - n-2:n ) ≤ d] , ∀d ∈ [0, ū] (10)
Proof. For all d on its support:

F Dn:n (d) = P [D n:n ≤ d] (by definition of CDF) ≥ P [2D n-1:n ≤ d]
(by decreasing normalized spacings due to IFR [START_REF] Barlow | Inequalities for linear combinations of order statistics from restricted families[END_REF]))

≥ P [2(U n-1:n -U - n-2:n ) ≤ d] (by definition of bound)
This result alone is therefore sufficient to bound π B and π S from above. Lower bounds could be derived by rewriting ( 6)-( 7) in terms of marginal value distribu-tions, and then using that an upper bound on F Un:n (.) is trivially identified as the distribution of the winning bid.

Sample spacings provide additional identifying information for the lower bounds on π B and π S . F + Dn:n (.) is based on the fact that uniform spacings, e.g. spacings of i.i.d. draws from a uniform distribution, are exchangeable random variables [START_REF] Pyke | [END_REF]), and that F U (.) ages slower than the uniform in the reliability theory sense (by the shape assumption 2 i).

Lemma 4. U n-1:n and U n-2:n identify F + D (n:n) (.) without knowing n:

F + Dn:n (d) ≡ P [(U n-1:n -U n-2:n ) ≤ d] , ∀d ∈ [0, ū] (11) Proof. Let Y ∼ i.i.d. G Y (.) = U nif [0, ū],
and {H i:n } i=1,..,n its sample spacings. By assumption 2 i, U ≤ disp Z. [START_REF] Bartoszewicz | Dispersive ordering and the total time on test transformation[END_REF] proves that this implies the stochastic ordering: D i:n ≤ st H i:n ∀i ∈ {1, .., n}. Therefore for all d on its support:

F D (n:n) (d) = P [D n:n ≤ d] (by definition of CDF) ≤ P [D n-1:n ≤ d]
(by increasing spacings due to LDTU, applying [START_REF] Pyke | [END_REF] and [START_REF] Bartoszewicz | Dispersive ordering and the total time on test transformation[END_REF])

This bound is based on the second-to-last spacing being pinned down exactly from bidding data, such as in the eBay auction model of [START_REF] Song | Nonparametric estimation of an eBay auction model with an unknown number of bidders[END_REF] or based on it [START_REF] Adams | Estimating demand from eBay prices[END_REF] and il [START_REF] Kim | Nonparametric estimation and testing of the symmetric IPV framework with unknown number of bidders[END_REF], and in any auction model where bids are equated to drop-out values. The presented model describing traditional English auctions with absentee bidding allows for additional incompleteness of the bid vector in terms of drop-out values. The result is presented regardless to provide a richer perspective on the use of sample spacings in the structural analysis of bid data. In the empirical application, I verify a necessary condition for applicability of this upper bound, based on the empirical and simulated sale probabilities. A comparison of estimation results based on the distribution of highest bid provides additional insight into these two approaches.

12

Electronic copy available at: https://ssrn.com/abstract=3622047 Electronic copy available at: https://ssrn.com/abstract=3622047

Simulations

Figure 1 shows how Lemma 3 and 4 apply to familiar distribution functions. Plots compare the true distribution of the spacing between the highest two order statistics with its bounds implied by the difference between U n-1:n and U n-2:n . Plotted are simple empirical CDF's based on 10.000 simulated sets of values, abstracting from bidding increments. 10 Plots a and b illustrate the simple basis for the presented identification results. Plot a is based on the Exponential distribution. F - Dn:n (.) as derived in Lemma 3 collapses to the true F Dn:n (.). For any distribution function that ages faster than the Exponential, the last spacing will therefore be stochastically dominated by F - Dn:n (.). Plot b is based on the Uniform distribution. F + Dn:n (.) as derived in Lemma 4 collapses to the true F Dn:n (.). For any distribution that ages slower than the Uniform, the last spacing will therefore be stochastically dominating F + Dn:n (.). Four other points are important to take away from these simulations. First, leveraging information revealed by sample spacings can indeed result in highly informative bounds. Second, F - Dn:n (.) is tighter the less IFR the underlying distribution function is (plot d versus e). This simultaneously means that the lower bound is less tight in those instances. Third, the larger the number of (unobserved) bidders, the tighter F - Dn:n (.) (plot f versus e). And fourth, IFR is sufficient but not necessary for the sample spacing method to identify F - Dn:n (.) (plot c). I find the excess wealth order intuitive to interpret the two shape restrictions in Assumption 2, especially because identification focuses on the upper tail of the distribution of F U . Definition: Excess Wealth Order. Random variable U is larger than Y in the excess wealth order (U ≥ EW Y ) if and only if:

∞ F -1 U (p) 1 -F U (t)dt ≥ ∞ F -1 Y (p) 1 -F Y (t)dt, ∀p.
For IFR distribution functions it holds that the dispersive ordering implies the excess wealth order (?), so that all IFR distributions satisfying

U ≥ EW Y ∼ U nif [a, b] also satisfies U ≥ disp Y .
It is easy to show that the excess wealth of E ∼ Exp(λ) at quantile p equals: p λ , e.g. in Figure 2 the thick solid line with slope -1 (exponential with rate λ = 1). IFR distributions (with F (0) = 0) have lower excess wealth at each q. So distribution 10 Generally, parametric distribution functions that have an increasing failure rate include the Normal, Exponential, Logistic, Extreme Value, Weibull (shape parameter ≥ 1), Gamma (shape parameter ≥ 1), and Beta (shape parameter ≥ 1)) [START_REF] Bagnoli | Log-concave probability and its applications[END_REF]).

functions with excess wealth in between of the U nif [0, 1] and Exp(1) distributions satisfy the two shape assumptions of this paper. This corresponds to the shaded area in the figure .   4 Application: wine auctions at Sotheby's I apply the sample spacing method to a unique dataset covering the 884 lots from the "Finest and Rarest Wines & Vintage Port" auction on November 19th 2014 at Sotheby 's London. 11 This provides a good and conservative test case as it is a relatively small dataset, with large bidding increments, and it does not contain the number of bidders. The dispersion of highest bids is large, even when normalizing by the number of bottles in the lot, and especially the upper tail is long. To reduce the impact of extremal values I therefore exclude auctions for which V - n:n or V + n:n exceed their 95th percentile.12 Descriptive statistics of the remaining sample are provided in Table 1. As common in traditional English auctions: increments are high at between 2 -16 percent of the winning bid.

The presence of absentee bidding requires bidders to be willing to announce their bids before others do, which is by itself a strong indication that bidders do not anticipate a "winners curse" and that the assumption of private values is justified. It is reasonable to expect that correlation in valuations is captured by auction observables, including the estimated value by Sotheby's Wine Department, if these specialists are unlikely to be outperformed by potential buyers in establishing the current value of the wines. The data supports this idea. The high predictive power of Sotheby's preauction value estimate is highlighted in Figure 3 plot a, showing its relation with the realized winning bid. Empirical tests based on Kolmogorov-Smirnov tests confirm that IPV is reasonable assumption when conditioning on Sotheby's value estimate, and that additional conditioning variables do not strengthen this conclusion.13

Nonparametric estimation

This section shows how to estimate bounds on π S and π B (equations 8 and 9) by applying the spacings identification approach. πS and πB are based on product Kernel estimators of conditional density functions (dF V n-1:n |Z (.), dF + Dn:n|Z , dF - Dn:n|Z ) and these results is warranted: the small sample size may also be a reason that the Kolmogorov-Smirnov tests do not reject the null, which is why the table with test results is not reported here.

conditional cumulative distribution functions (F + Dn:n|Z (.), F - Dn:n|Z (.)), such as:

dF + Dn:n|Z (d|z) = 1 h D- K t∈T - D t- n:n -d h D- K Z t--z h Z- t∈T -K Z t--z h Z- (12)
F -Dn:n|Z (d|z) =

t∈T + L D t+ n:n -d h D+ K Z t+ -z h Z+ t∈T + K Z t+ -z h Z + , with: L(x) = x -∞ K (u) du (13)
∀(d, z) on their supports. The minus (plus) superscripts on T , D n:n , Z, and h indicate that they relate to the lower (upper) bound on the last spacing and the therefore relevant observations. K(.) indicates the Epanechnikov kernel function, h the cross-validated variable-specific bandwidths as functions of the relevant sample sizes. Moreover, let T -= |T -| (T + = |T + |) denote the total number of auctions in which D - n:n (D + n:n ) is identified, with the caligraphic script denoting sets of such auctions.14 For a meaningful analysis of these estimators, estimated spacings CDF's are first applied to bound consumer surplus in the data (CS t ): ĈS t differs from counterfactual surplus as defined in (7) as it does not involve the sale probability: it conditions on the event of a sale given the current (unknown) reserve price. The upper bound on CS t is also compared against one derived from the distribution of B b:b itself, as anticipated on page 12:

ĈS t ≥ x dF + Vn:n|Z (x|z)dx -(b t + ∆) (15)
After estimating the three conditional densities, ĈS t is approximated numerically on a fine grid of x, for all sold auctions. [50.471,61.768] [112.477,123.637] [254.843,291.649] [141.767,162.380 [68.655,85.092] [191.840,237.115] [342.632,432.390] [201.082,233.435] Estimates are in pounds per bottle. Confidence intervals are calculated from 100 bootstrap samples. Z = Sotheby's low value estimate.

(a) Bounds how informative the estimated bounds are, even in this "worst-case scenario": a small sample with large bidding increments and without knowing the number of bidders. The (point estimate of the) lower bound on expected consumer surplus equals 75 percent of the average winning bid (87 percent when based on V - n:n instead of D - n:n ), not far removed from the upper bound at 125 percent. Estimated bounds are wider at the tertile level, based on only a third of the sample and about 200 observations. The middle tertile for instance has estimated bounds between 121 and 193 percent of the average winning bid, and the first tertile between 137 and 218 percent of the winning bid. Another clear result is that consumer surplus is higher for higher-end wines, with even the upper bound of the estimated surplus for the first tertile being less than the lower bound for the second tertile, and similarly for comparing the second to the third tertile. This goes through even when taking the 95% bootstrapped confidence intervals into account.

Figure 4 explores further auction-level heterogeneity of ĈS t . Plot a) displays heterogeneity of estimated bounds by decile of the conditioning variable. The previous result that higher-value lots deliver more consumer surplus (which is not trivially so) is reinforced. The boxplots also reveal that there is more heterogeneity for higher-end wines, insofar as this is reflected by Sotheby's pre-auction value estimate. Plot b) plots the difference between the point estimate of high and low bound on CS t , in this case reported as a share of the winning bid, and plotted by decile of the winning bid. There is again a remarking auction-level variation, with the sample spacing method resulting in bounds being more informative in terms of this outcome for higher-end wines.

Policy simulation: optimal reserve price

It is an official policy of Sotheby's to allow only reserve prices at or below the low bound of the value estimate.15 Also empirically, Figure 3 plot b shows that secret reserve prices are set at a fixed share of Sotheby's low value estimate. The highest bids are below this estimate for all unsold lots (between 75-100%). Together with the high sale probability of 90 percent, this begs the question whether and how much sellers would benefit from adopting an optimal reserve price; a counterfactual of primary interest in empirical auction studies. 16To bound the increase in expected seller revenue from adopting a higher reserve price, I run the following simulation exercise. I consider counterfactual reserve prices rt = αZ t for a range of α ∈ (1, 2), thereby relaxing the constraint on reserve prices insofar as the current policy is binding. To evaluate the impact on sellers, I estimate bounds on π S by applying the results from Lemma's 3 and 4 to (8). With reserve prices being secret, the highest bid of the bid vector is the floor of the counterfactual bid realizations and the trade-off of increasing the price conditional on a sale with a lower sale probability drives the results.

Specifically, the expectation over U n-1:n in ( 8) is done over realizations of B b:b in the data (not excluding unsold lots). Suggestive evidence that the upper bound F + Dn:n (.) in Lemma 4 applies is that the simulated lower bound on the sale probability indeed is lower than the empirical sale probability, at 93.7 percent versus 94.6 percent. All counterfactual results are simulated only for auctions in which D + n:n is identified (e.g. where B x is identified), to make sure the lower and upper bounds relate to the same primitive. F + Dn:n (.) and F - Dn:n (.) are estimated as described in the previous section, with the addition that the larger cross-validated bandwidths from the upper bound are used in both estimators to guarantee that the bounds don't cross in the even smaller sample (167 observations).

The benchmark policy is that Sotheby's determines Z and sellers choose a reserve price less than Z, so this counterfactual sheds light on two questions: 1) whether sellers on the whole would be better off when adopting the simple policy of a standardized and reserve price rule, and 2) whether sellers and Sotheby's gain from allowing the reserve price (rule) to exceed Z.

Resulting πS (r) are plotted in Figure 5. Despite not point-identifying it, the simulations reveal interesting facts. Moving from a system where sellers set their secret reserve individually between 75-100% of Z to one where there is a common reserve price rule at any level between 75-100% of Z is clearly beneficial to sellers. This suggests that even conditional on the reserve price constraint r ≤ Z, individual sellers set suboptimal reserve prices. They would gain at least 2.5 percent by setting a common reserve price rule α ∈ [0.75, 1], and the upper bound for such a policy is estimated at 6 percent gain.

The real kicker comes for higher values of α meaning that reserve prices are set sub-optimally low, and that the reserve price constraint is indeed binding. The policy to set rt = 1.2z t increases seller revenue up to 13 percent.

Recall that Sotheby's provides a value bracket for each lot, and it is notable that α = 1.2 would set the secret reserve near the upper end of this bracket (the median upper bound is 27% higher than Z). In other words, by restricting α ≤ 1, 20 Electronic copy available at: https://ssrn.com/abstract=3622047 Sotheby's does not fully allow for reserve prices to satisfy the exclusion principle of optimal reserve prices [START_REF] Krishna | Auction theory[END_REF], based on [START_REF] Myerson | Optimal auction design[END_REF] and [START_REF] Riley | Optimal Auctions[END_REF]). It places too much weight on the sale probability and too little on the expected revenue conditional on a sale. It is even suboptimal for Sotheby's themselves as most of their income comes from commissions.At higher values of α, the lower bound on πS (r) becomes negative so can't be said with certainty that higher levels are beneficial.

In terms of monetary values and extrapolating to the whole sample, the wine auctions would generate up to 25,694 pounds in additional seller surplus when setting a common reserve price rule of rt = 1.2z t . This is for just one day of auctions; Sotheby's holds over 20 of these "Finest and Rarest Wine auctions" per year.

Conclusion

This paper proposes a new approach to identify policy counterfactuals from limited Enlisch auction data. It shows previously unexplored identifying power of the spacing of order statistics in combination with weak shape restrictions. Simulations furthermore highlight the simple identification approach based on ageing properties of the extremal cases of exponentially and uniformly distributed latent values. A particu-lar benefit of the approach is that it provides a feasible solution to not knowing the number of bidders, as in the motivating example of English auctions with absentee bidding. While the results apply more generally, this setting is exploited to identify additional information about the drop-out value of one absentee bidder such that the distribution of the second-to-last spacing is bounded. Combined with results from the statistics literature about the stochastic ordering of adjacent spacings, the paper shows that this delivers an upper bound on the last spacing and hence overcomes the censuring problem of English auctions.

The method is applied to a new dataset of fine wine auctions in which the number of bidders is unknown. Results highlight that even in small samples with large bidding increments, sample spacings allow for the estimation of informative bounds on structural features of interest. For example, consumer surplus is estimated to be between 75-125 percent of the average highest bid. It also turns out that surplus is higher for higher-end wines. Structural estimates are used to evaluate the benefit of relaxing Sotheby's policy that the reserve price has to be lower than the pre-auction value estimate. Results show that the restriction is indeed binding: sellers benefit by up to 13 percent from setting a reserve price equal to 120 percent of the value estimate. This suggests that the current policy limits sellers to fully leverage the exclusion principle of optimal reserve prices.

Assumption 3 .

 3 If n varies across auctions with identical covariates, F U (.|n) = F U (.|n ) ∀n = n .

  B b:b is the highest submitted (conditional) bid, B b-1:b the second-highest, and B x the highest observed drop-out value less than B b-1:b . As used in Haile and Tamer (2003); the second-highest valuation is bounded between B b-1:b and B b:b + ∆, and the highest valuation must clearly also exceed B b:b :

Figure 1 :Figure 2 :

 12 Figure 1: Informativeness of bounds on F Dn:n (.))Based on Lemma 3 and 4 applied to different data generating processes, plotted for spacings between the 1st percentile of D n-1:n and the 99th percentile of Dn:n (x-axis). Di in legends refers to D i:n .

Figure 3 :

 3 Figure 3: The role of Sotheby's pre-auction value estimate. Plot a) normalized values are divided by the number of bottles in the lot; dot sizes reflect non-normalized values.

  (x|z)dx-(b t +∆), (b t +x) dF + Dn:n|Z (x|z)dx-(b t +∆)], (14) for all t sold, and 0 otherwise, and with B b-1:b = b t and B b:b = b t + ∆ respectively the observed second-highest and highest "winning bid" equal to b t plus bidding increment.

Figure 4 :

 4 Figure 4: Heterogeneity of auction-level surplus CS t Plot a: point estimates of bounds, by decile of Z=Sotheby's pre-auction value estimate. Plot b: differences in point bounds as a share of the winning bid, by decile of Z. Vertical lines indicate sample medians.

Figure 5 :

 5 Figure 5: Policy simulation: π ∆ S (r) with reserve price rule r t = αz t

Table 1 :

 1 Descriptive statistics for sample of fine wine auctions The vintage is missing for non-vintage champagnes and for mixed lots of various vintages.

		N	Mean	St. Dev.	Min	25th pct. Median 75th pct.	Max
	Opening bid (pounds)	697 846.255	847.841	50	280	520	1,150	9,000
	Highest bid (pounds)	697 999.857	993.776	90	340	620	1,300	9,200
	Number bottles per lot	697	8.624	4.721	1	6	8	12	36
	Highest bid per bottle (pounds)	697 166.434	183.561	5	40	87.50	225	933.33
	Increment at highest bid (%)	697	5.623	1.949	2.041	4.348	5.263	6.667	16.000
	Is sold	697	0.902	0.297	0	1	1	1	1
	Number of bids	697	4.184	2.936	2	2	3	5	25
	Sotheby's low estimate (pounds)	697 917.805	980.857	80	300	550	1,200	13,500
	Sotheby's high estimate (% above low estimate) 697	27.418	6.582	9.091	23.077	27.273	30.769	62.500
	Special format (0.5l, magnum, double mag, ...)	697	0.063	0.243	0	0	0	0	1
	Vintage †	647	1999	9.787	1929	1995	2002	2006	2012
	Mixed lot (different wines)	697	0.088	0.283	0	0	0	0	1
	†:

Table 2 :

 2 Table 2 reports estimated bounds on E[CS t ], both by tertile of the conditioning variable and for the whole sample. Results highlight Estimated bounds on E[CS t ], by tertile of Z

		1st tertile	2nd tertile	3rd tertile	all auctions
	Observed:
	N (sold)	225	195	209	629
	Z, mean	29.981	96.546	341.194	154.025
	Winning bid	35.087	111.218	385.726	175.197
	Estimated bounds on E[CS t ]:
	Lower bound (based on V -n:n )	55.392	118.029	276.335	151.592
	95% Confidence interval

See e.g.[START_REF] Akbarpour | Credible auctions: A trilemma[END_REF]

Absentee bidding in English auctions is discussed previously in: Ginsburgh (1998),[START_REF] Rothkopf | Why are Vickrey auctions rare?[END_REF] and[START_REF] Thiel | Bidding behaviour in second-price auctions: Rare stamp sales, 1923-1937[END_REF].[START_REF] Lucking-Reiley | Vickrey auctions in practice: From nineteenth-century philately to twenty-first-century e-commerce[END_REF] finds that this practice has been used since at least 1878 for stamp auctions.

Electronic copy available at: https://ssrn.com/abstract=3622047

Specifically, when G(.) = U nif [0, ū], F -1 U (b) -F -1 (a) ≤ G -1 Y (b) -G -1 Y (a), ∀0 ≤ a ≤ b ≤ 1.

[START_REF] Bagnoli | Log-concave probability and its applications[END_REF] provide references to many papers that use the slightly stronger log-concavity shape restriction in the economics literature.

Source: http://www.sothebys.com/en/auctions/2014/finest-rarest-wines-l14711.html (last accessed June 11 2020).

 6 Otherwise, rational bidders would not find it optimal to place absentee bids. An auctioneer not representing (absentee) bidders truthfully would not go undetected as the distribution of winning bids for absentee bidders would be stochastically dominated by the winning bid distribution for live bidders. This goes beyond the detectability property of single deviations from truthful auctioneer behavior in[START_REF] Akbarpour | Credible auctions: A trilemma[END_REF] and would provide an empirical test of the protocol being truthful in practice (when the number of bidders is known).

See Hasker and Sickles (2010) and the references therein.

Also note that in Auction 1 there must be more than one absentee bidder as there are multiple absentee bids placed before live bidding starts, and any other explanation would violate that the auctioneer acts in the best interest of the absentee bidder. According to the stylized bidding sequence, both A 1 and A 3 are drop-out values of absentee bidders. Practically, due to the fixed bidding increments and rounding of absentee bids in my data, all these drop-out values are lower bounds on the valuations of the corresponding bidders.

This is for example done in[START_REF] Aradillas-López | Identification and inference in ascending auctions with correlated private values[END_REF]. They also focus directly on overcoming the censoring problem, in an incomplete model that allows for correlated private values. Their clever identification strategy exploits observed exogenous variation in the number of bidders.

I collected the data by simply registering as an online bidder, recording the complete auction, translating the video material into a dataset of bids, and adding lot descriptors from the catalogue.

This is done unconditional on Z but normalizing the variables by the number of bottles. The 99th (95th) percentiles of unconditional per-bottle V - n:n and V + n:n are respectively: 945.83 (3356.67) and 2419.50 (6540.00) pounds

.13 The test is done under the equilibrium assumptions that the distribution of the lowest (secondhighest) bid equals the distribution of lowest (second-highest) valuation, and performed separately for 3-20 bidders. Resulting high p-values when at least conditioning on Sotheby's low value estimate indicate the unlikeliness that the resulting parent distribution is not the same, justifying the IPV framework up to the equilibrium assumptions being valid. Some additional caution in interpreting

The former requiring at least two bids. The latter requiring at least three bids, and identification of an absentee drop-out value less than the second-highest bid. Exogenous variation in the unobserved number of bidders (if there is variation) and IPV guarantee that these selection criteria are inconsequential. Uniform consistency of the PDF requires h D-→ 0, h Z-→ 0, and T -h D-h Z-→ ∞ as T -→ ∞, and for the CDF that h Z+ → 0 and T + h Z+ → ∞ as T + → ∞.

Source: https://www.sothebys.com/en/glossary (last accessed June 11 2020).

Including in:[START_REF] Paarsch | Deriving an estimate of the optimal reserve price: An application to British Columbian timber sales[END_REF],[START_REF] Haile | Kevin and Robin Sickles. eBay in the economic literature: Analysis of an auction marketplace[END_REF],[START_REF] Tang | Bounds on revenue distributions in counterfactual auctions with reserve prices[END_REF][START_REF] Aradillas-López | Identification and inference in ascending auctions with correlated private values[END_REF],[START_REF] Coey | Ascending auctions with bidder asymmetries[END_REF][START_REF] Coey | Discounts and deadlines in consumer search[END_REF], either first estimating the latent value distribution or identifying this structural feature directly (within bounds) as in this paper.

Acknowledgement

I would like to thank Adam Rosen, Andrew Chesher, Phil Haile, Suehyun Kwon, Lars Nesheim, Áureo de Paula, and conference and seminar participants, for helpful comments. Thanks also to Sebastian Fahey for going over practicalities of Sotheby's auctions and to Alan Crawford for introducing us. All errors are my own. This work was support by a PhD Studentship from the Economic and Social Research Council.