Nathalie Fabbe-Costes
email: nfc@romarin.univ-aix.fr

Information management in the logistics service industry : A strategic response to the re-integration of logistical activities

Keywords: Logistics management, logistics information systems, logistics service providers, reintegration of logistical activities, information and skills management

The logistics services market has evolved over the last 30 years and the LSPs has been expanding their role in the logistics chains, but shippers benefit from considerable negotiating power and constantly evaluate the relative advantages of subcontracting or performing the work themselves in-house. The threat that shippers may reintegrate logistics activities, which has always existed, has become more acute in Europe over the past two years.

The management of information, expertise and knowledge is according to us a major strategical factor for LSPs to avert a major swing back to in-house operation. Information management is vital as shippers need to exert tight control over their logistical operations and to maintain their expertise in this area. Skills management is important for LSPs either to differentiate their services and/or to develop complex or innovative logistics services.

The management of LSPs, like that of all businesses, depends on three inter-related dimensions of competence : action, expertise and knowledge; and the need for effective collection, communication and processing of information exists on each dimension. In managing information, LSPs must establish congruence with the shippers' systems that can be achieved on one or more of the dimensions. This highlights the importance of the design of LICS and the supporting information and communication technologies to the furure competitiveness of LSPs. Close integration of the information systems of LSPs and shippers will be essential to maintain the mutual trust necessary for long term relationship and to maintain their ability to adapt to the changing business environment.

INTRODUCTION

Nobody would contest that since the beginning of the 1980s, logistics has become a real industry and that logistics suppliers have been playing an ever increasing role within logistics chains. This evolution had led to the development of partnerships between shippers and logistics service providers (LSPs) and alliances between LSPs [START_REF] Van Laarhoven | Logistics alliances: The European experience[END_REF]. Most of the research on this suject assumes that the general trend towards greater subcontracting of logistics activities will continue. Interviews recently undertaken accross Europe in association with CELO and McKinsey [START_REF] Celo | Operational Alliances in Logistics[END_REF] have suggested that logistics managers in some major corporations are considering reversing this trend and reintegrating logistical activities which they currently contract out. They also presented arguments to justify such action.

This finding has prompted further research involving a new programme of interviews with logistics managers, designed to provide answers to the following questions :

(1) Why should shippers be tempted to reintegrate certain logistics activities?

(2) What kinds of activities are affected?

(3) What actions might LSPs take to counter the return to in-house operations?

In an earlier paper [START_REF] Fabbe-Costes | Strategies developed by logistics suppliers facing the temptation for shippers to reintegrate logistics operations[END_REF], presenting the preliminary results of this research, three strategies that LSPs might adopt were identified: (1) building a coherent and homogeneous 'portfolio of strategic logistical services'; [START_REF] Celo | Operational Alliances in Logistics[END_REF] improve the management of three inter-related competencies (action, expertise and knowledge); (3) under certain circumstances, establish 'partnerships' with clients and/or 'alliances' with other LSPs.

In this paper, we explore in greater depth how LSPs can use the management of information and expertise to adress the threat to their business posed by shippers wishing to reinternalize logistics functions.

The paper begins by examining the main characteristics of the logistics supply industry and the reasons why shippers are expressing a desire to take some of their logistics operations back in-house. Later sections focus on the strategic importance for logistics suppliers of managing information at the action, expertise and knowledge levels.

CHARACTERISTICS OF THE LOGISTICS SERVICES INDUSTRY

Division of responsibility for logistical activities

The logistics services market has evolved over the last 30 years and during this period it has been contested by different types of organization. In addition to the LSPs which provide a third-party logistics service, many manufacturers and retailers have retained direct involvement in logistics, often using it as an extension of their core business. In some cases, the logistical activities of these different organizations are in competition, in others they are essentially complementary. The aspiration to extend their responsibility for logistical activities can be broadly summarized as follows.

Manufacturers often control their inbound logistics (supplies to the manufacturing plant) and their outbound logistics (supplies to customers or retail networks). At a time when stocks are being reduced, service levels improved and 'just-in-time' methods becoming more prevalent, manufacturers see powerful and structured logistics as a way of ensuring efficiency, reliability and durability.

Retailers whose main concern is the management of sales outlets have, in many cases, seized control of the inward flow of supplies. This enables them, among other things, to regulate deliveries and reduce the level of inventory. The ever-increasing size and number of their retail outlets plus their purchasing power allow them to take full advantage of the economies of scale in transport and storage. By integrating logistical activities, retailers can improve their productivity and increase the reliability of shop replenishment.

Logistics service providers, acting as an interface between the logistics systems of both manufacturers and retailers, tend to exploit this potentially favourable situation by building up a 'neutral' multi-manufacturing and/or multi-retailer logistical service, able to operate shared distribution operations. In theory, the logistics supplier is then in a strong position to co-ordinate flows. A significant proportion of third party logistics services are, nevertheless, provided on a dedicated contract basis.

The same logistics activity can be performed by any one of the three partners in the logistics chain, depending on how far they choose to extend their original activity. It is not possible, on an a priori basis, to determine the optimal allocation of responsibility for logistics between the three agencies. Nevertheless, the LSPs has been expanding their role in the logistics chains, promoting themselves as offering greater specialization and professionalism.

Evolution and characteristics of the logistics supply industry

In France and several other European countries, the logistics supply industry has developed through four main stages.

Stage 1. The 1970s saw the arrival of the first generation of logistics suppliers resulting, in some cases, from some manufacturers hiving-off their in-house distribution operations. These companies undoubtedly pioneered the development of logistics skills, initially in the areas of warehousing and forwarding. Many subcontracted their transport needs to small hauliers.

Stage 2. Logistics know-how gradually spread to road transport companies many of which offered only regional coverage. They took advantage of this opportunity to diversify, by supplementing their original transport services with other services. National networks began to appear and consolidated the services of several regional companies. These operators, which, in many cases, became logistics suppliers to large manufacturers, combined their respective flows to provide consignee groupages, mainly for supermarkets. This period also saw some LSPs being 'spun-out' of retail organizations, which thereafter not only serviced the logistical needs of the parent company but also those of wholesale affiliated.

Stage 3. At the end of the 1980s the demand for third-party logistical services was extreme buoyant. Many LSPs began to specialize and provide state-of-the-art expertise in particular types of logistic services. Some catered for the needs of large-scale retail companies for which they set up customized solutions. Others, taking advantage of the know-how they had acquired at higher levels in the supply chain concentrated on supplying industrial plants often on a just-in-time basis. They could pre-assemble components to deliver a complete module to an assembly plant. Some of these LSPs focused their attention on the upper reaches of logistics chains and became involved primarily in the movement and storage of spare parts. Finally, some wholesalers began to provide 'pure' third-party logistic services (e.g. pharmaceutical products wholesalers), and a number of them even found themselves devoting equal time to logistics services and their traditional wholesale activities.

Stage 4. The most recent phase of this progression has been the quest for new markets. This is well exemplified by the growth of third-party reverse logistics services in the collection and recycling of waste.

The logistics services industry today appears to be particularly complex and segmented into several distinct sectors. The wide differences between LSPs reflect their differing origins and subsequent diversification and specialization. They offer broad range of experience, management culture, knowledge and know-how. The industry is undoubtedly maturing, though is still highly fragmented. This fragmentation makes the industry intensively competitive. Some LSPs that at the outset operated in separate markets segments are now to a certain extent in competition. 'Hierarchism' has also developed within the industry, with the 'leading' logistics suppliers organizing the sections of the chain under their control and subcontracting the activities they consider less profitable to smaller operators that usually have less room to manoeuvre. Where market niches exist they tend to be risky and temporary.

The high degree of fragmentation in the industry is largely a consequence of the fact that there are few barriers to entry. These barriers are higher, however, where the service is more complex and capital-intensive, where there is a rapid rate of innovation or where firms are required to provide wide geographical coverage. Where barriers are low, services are usually substituable (e.g., between modes of transport) and shippers can often switch traffic between carriers at minimal cost. As the slightest failure may rapidly prove disastrous, the position of the LSPs operating in 'spot' markets is always precarious.

Shippers benefit from considerable negotiating power, and their management expertise is often greater than that of the LSPs on whom they impose their logistics solutions. They are also loath to commit themselves over long periods and prefer short-term contracts (especially in France). Their experience of controlling logistics activities encourages them constantly to evaluate the relative advantages of subcontracting or performing the work themselves in-house.

This industry, which was able to find a certain equilibrium at the end of the 1980s, is again restructuring. Some factors threatening to upset the existing balance are inherent in the logistics supply industry, in particular the increased competition and the hierarchical organization within the industry. Other more significant factors are linked to the development of shipper strategy, in particular the threat that shippers may reintegrate logistics activities. This threat, which has always existed, has become more acute in Europe over the past two years even though it is still only psychological and not yet manifest in company behaviour.

POTENTIAL REINTEGRATION OF LOGISTICS OPERATIONS

The strategic nature of logistics operations for shippers

According to a study carried out in France by BIPE [START_REF] Bipe Conseil | La logistique dans l'industrie et la grande distribution en France[END_REF] 'a product spends 15% of its time being manufactured and the remaining 85% in flow operations (storage, transfer, transport, etc.).... and the cost of logistics can be estimated at approximately 12% of turnover'. Controlling these logistics times and costs is a strategic challenge for shippers. The impact that logistics has on a company is measured not only in cost terms but also, to an ever increasing extent, by the quality of the logistics service. The 'logistics service' is an integral part of firms' market offering and strongly influences their competitiveness. Two surveys carried out by the American consultancy firm A.T. Kearney show there have been improvements in the quality of logistic services in large European companies: an average reduction of 31% of service failures between 1987 and 1992 and an expected reduction of 60% between 1992 and 1997. It also has been found that logistics costs are increasing as a proportion of value added [START_REF] Kearney | Logistics Productivity: The Competitive Edge in Europe[END_REF][START_REF] Kearney | Qualité et productivité logistique en Europe[END_REF].

This raises a difficult question. Should companies continue to subcontract to take advantage of the skills of a service industry that is becoming increasingly professional or should they reintegrate (or develop) at least partly an activity that has such an important influence on company performance? According to the study carried out by BIPE, 85% of manufacturing companies in France use logistics subcontractors, the cost of subcontracting representing 55% of logistics costs.

One firm which has decided to retain a large direct involvment in logistics is Philips-Eclairage. It decided to make a substantial capital investment (several hundred million francs) in a highly automated warehouse. It supplies customers (70% wholesalers, 15% supermarkets, 15% fitters) located in France and in certain areas of bordering countries with central supply facilities. Philips-Eclairage has managed to enhance its responsiveness and reduce delivery time (from a maximum 72 to 48 h). Although still heavily engaged in warehousing operations, the firm contracts out the final delivery of its products.

Possible motives for the reintegration of logistics functions

Considering the importance of the logistics service, it seems logical that shippers should weigh up the advantages of maintaining their subcontractors or reintegrating all or part of their subcontracted operations. Let us consider the motives for reintegration in greater detail.

The main temptation for shippers to reintegrate logistics operations is often the desire to assert direct control over the cost and quality of these activities. When activities are subcontracted, this control is linked directly to knowing how to get things done. In view of the rapid developments made in the logistics profession, a number of shippers, that have relinquished the management of logistical activities, are concerned that they are not updating their expertise in this area and becoming uncertain about the net benefits of continuing to contract out their logistics. They are tempted to reintegrate, at least in part, some logistics activities if only to reacquaint themselves and establish a firmer basis for evaluating logistical performance. They are even prepared to subcontract this work out again at a later stage, applying new criteria.

An example of this is J. Sainsbury, the British supermaket chain, which uses contractors to operate twenty of their twenty-four distribution centers. As stated by Quarmby, the former Managing Director of the company, 'there are two reasons for continuing to own these four centers. Firstly we can continue to obtain detailed knowledge of this profession and experience in managing operational tasks. Secondly, it provides us with an opportunity to try out new techniques and new methods without having to persuade a subcontractor to perform what could for him prove a risky investment'.

Another motive that shippers might have for reintegrating logistics operations is to reduce the strategic risk they take when they entrust to logistics suppliers activities from which they gain a significant competitive advantage. The level of risk is determined by the operating reliability of the logistics supplier which, in turn, influences the loyalty of the shipper's customers. The level of risk is also determined by the strategic 'discretion' (i.e. ability to withhold from competitors the same services and/or not disclose the relevant know-how) that influences the ability of the other firms to compete. The LSP's reliability and discretion are key factors affecting the sustainability of the shipper's competitive advantage.

Another important factor when choosing between subcontracting, integration or reintegration, is the level of investment required to perform logistics activities, and the ability of companies to make an adequate return on these investments before they are rendered obsolete by new logistics developments. For quite some time, this factor gave manufacturers and retailers a good reason to reduce their direct involvement in logistics, particularly for transport. In restructuring their logistical operations at a European scale, some manufacturers are now finding that the economies of scale they can achieve in consolidating their operations are making direct investment in logistical facilities more attractive. They can show a healthy return on these large scale investments, particularly as their size and status enables them to borrow from financial institutions at somewhat lower rates than those obtained by logistics suppliers.

The final factor influencing the decision to reintegrate logistics activities is that of personnel. Since the 1980s, the increasing complexity of logistics facilities has been combined with rapid developments in logistics information and communication systems (LICS). At the same time, the qualifications of logistics staff have risen, particularly in warehousing, reflecting the development of more advanced handling and storage technologies. The differences in technical skills required by manufacturing, retailing and logistics activities are lessening, which makes manufacturing and retailing companies more inclined to reintegrate these operations. Moreover, the management of logistics personnel is no longer a hindrance to the reintegration of these activities, as the mechanization of warehouses has reduced the need to employ large numbers of poorly qualified workers. However, this is only feasible for companies handling with sufficient volumes to justify investment in highly mechanized systems, and for companies with financial clout, which is often not the case for LSPs.

Logistical operations most likely to be reintegrated

Discussions with European shippers have revealed that logistics operations are more likely to be reintegration than others. The following examples were given. They are neither mutually exclusive nor presented in order of importance. Shippers only envisage reintegration for : • operations that directly concern customer services. The more an operation affects the shipper's customer, the more it is seen to be sensitive. This confirms that logistics has become a strategic marketing element for many manufacturing and retailing companies. • operations that require the handling of information thought to be 'strategic' (usually pertaining to customers and/or cost structure). • operations considered to be 'critical' in the shipper's logistics process (i.e. in terms of their service and/or logistics costs), or more precisely, operations in which security is more important than flexibility. • operations that do not require a large amount of new investment, particularly those for which physical or information systems have already been set up within the company. • operations that offer a rapid rate of return on investment in logistics assets.

• operations that require highly specific investment and which LSPs, especially those providing mainly shared-user services, find difficult to run profitably.

• operations that do not require staff with a different culture than that already prevalent in the company. • operations that require highly specific skills, not available from a logistics supplier.

It can be seen that the increasing need for either dedicated investments and specialist skills can act both to encourage and discourage reintegration. Where these skills are innovative and/or closely tailored to a shippers' operations, a move back to in-house operation is favoured. Other shippers can obtain the specialist skills they require from existing LSPs and hence are more inclined to continue contracting out their operations.

The decision to continue subcontracting or, on the contrary, to reintegrate logistics activities is in any event contingent. First it depends on the situation of the logistics supply industry at the time when the issue arises. It also depends on the shipper's perception of the relative risks and benefits of the two options. Moreover, this dilemma usually only arises when a shipper is conducting a strategic review of its logistical system. Reinternalization can then occur as part of a wider restructuring of a firm's logistics. Van Laarhoven [START_REF] Van Laarhoven | The new logistics challenge : battle for value-added[END_REF] indicates that 43% of the fifty European LSP shipper contracts that he examined in 1994 were renegotiated in that year. The result of the renegotiations were that 82% of shippers renewed the contract with the same LSP, 9% changed LSP, and 9% decided to bring logistics back in-house.

The reintegrating of these activities poses an obvious threat to LSPs. The threat is all the more important because it is the larger companies that are today considering this option. They demonstrate that, where logistics is concerned, no decision is ever permanent. The fact that shippers do not have logistics infrastructure currently in place does not guarantee a future role for LSPs. Shippers, in particular manufacturing companies, often have the financial resources required to make the necessary investments and, in many cases, either have or can acquire personnel to run their logistical operations.

While there may be an increase in the proportion of logistics activity performed in-house, much logistics work will continue to be contracted out, particularly in the case of transport operations. Nevertheless, the potential for reintegration is now great enough to require LSPs to give serious thought to how they might counter this trend. Indeed the likehood of this trend becoming established will partly depend on the measures that LSPs adopt. It is important, therefore, that they consider what strategic actions will be needed to avert a major swing back to in-house operation. Our purpose here is not to formulate contingency plans for LSPs, but to examine the factors that are likely to figure prominently in their future strategies, namely the management of information, expertise and knowledge.

MANAGEMENT OF INFORMATION, EXPERTISE AND KNOWLEDGE

Why should logistics suppliers focus on information and skills management ?

Because of competition in the logistics supply industry, shippers can now expect high standards when they contract out their logistical activities. Those standards relate both to physical operations and to the processing of information. The quality of information support is vital as shippers need to exert tight control over their logistical operations. Shippers also want detailed operational information to acquaint themselves with physical operations and maintain their expertise in this area. They are also interested in the security of an LSP's information system, because they do not want anyone to have access to information they consider commercially sensitive. Information management is also fundamental to efficient and reliable operation of LSP's business. Today, logistic organizations are working with much smaller safety stocks and, as a result, the effects of any disruption are rapidly transmitted along the logistics chain, often to the end customer. LSPs need an information and communication system directly linked to the physical flow of goods to forecast, plan, organise and follow the operational process and react against any disruption. Information management is thus a core activity and basic offering of LSPs.

In reviewing their strategies, logistics suppliers must also take account of two other factors : (1) the fiercer the competition, the greater will be the risk of shippers switching to another LSP, but the risk of shippers reintegrating logistical activities will be reduced; and

(2) the greater the degree of expertise required, the greater will be the risk involved in providing the service, the higher will be the barriers to entry and hence the less intense will be the competition.

These factors highlight the importance of skills management for LSPs either to differentiate their services and/or to develop complex logistics services. To provide shippers with specialized or innovative services, logistics suppliers must be able to develop state-ofthe-art expertise, to convince shippers of their ability to provide a superior service and to avoid divulging the relevant know-how. Evolution of skills is another key factor to consider. As specialist niches are risky and/or temporary, logistics suppliers must be able to adapt, or to develop, the expertise needed to exploit these niches very quickly.

Advances in technology are increasing the skills required to operate logistical systems. Skills management is thus crucial for LSPs who used to employ poorly qualified workers and experienced a relatively high turnover of staff. They need to enhance skills and learn how to consolidate experience if they want to perform sophisticated logistics operations.

Finally, it should be recognized that LSPs are becoming increasingly involved in consultancy and engineering services. In tendering for contracts they are often required to redesign a client's logistical system, providing an important value-added service, for which they are only compensated if they secure the contract.

Skills management

Most of the research so far done on LSPs has concentrated on their mode of operation in terms of physical activites and information processing. Few attempts have been made to assess their overall competence. It is true that logistics suppliers are largely percieved through the logistics operations they perform for their contractors. These operations are the manifestation of higher-level organizational structures and processes.

The management of LSPs, like that of all businesses, depends on three inter-related dimensions of competence : action, expertise and knowledge. These provide LSPs with many ways of differentiating themselves from competitors and strategically positioning themselves vis-à-vis shippers :

Action relates to the way in which logistics services are actually performed. The nature of this physical service largely determines the entry barriers.

Expertise relates to all the resources directly associated with the Action, including methods, procedures, technologies and engineering. They strongly influence the quality, efficiency, durability and reliability of the logistics services and are a major source of competitive advantage.

Knowledge represents the highest level of abstraction in the management of LSPs. It is high-level information that must be collected and assimilated in the course of formulating strategy. It also incorporates the experience of senior managers and the general management culture of the business. To survive, an LSP must evolve and adapt. This can only be achieved by mobilising knowledge. It is important, therefore, that LSPs invest in high level management skills and in R&D, and try to convert into knowledge everything that is experienced and perceived by people in the company.

Figure 1 illustrates the inter-relationship between the three dimensions of competence. An upward arrow signifies a demand from a lower dimension, while a downward arrow represents the effect of higher level knowledge on expertise and, in turn, operating procedures. A strengthening of an LSP's knowledge and expertise can result in the development of more complex services, which may be difficult for shippers to provide on an in-house basis. Unfortunately, however, many LSPs have been neglecting their knowledge base. The restructuring of some of these businesses has resulted in the down-sizing of planning departments and loss of senior management staff with extensive knowledge of the industry.

Intrinsic to the development of more complex logistics services is information management [START_REF] Fabbe-Costes | Les systèmes d'information et de communication au coeur de la stratégie des prestataires logistiques?[END_REF][START_REF] Fabbe-Costes | Logistic information and communication systems (LICS) are producing organization meanings[END_REF]. The need for effective collection, communication and processing of information exists on each of the dimensions. Information management has therefore become a strategic factor for LSPs, especially those that wish to add value to the service offering.

ACTION

Scope for improving information management

Before examining information management in detail, it is necessary to consider how changes in relationship between LSPs and shippers are affecting the management of information in the logistics chain, particular for those logistics operations that have been identified as being 'sensitive' i.e. most likely to be re-integrated.

• Shippers are currently reducing the number of separate LSPs that they use. This is encouraging the formation of stronger inter-company links, particularly at the Action level where day-to-day operations are performed and the operating staff on both sides are in close contact. • When 'sensitive' logistics operations are subcontracted, shippers often seek a high degree of transparency to be able to audit and control the operations closely. This is conducive to the development of joint methods and procedures at the Expertise level [START_REF] Fabbe-Costes | Formulating logistics strategy[END_REF]. • The development of closer, more open relationships results in greater willingness to pool information and knowledge. This can lead to the establishment of a joint frame of reference at the Knowledge level, a body of logistics knowledge common to both organizations.

Despite the increasing temptation to reintegrate logistics operations examined above, all of these trends indicate that shippers are not against contracting out their logistics in principle, so long as information management enables them to control the logistics service and maintain their ability to adapt to the changing business environment.

For LSPs, information management offers a means of raising 'exit barriers' for shippers by increasing their substitution costs and creatinga means of adding value to their services. In managing information, LSPs must establish congruence with the shippers' systems. Depending on the degree of co-operation, this congruence can be achieved on one or more of the dimensions outlined above. LSPs must then develop LICS that are capable of dealing with the three dimensions of competence and can be applied to all partners in the chain, in particular shippers.

In the Action dimension, LICS should help LSPs to improve the responsiveness of their operations. This is the basis of service performance, quality and reliability. At this level, LICS must capture and record all data produced by the logistics process (i.e. information directly generated at the operational level). The main objective is to build a complete database, if possible updated in real time, of what has happened and is happening in the logistics chain. It allows logistics suppliers to inform shippers of ongoing activities (to control operations and track physical flows) and to provide them with measures of the effectiveness and efficiency of the subcontracted activities.

LICS have been traditionally confined to the Action level. As explained above, however, this is no longer sufficient. LICS must also be designed as 'knowledge-based management systems'. At the Expertise level, LICS can help to formalize procedures and control systems, in line with the current move towards ISO 9002 certification. For this purpose, LICS can help to monitor logistics indicators, and can keep records of service failures on the part of both the LSP and the shipper. The analysis of these indices can lead to a significant improvement in existing systems. It also can help LSPs and their clients to co-operate in the development of new logistics expertise. At this level, LICS not only collect and record raw operating data but also are enriched with 'contextual information' that help managers to undertake statistical analysis and identify trends. By acting as a decision-support tool, LICS can help LSPs formulate 'behavioral rules' in their business, improve their procedures and knowledge, and detect trends in the business environment.

In addition to providing shippers with detailed operating data, LSPs can also help them understand the workings of the logistics system, updating their logistics know-how without tempting them to reinternalize the logistical function. This second level of LICS is currently developing, but there is still little sign of LICS being developed at the higher Knowledge level. At this level LICS could play three different roles.

The first role, which is quite widely recognised though insufficiently developed, is to furnish senior management with high-level strategic information, much of it relating to business trends and forecasts.

The second is to assist the corporate learning process, helping firms assimilate the information and adapt their systems accordingly. A key element in logistics competence is its capacity to adapt the planning process to what is happening in the external business environment. If LICS provide a better (i.e. more adaptative, more evolutive) knowledge about logistics, it can help companies to find new management responses to changes in the business environment.

The third role is to capitalise on the collective knowledge on logistics within the organization. In theory, communication networks should be created internally to permit the assembly and exchange of all knowledge relating to the physical flow of goods. Once structured and distilled, much of this knowledge could be shared with shippers.

This highlights the importance of the design of LICS and the supporting information and communication technologies to the furure competitiveness of LSPs. Close integration of the information systems of LSPs and shippers will be essential to maintain the mutual trust upon which secure, long term relationship are built. The growth of Electronic Data Interchange (EDI) and Internet use is evidence that this integration is well underway. LSPs now need to build up their skills in the areas of information and communication design and in the associated technologies.

CONCLUSIONS

The logistics supply industry is currently maturing but remains relatively segmented. This competitive market is now facing a potentially threatening trend as major shippers that currently contract out their logistics activities are considering reintegrating some of them. Analysis of shippers' motives for reintegrating reveals that LSPs must recognize and react to this risk, even though the general principle of contracting out is still generally accepted. The extent to which the move back to in-house operations occurs will be directly linked to the attitudes and strategies that LSPs now adopt. Faced with this potential loss of business, LSPs must consider taking preventative measures. The main focus of these measures should be on information and skills management. We have identified three inter-related dimensions of competence in which information need to be more effectively managed : action, expertise and knowledge. LSPs should invest substantially in the development of information management at the upper two levels which have so far, in many firms, been relatively neglected. LSPs must develop LICS that deal with the three dimensions of competence. By doing so, they should be able to secure their long term position by addressing shippers current concerns about contracting out their logistics.

  and/or effect between two dimensions of competence the logistical operating process and physical flow the cross dimension of information management

Figure 1 .

 1 Figure 1. Inter-relationship between the three dimensions of competence

Acknowledgments

I am grateful to Professor Jacques Colin of CRET-LOG for allowing me to use the results of our collective research in this paper. I also thank Professor Alan C. McKinnon for his careful rereading of the paper.


L Galtchouk
email: galtchou@math.u-strasbg.fr

S Pergamenshchikov
email: serge.pergamenchtchikov@univ-rouen.fr

Efficient adaptive nonparametric estimation in heteroscedastic regression models *

Keywords: adaptive estimation, asymptotic bounds, efficient estimation, heteroscedastic regression, nonparametric estimation, non-asymptotic estimation, Oracle inequality, Pinsker's constant AMS (1991) Subject Classification : primary 62G08; secondary 62G05, 62G20

An adaptive nonparametric estimation procedure is constructed for the estimation problem of heteroscedastic regression. A non-asymptotic upper bound for the quadratic risk (the Oracle inequality) is obtained. Asymptotic efficiency of this procedure is proved, i.e. Pinsker's constant is found in the asymptotical lower bound for the risk. It is shown that the asymptotical quadratic risk for the constructed procedure coincides with this constant.

Introduction

Suppose we are given observations (y j ) 1≤j≤n which obey the heteroscedastic regression equation y j = S(x j ) + σ j (S) ξ j ,

where design points x j = j/n, S(•) is an unknown function to be estimated, (ξ j ) 1≤j≤n is the sequence of i.i.d. random variables, (σ j (S)) 1≤j≤n are unknown scale functionals depending on unknown regression function S and the design points. Our goal is to estimate S in the mean quadratic sense in the both non-asymptotic and asymptotic setups when the smoothness of S is unknown. Note that heteroscedastic regressions with this type of scale functionals has been encountered in consumer budget studies utilizing observations on individuals with diverse incomes and in analyses of the investment behavior of firms of different sizes (see, for example, [START_REF] Goldfeld | Nonlinear Methods in Econometrics[END_REF]). Such kind of models was considered by [START_REF] Gunst | Regression analysis and its applications: a data oriented approach[END_REF], [START_REF] Efroimovich | Nonparametric curve estimation. Methods, theory and applications[END_REF], [START_REF] Akritas | ANCOVA methods for heteroscedastic nonparametric regression models[END_REF] as well.

The problem of minimax estimation of nonparametric regression function in homoscedastic case (i.e. σ j ≡ σ) in the asymptotic setup has been studied in a number of papers. The optimal rate for L 2losses and regression functions from a L 2 -Sobolev or a Hölder space was studied by [START_REF] Ibragimov | Bounds for the risk of nonparametric regression estimates[END_REF], [START_REF] Speckman | Spline smoothing and optimal rate of convergence in nonparametric regression models[END_REF], [START_REF] Donoho | Wavelet shrinkage : asymptotia?[END_REF]; the case of regression functions from Triebel and Besov space was investigated by Donoho and Johnston (1998); the optimal rate for model (1.1) in L q -losses and adaptive estimate for Sobolev spacial regression was considered by Nemirovskii (2000). Efficient linear estimators for L 2 -risk over linear estimators and some other risks was given by [START_REF] Donoho | Geometrizing rates of convergence[END_REF], [START_REF] Donoho | Statistical estimation and optimal recovery[END_REF].

The notion of asymptotic optimality is usually associated with convergence rate of the minimax risk (see for example [START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF], [START_REF] Stone | Optimal global rates of convergence for nonparametric regression[END_REF]). An important question in the development of nonparametric estimation is to study the exact asymptotic behaviour of minimax risk and to find an efficient estimator, i.e. an estimator which achieves this asymptotics.

The optimal constant and efficient estimators for L 2 -losses was obtained by [START_REF] Nussbaum | Spline smothing in regression models and asymptotic efficiency in L 2[END_REF], [START_REF] Golubev | Asymptotic minimax estimation of regression function in additive model[END_REF], [START_REF] Golubev | Adaptive spline estimates in a nonparametric regression model[END_REF], for sup-norm losses by [START_REF] Korostelev | Exact asymptotically minimax estimator for nonparametric regression in uniform norm[END_REF]. For the absolute error loss and the estimation of regression function S at a fixed point, the optimal constant and efficient estimators was found by [START_REF] Galtchouk | Asymptotically efficient estimates for nonparametric regression models[END_REF].

A non-asymptotic approach for nonparametric estimation problem in the model (1.1) with σ j ≡ σ was studied in a few papers. A non-asymptotic upper bound for quadratic risk over thresholding estimators is given by [START_REF] Kalifa | Thresholding estimators for linear inverse problems and deconvolutions[END_REF]. [START_REF] Barron | Risk bounds for model selection via penalization[END_REF], [START_REF] Massart | A non-asymptotic theory for model selection[END_REF] have constructed an adaptive procedure of model selection based on least squares estimators and have obtained a non-asymptotic upper bound for quadratic risk which is best in the principal term for given class of estimators. This type of upper bounds is called the Oracle inequalities.

It should be noted that to obtain efficient estimators (in asymptotic setup), in contrast with least squares estimators [START_REF] Nussbaum | Spline smothing in regression models and asymptotic efficiency in L 2[END_REF] and later [START_REF] Golubev | Adaptive spline estimates in a nonparametric regression model[END_REF] have used some special class of weighted least squares estimators, where the weights are depending on regularity of unknown function and was chosen specially to obtain the optimal constant in quadratic risk (so-called the Pinsker constant).

By reason of asymptotic efficiency, we make use of Golubev-Nussbaum estimators to obtain non-asymptotic upper bound for quadratic risk in this paper. The main distinction of our approach from that of Barron-Birge-Massart is the following one. We choose a family of estimators such that the every estimator is optimal when the regularity of unknown function is fixed and a basis is given. In contrast with our approach, in the selection model theory one chooses a family of basis (models) for which a least squares estimator will be constructed.

In this paper an adaptive procedure is proposed which is based on weighted least squares estimators and the non-asymptotic upper bound for the quadratic risk is obtained which is best in the principal term for the chosen family of estimators. Moreover, at the same time the asymptotic properties of the proposed procedure are studied as n → ∞. It turns out that this procedure is asymptotically efficient, that is the asymptotic quadratic risk coinsides with the lower bound for all estimators, i.e. with the Pinsker constant.

The paper is organized as follows. In the next section we formulate the problem and give main results. In section 3 we construct an adaptive estimation procedure based on Nussbaum's estimators and we obtain a nonasymptotic upper bound for the quadratic risk of this procedure. In section 4 we prove the sharp asymptotic upper bound for the quadratic risk of this estimator. In section 5 the sharp asymptotic lower bound for the minimax risk is obtained. An appendix contains some technical results.

Problems and Results

First we consider the model (1.1) in which the sequence (ξ j )

1≤j≤n is i.i.d. with E ξ 1 = 0 , E ξ 2 1 = 1 and E ξ 4 1 = ξ * < ∞ . (2.1)
We introduce the Sobolev class W q r as

W q r = {S ∈ C q-1 [0, 1] : q j=0 S (j) 2 ≤ r} , (2.2)
where S (j) is the j-th derivative of S, r is some positive constant, q ≥ 1 is an integer,

S 2 = 1 0 S 2 (x)dx .
For any estimate Ŝn of S, based on observations (y j ) 1≤j≤n , we use the following quadratic risk

R n ( Ŝn , S) = E S Ŝn -S 2 n , (2.3)
where

S 2 n = 1 n n j=1 S 2 (x j ) .
In the next section we construct an adaptive procedure (see below (3.8)) for which we obtain a non-asymptotic upper bound for the quadratic risk. To construct this procedure, we make use of the estimator family introduced by [START_REF] Nussbaum | Spline smothing in regression models and asymptotic efficiency in L 2[END_REF], see below (3.4)- (3.5), and the approach proposed by [START_REF] Golubev | Adaptive spline estimates in a nonparametric regression model[END_REF] for the homoscedastic case, i.e. σ l ≡ σ. The key idea in the construction of this procedure is the following one. We replace the unknown variance σ 2 in the Golubev-Nussbaum estimation procedure by some estimator for n -1 n j=1 σ 2 j . For this procedure we obtain the nonasymptotic upper bound for the risk (2.3) which is best for the choosen family of estimators (see Theorem 3.3). This type of upper bound is called Oracle inequality.

To study asymptotic properties, we need to impose some additional conditions on the sequence (σ j ) in (1.1).

H 1 ) σ j = g(x j , S) for some unknown function g :

[0, 1] × L 1 [0, 1] → R + ,
which is square integrable with respect to x such that

lim n→∞ sup S∈W q r 1 n n j=1 g 2 (x j , S) -ς(S) = 0 , (2.4)
where ς(S) := 1 0 g 2 (x, S)dx. Moreover there exist some reals 0 < g * ≤ g * < ∞, for which

g * ≤ inf 0≤x≤1 inf S∈W r q g(x, S) ≤ sup 0≤x≤1 sup S∈W r q g(x, S) ≤ g * .
(2.5)

H 2 ) The function g(x, S) is differentiable in the Frechet sense with respect to S in L 1 [0, 1] uniformly over 0 ≤ x ≤ 1, i.e. for any S, S 0 from

L 1 [0, 1] g(x, S) = g(x, S 0 ) + L x,S 0 (S -S 0 ) + Υ(x, S 0 , S) ,
where the linear operator L x,S 0 (the Frechet derivative) is bounded uniformly over 0 ≤ x ≤ 1 in L 1 [0, 1], i.e. for any S 0 from L 1 [0, 1] there exists some positive constant C * = C * (S 0 ) such that

sup 0≤x≤1 sup S∈L 1 [0,1] , S =0 | L x,S 0 (S) | S 1 ≤ C * (2.6)
and the residual term Υ(x, S 0 , S) satisfies the following property

lim S 1 →0 sup 0≤x≤1 |Υ(x, S 0 , S)| S 1 = 0 ,
where

S 1 = 1 0 |S(x)| dx. H 3 ) The function g 0 (x) = g(x, 0) is continuous on the interval [0, 1].
Now we formulate the main asymptotic results.We set

γ q (S) = C * q r 1/(2q+1) (ς(S)) 2q/(2q+1) , (2.7)
with

C * q = (2q + 1) 1/(2q+1) q π (q + 1) 2q/(2q+1)
.

It is well known (see, for example [START_REF] Nussbaum | Spline smothing in regression models and asymptotic efficiency in L 2[END_REF]) that the optimal rate is ϕ n = n 2q/(2q+1) in the case when S ∈ W q r . Theorem 2.1. Assume that in the model (1.1)-(2.1) the sequence (σ j ) fulfils the condition H 1 ). Then the estimator S * n defined by (3.8) with ε = 1/ ln(n + 1) satisfies the inequality

lim sup n→∞ ϕ n sup S∈W q r 1 γ q (S) R n (S * n , S) ≤ 1 (2.8)
Moreover we will show that the estimator S * n is efficient in the following sense.

Theorem 2.2. Assume that in the model (1.1) the sequence (ξ j ) is i.i.d. ∼ N (0, 1) and the sequence (σ j ) fulfils the conditions H 1 )-H 3 ). Then the risk (2.3) admets the following asymptotic lower bound

lim inf n→∞ inf Ŝn ϕ n sup S∈W q r 1 γ q (S)
R n ( Ŝn , S) ≥ 1 .

(2.9)

Remark 2.1. Inequalities (2.8) and (2.9) imply that γ q (S) is the so-called Pinsker's constant, i.e. the sharp asymptotic lower bound for the quadratic risk.

3 Non-asymptotic estimation

In this section we construct the estimation procedure for the model (1.1)-(2.1). We use the trigonometric basis {φ j , j ≥ 1} in L 2 [0, 1] with

φ 1 (x) = 1, φ j (x) = √ 2T r j (2π[j/2]x) , j ≥ 2 , (3.1)
where [a] is the integer part of a real a, T r j (x) = cos x for even j , sin x for odd j .

It is easy to see that for this basis

(φ i , φ j ) n = 1 n n l=1 φ i (x l ) φ j (x l ) = 0 for i = j and 1 ≤ i, j ≤ n. Moreover, φ j 2 n = 1 for j ≤ n -1 and φ n 2 n = ν n with ν n = 2 for even n , 1 for odd n . (3.2)
Now by making use of the discrete Fourier transformation we reduce the model (1.1) to the model

θj,n = θ j,n + 1 √ n ξ j,n (3.3)
with θj,n = (y, φ j ) n , θ j,n = (S,

φ j ) n , ξ j,n = 1 √ n n l=1 σ l ξ l φ j (x l ) .
For any given ε > 0, let us define a 2-dimensional parameter τ = (β, t) with values into the set A ε = {1, . . . , β * } × {t 1 , . . . , t m }, where

β * = [1/ √ ε], t i = iε for 1 ≤ i ≤ m = [1/ε 2 ]. Notice that the number of elements in the set A ε is υ ε = card(A ε ) = β * • [1/ε 2 ].
For any τ ∈ A ε we set

ρ τ (j) = 1 for 1 ≤ j ≤ k 0 (τ ) Ψ β (j w -1 τ ) for j > k 0 (τ ) , (3.4)
where

Ψ β (z) = (1 -z β )1 (|z|≤1) , w τ = n 1/(2β+1) c τ , c τ = π -2β/(2β+1) (t/ι β ) 1/(2β+1) , ι β = β (β + 1)(2β + 1) , k 0 (τ ) = w τ / ln(n + 2) .
For every τ ∈ A ε , we define the following estimator

Ŝτ (x) = n j=1 ρ τ (j) 1 φ j 2 n θj,n φ j (x) . (3.5)
In the sequel we use the following weighted Fourier transformation of S:

S τ (x) = n j=1 ρ τ (j) 1 φ j 2 n θ j,n φ j (x) .
To estimate the unknown function S by estimators (3.4), we should find τ from A ε which minimizes the loss function

Ŝτ -S 2 n = n j=1 ρ 2 τ (j) θ2 j,n φ j 2 n -2 n j=1 ρ τ (j) θj,n θ j,n φ j 2 n + n j=1 θ 2 j,n φ j 2 n (3.6)
or equivalently, minimizes the function

n j=1 ρ 2 τ (j) θ2 j,n φ j 2 n -2 n j=1 ρ τ (j) θj,n θ j,n φ j 2 n .
Since the coefficients θ j,n are unknown, we will find τ which minimizes the function

J n (τ ) = n j=1 ρ 2 τ (j) θ2 j,n φ j 2 n -2 n j=1 ρ τ (j) θ2 j,n -ςn /n φ j 2 n ,
where ςn is some estimator of

ς n = 1 n n l=1 σ 2 l . (3.7)
The estimator ςn will be constructed below. Denote by τ = ( β, t) the argmin of the function J n (τ ), i.e.

J n (τ ) = min τ ∈Aε J n (τ ) .
We denote

S * n,ε = Ŝτ . (3.8)
In this section we study this estimator in the non-asymptotic setup.

Theorem 3.1. Assume that in the model (1.1)-(2.1) the function S belongs to ∪ β≥1 , 0<r≤r * W β r , where r * is an unknown constant. Then, for any n ≥ 1 and ε > 0, the estimate S * n,ε satisfies the following inequality

E S S * n,ε -S 2 n ≤ 1 1 -4 σ * inf τ ∈Aε E S Ŝτ -S 2 n + 2 c * υ ε B 1 ( ) n 2/3 E S |ς n -ς n | + B n ( , ε) , (3.9)
where 0 < < 1/4σ * is an arbitrary constant and

B n ( , ε) = 2 2(ξ * -1) σ * c * B 1 ( ) n 7/6 υ ε + 7 σ * 2 β * B 1 ( ) + B 2 ( ) n υ ε , B 1 ( ) = 2 1 -2 σ * ν n 1 -4 σ * ν n , B 2 ( ) = 2 (1 + ξ * )(1 -2 σ * ν n ) (1 -4 σ * ν n ) , (3.10)
σ * = max 1≤l≤n σ 2 l , c * = sup τ ∈Aε c τ .
To estimate ς n , we make use of the following estimator

ςn = n j=l n +1 θ2 j,n φ j 2 n , n ≥ 3 , (3.11)
with

l n = [n 1/3 + 1]
. This estimator satisfies the following inequality.

Proposition 3.2. sup β≥1 ,r≤r * sup S∈ W β r E S |ς n -ς n | ≤ 1 √ n T * n ,
where

T * n = 2(ξ * -1)ν n σ * + 2r * + σ * n 1/6 + 2 √ 2r * σ * ν n n 1/3 + 5σ * √ n .
The proof of this proposition is given in Appendix. Theorem 3.1 and Proposition 3.2 imply immediately the following result.

Theorem 3.3. Assume that in the model (1.1)-(2.1) the function S belongs to ∪ β≥1 , 0<r≤r * W β r , where r * is an unknown constant. Then, for any n ≥ 1 and ε > 0, the estimate S * n,ε satisfies the following inequality

E S S * n,ε -S 2 n ≤ 1 1 -4 σ * inf τ ∈Aε E S Ŝτ -S 2 n + D n ( , ε) , (3.12)
where

D n ( , ε) = 2 c * υ ε B 1 ( ) n 7/6 T * n + B n ( , ε) and 0 < < 1/4σ * is an arbitrary constant.
-S 2 n = J n (τ ) + 2 n j=1 ρ τ (j) φ j 2 n ( θ2 j,n - ς n -θj,n θ j,n ) + n j=1 θ 2 j,n φ j 2 n .
Further, from the model (3.3) we obtain that

θ2 j,n -ςn n -θj,n θ j,n = θ j,n ξ j,n √ n + ξj,n n + mj,n n + 1 n (ς n -ςn ) , where ξj,n = ξ 2 j,n -m j,n , m j,n = E ξ 2 j,n = 1 n n l=1 σ 2 l φ 2 j (x l ) , mj,n = m j,n -ς n = 1 n n l=1 σ 2 l φ j (x l ) , φ j (x l ) = φ 2 j (x l ) -1 . (3.13) Therefore, for any τ ∈ A ε , Ŝτ -S 2 n = J n (τ ) + 2 √ n n j=1 ρ τ (j) θ j,n φ j 2 n ξ j,n + 2N n (τ ) + n j=1 θ 2 j,n φ j 2 n , (3.14)
where

N n (τ ) = 3 i=1 N i,n (τ ) with N 1,n (τ ) = n -1 n j=1 ρ τ (j) φ j 2 n ξj,n , N 2,n (τ ) = n -1 n j=1 ρ τ (j) φ j 2 n mj,n , N 3,n (τ ) = n -1 n j=1 ρ τ (j) φ j 2 n (ς n -ςn ) .
Let τ 1 be some fixed value of parameter from A ε . Then by (3.14) we find

Ŝτ -S 2 n -Ŝτ 1 -S 2 n = J n (τ ) -J n (τ 1 ) + 2Z 1,n (τ ) + 2 V n (τ ) , (3.15)
where

Z 1,n (τ ) = 1 √ n n j=1 ρ τ (j) -ρ τ 1 (j) φ j 2 n θ j,n ξ j,n , V n (τ ) = N n (τ ) -N n (τ 1 ) . Lemma 6.4 implies that E Z 4 1,n (τ ) ≤ ξ * d 4 1,τ (3.16)
where

d 2 1,τ = E Z 2 1,n (τ ) ≤ σ * ν n n n j=1 (ρ τ (j) -ρ τ 1 (j)) 2 φ j 4 n θ 2 j,n ≤ σ * ν n n S τ -S τ 1 2 n , ν n is defined by (3.2). Taking into account in (3.15) that J n (τ ) -J n (τ 1 ) ≤ 0, we obtain Ŝτ -S 2 n -Ŝτ 1 -S 2 n ≤ 2 Z 1,n (τ ) + 2 V * n ,
where

V * n = max τ ∈Aε |V n (τ )|. We set, for x > 0, > 0 and fixed τ ∈ A ε , Z1,n (τ, x) = 2 Z 1,n (τ ) n ( n) 2 d 2 1,τ + x and Γ 1,n (x) = sup τ | Z1,n (τ, x)| ≤ 1 . Notice now that ( n) 2 d 2 1,τ + x ≥ 2d 1,τ √ x n.
Therefore by Chebyshev's inequality and (3.16), we obtain, for any x > 0 and τ ∈ A ε ,

P | Z1,n (τ, x)| > 1 ≤ P |Z 1,n (τ )| > √ x d 1,τ ≤ e * (x)
,

where e * (x) = 1 {x≤1} + ξ * x -2 1 {x>1} . Hence P Γ c 1,n (x) ≤ τ =τ 1 P |Z 1,n (τ, x)| > √ x d 1,τ ≤ υ ε e * (x) . (3.17) Thus on the set Γ 1,n (x) Ŝτ -S 2 n ≤ Ŝτ 1 -S 2 n + nd 2 1,τ + x n + 2V * n ≤ Ŝτ 1 -S 2 n + σ * ν n S τ -S τ 1 2 n + x n + 2 V * n . (3.18)
Let us estimate now S τ -S τ 1 2 n . We have

S τ -S τ 1 2 n = Ŝτ -Ŝτ 1 2 n + ( S τ -S τ 1 2 n -Ŝτ -Ŝτ 1 2 n ) = Ŝτ -Ŝτ 1 2 n + n j=1 (ρ τ (j) -ρ τ 1 (j)) 2 φ j 2 n (θ 2 j,n -θ2 j,n ) ≤ Ŝτ -Ŝτ 1 2 n -2 Z 2,n (τ ) ,
where

Z 2,n (τ ) = 1 √ n n j=1 (ρ τ (j) -ρ τ 1 (j)) 2 φ j 2 n θ j,n ξ j,n with d 2 2,τ = E Z 2 2,n (τ ) ≤ σ * ν n n n j=1 (ρ τ (j) -ρ τ 1 (j)) 4 φ j 4 n θ 2 j,n ≤ 2σ * ν n n S τ -S τ 1 2 n .
By the same way as before, for x > 0, we set

Γ 2,n (x) = {sup τ | Z2,n (τ )| ≤ 1} , where Z2,n (τ, x) = 2Z 2,n (τ ) n ( n) 2 d 2 2,τ + x .
Similarly to (3.17), one shows that

P(Γ c 2,n (x)) ≤ υ ε e * (x) .
On the set Γ 2,n (x) we obtain that

S τ -S τ 1 2 n ≤ Ŝτ -Ŝτ 1 2 n + nd 2 2,τ + x n ≤ Ŝτ -Ŝτ 1 2 n + 2 σ * ν n S τ -S τ 1 2 n + x n .
Hence

S τ -S τ 1 2 n ≤ v * ( ) Ŝτ -Ŝτ 1 2 n + v * ( ) n x , (3.19)
where

v * ( ) = (1 -2 σ * ν n ) -1 . Thus on the set Γ n (x) = Γ 1,n (x) ∩ Γ 2,n (x), in view of (3.18),(3.19), we get Ŝτ -S 2 n ≤ Ŝτ 1 -S 2 n + σ * ν n v * ( ) Ŝτ -Ŝτ 1 2 n + 1 + σ * ν n v * ( ) n x + 2 V * n .
Estimating here the term Ŝτ -Ŝτ

1 2 n by 2 Ŝτ -S 2 n + 2 S -Ŝτ 1 2 n yields the inequality Ŝτ -S 2 n ≤ 1 1 -4 σ * ν n Ŝτ 1 -S 2 n + 2 1 -2 σ * ν n 1 -4 σ * ν n V * n + 1 -σ * ν n (1 -4 σ * ν n ) n x .
Applying Lemma 6.5 with M (x) = 2e * (x)υ ε gives

E S Ŝτ -S 2 n ≤ 1 1 -4 σ * ν n E S Ŝτ -S 2 n + B 1 ( )E S V * n + υ ε B 2 ( ) 1 n , (3.20)
where B 1 ( ) and B 2 ( ) are defined by (3.10). In Appendix 6.2 we show that

E S V * n ≤ υ ε 2 2(ξ * -1)σ * c * n 7/6 + 7σ * 2 β * n + 2 c * n 2/3 E S |ς n -ς n | . (3.21)
Therefore the inequality (3.9) follows immediately from (3.20) and (3.21).

Asymptotic upper bound

In this section we prove Theorem 2.1.

We start with the estimation problem (1.1) under the condition that S ∈ W q r with the known parameters q, r and ς(S) defined in (2.4). In this case we use the estimator Sn = Ŝτε defined in (3.5) with τ ε = (q, r ε ) from A ε ,

r ε = inf{i ≥ 1 : iε ≥ r/ς(S)} and ε = ε n = 1/ ln(n + 1). Theorem 4.1. Under the condition H 1 ) lim sup n→∞ ϕ n sup S∈W q r 1 γ q (S) R( Sn , S) ≤ 1 , (4.1)
where ϕ n = n 2q/(2q+1) .

Proof. First, note that from (3.5), taking into account that φ j n ≥ 1, we obtain that

E S Sn -S 2 n ≤ n j=1 (1 -ρj ) 2 θ 2 j,n φ j 2 n + 1 n n j=1 ρ2 j m j,n ,
where ρj = ρ τε (j) and m j,n is defined in (3.13). Moreover, puting k0 = [k 0 (τ ε )] and k1 = [ω τε • ln(n + 1)], we estimate the risk R( Sn , S) as

R( Sn , S) ≤ k1 -1 j= k0 (1 -ρj ) 2 θ 2 j,n + ς n 1 n n j=1 ρ2 j + ∆ 1 (n) + ∆ 2 (n) with ∆ 1 (n) = n j= k1 θ 2 j,n φ j 2 n and ∆ 2 (n) = 1 n n j=1 ρ2 j mj,n = 1 n 2 n d=1 σ 2 d n j=1 ρ2 j φ j (x d ) .
The last inequality implies that, for any fixed 0 < δ < 1,

R( Sn , S) ≤ (1 + δ) k1 -1 j= k0 (1 -ρj ) 2 θ 2 j + ς n 1 n n j=1 ρ2 j + ∆ 1 (n) + ∆ 2 (n) + (1 + 1/δ) ∆ 3 (n) , where ∆ 3 (n) = k1 -1 j= k0 (θ j,n -θ j ) 2 , θ j = 1 0 S(x) φ j (x) dx . (4.2)
Lemmas 6.1-6.3 imply immediately that

lim n→∞ sup S∈W q r ϕ n 3 l=1 |∆ l (n)| = 0 .
Therefore denoting ϕ n ∆ l (n) as o(1) yields

ϕ n R( Sn , S) ≤ (1 + δ)ϕ n k1 -1 j= k0 (1 -ρj ) 2 θ 2 j + ς n n -1/(2q+1) n j=1 ρ2 j + o(1) .
Further, for any S ∈ W q r denoting a j = q i=0 (2π[j/2]) 2i , we have that

j≥1 a j θ 2 j ≤ r . (4.3)
Therefore, in view of (3.4)

lim sup n→∞ sup j≥ k0 ϕ n (1 -ρj ) 2 a j ≤ 1 π 2q c 2q 0 ,
where c 0 = c τ 0 with τ 0 = (q, r/ς(S)). From here, by taking into account the conditions (2.4)-(2.5), we obtain, for sufficiently large n, the following upper bound for the quadratic risk

ϕ n R( Sn , S) ≤ (1 + 2δ) r π 2q c 2q 0 + ς(S) c 0 1 0 Ψ 2 q (z)dz + o(1) = (1 + 2δ) γ q (S) + o(1) .
This implies the inequality (4.1). Hence Theorem 4.1. Now Theorem 3.1 and Theorem 4.1 imply immediately Theorem 2.1.

Asymptotic lower bound

In this section we prove Theorem 2.2. For this we need the next auxiliary result.

Lemma 5.1. For any 0 < ε < 1 and any estimate Ŝn of S,

Ŝn -S 2 n ≥ (1 -ε) T ( Ŝ) n -S 2 -(ε -1 -1) r/n 2 ,
where

T ( Ŝ) n (x) = n k=1 Ŝn (x k )1 (x k-1 ,x k ] (x).
Proof of this Lemma is given in Appendix 6.5.

From this Lemma we deduce that to prove (2.9), it suffices to show that lim inf

n→∞ inf Ŝn ϕ n R 0 ( Ŝn ) ≥ 1 , (6.1)
where

R 0 ( Ŝn ) = sup S∈W q r E S Ŝn -S 2
γ q (S) .

For η > 0 and x ∈ R, denote

I η (x) = η -1 R 1 (|u|≤1-η) G u -x η du , where 1 A is the indicator of a set A, the kernel G ∈ C ∞ (R) is such that G(-u) = G(u) for |u| ≤ 1 , G(u) = 0 for |u| ≥ 1 , 1 -1 G(u) du = 1 .
It is easy to see that the function I η (x) possesses the properties : I η is symmetric,

I η (x) =    1 for |x| ≤ 1 -2η ; 0 for |x| ≥ 1 ; and R f (x)I η (x) dx → 1 -1 f (x) dx , η → 0 , R f (x)I 2 η (x) dx → 1 -1 f (x) dx , η → 0 , (6.2
S z (x) = M m=1 N j=1 z m,j D m,j (x) , (6.4)
where

D m,j (x) = e j (v m (x)) I η (v m (x)) , v m (x) = (x -a m )/h , a m = 2mh , h = L n (n) -1/(2q+1) , L n = ln(n + 1) .
To construct a priori distribution on the family of arrays, we choose the following random array

ϑ = {ϑ m,j , 1 ≤ m ≤ M, 1 ≤ j ≤ N } with ϑ m,j = δ m,j ξm,j , δ m,j = g 0 (a m )κ j √ nh , (6.5)
where

κ 2 j = N j q -1 + , N = [L n N * ε ], g 0 (•) = g(•, 0) , N * ε = 2 rε (q + 1)(2q + 1) qπ 2q 1/(2q+1) , rε = (1 -ε) r ς(0) , (6.6)
ε is a small positive real, 0 < ε < 1, ς(0) = 1 0 g 2 (x, 0)dx, ( ξm,j ) are i.i.d. bounded random variables with a density ρ d (x) such that

E ξm,j = 0, E ξ2 m,j = 1, | ξm,j | ≤ d , i d = d -d ( ρd (x)) 2 ρ d (x) dx → 1 as d → ∞ . (6.7)
The construction of a such density ρ d is given in Appendix 6.6.

For any estimator Ŝn , we denote by Ŝ0 n its projection on W q r , i.e. Ŝ0 n = Pr W q r ( Ŝn ). Since W q r is convex set we get that Ŝn -S 2 ≥ Ŝ0 n -S 2 . Therefore, for sufficiently large n,

R 0 ( Ŝn ) = sup S∈W q r E S Ŝn -S 2 γ q (S) ≥ sup S∈W q r E S Ŝ0 n -S 2 γ q (S) ≥ {z:Sz∈W q r } E z Ŝn -S z 2 γ q (S z ) µ ϑ (dz) ≥ 1 γ q (ε) {z:Sz∈W q r } E z Ŝ0 n -S z 2 µ ϑ (dz) ,
where γ * q (ε) = sup S ≤ε γ q (S) (the last inequality is true since S ϑ → 0 as n → ∞). Here we denote by E z the expectation with respect to the distribution of the process (1.1) with S = S z and the measure µ ϑ denotes the distribution of ϑ in R ι (ι = M N ) which is a priori distribution for the Bayes risk. The last inequality implies that

R 0 ( Ŝ0 n ) ≥ 1 γ * q (ε) R0 ( Ŝ0 n ) -2 1 γ * q (ε) ω n , (6.8)
where

R0 ( Ŝ0 n ) = E z Ŝ0 n -S z 2 µ ϑ (dz) , ω n = {z:Sz / ∈W q r } (r + S z 2 ) µ ϑ (dz) .
To reduce the nonparametric problem to a parametric one, we replace the functions Ŝ0 n and S by their Fourier series with respect to the basis (ẽ m,i ), where

ẽm,i (x) = 1 √ h e i (v m (x)) I(v m (x)) , 1 ≤ m ≤ M , 1 ≤ i ≤ N and I(x) = 1 (|x|≤1) . We estimate the term Ŝ0 n -S z 2 as Ŝ0 n -S z 2 ≥ M m=1 N j=1 ( λm,j -λ m,j (z)) 2 .
Here λm,j and λ m,j mean the Fourier coefficients for the functions Ŝ0 n and S, respectively, i.e. λm,j = Due to the definition of the function λ m,j (z) we obtain that

λ m,j (z) = M l=1 N i=1 z l,i 1 0 D l,i (x)ẽ m,j (x) dx = M l=1 N i=1 z l,i √ h 1 0 e i (v l (x)) e j (v m (x)) I η (v l (x))I(v m (x)) dx = √ h N i=1 z m,i 1 -1 e i (u)e j (u)I η (u) du . Therefore Λ m,j = ∂λ m,j ∂z m,j = √ h 1 -1 e 2 j (v)I η (v) dv = √ h (e 2 j , I η ) .
By the van Trees inequality (see Appendix 6.7) one gets

R0 (S 0 n ) ≥ M m=1 N j=1 Λ 2 m,j A m,j + B m,j + I m,j = M m=1 N j=1 h(e 2 j , I η ) 2 A m,j + B m,j + (δ m,j ) -2 i d , (6.9)
where I m,j is the Fisher information relative the random variable ξ m,j ,

A m,j = n k=1 R ι 1 g 2 (x k , S z ) ∂S z (x k ) ∂z m,j 2 µ ϑ (dz) , B m,j = n k=1 R ι Lm,j (x k , S z ) g(x k , S z ) 2 µ ϑ (dz) , Lm,j (x, S z ) = L x,Sz ∂S z ∂z m,j .
First note that from (6.4) we obtain that

A m,j = n k=1 D 2 m,j (x k ) R ι 1 g 2 (x k , S z ) µ ϑ (dz) = (1 + o(1)) n k=1 1 g 2 0 (x k ) D 2 m,j (x k ) = n h (1 + o(1)) 1 g 2 0 (a m ) 1 -1 e 2 j (x) I 2 η (x)dx = n h (1 + o(1)) 1 g 2 0 (a m ) (e 2 j , I 2 η ) .
Now we study the behaviour of B m,j . Due to the inequality (2.6) and the definition (6.4), we obtain that

| Lm,j (x, S z ) | ≤ C * ∂S z ∂z m,j = C * D m,j ≤ C * h .
Therefore by the condition H 3 ) we get

|B m,j | ≤ C * 1 g 2 * n h 2 .
Taking into account this inequality, the inequality (6.9) and the definition of δ m,j in (6.5), we evaluate the Bayes risk as

1 γ * q (ε) R0 ( Ŝ0 n ) ≥ 1 n γ * q (ε) M m=1 g 2 0 (a m ) N j=1 α j (η) (1 + o(1)) β j (η) + (κ j ) -2 i d ≥ (1 + o(1)) 2nh γ * q (ε) ς(0) N j=1 α j (η) (1 + o(1)) β j (η) + (κ j ) -2 i d = (1 + o(1)) ς(0) 2 γ * q (ε) n 2q/(2q+1) 1 L n N j=1 α j (η) (1 + o(1)) β j (η) + (κ j ) -2 i d ,
where we denote α j (η) = (e 2 j , I η ) 2 and β j (η) = (e 2 j , I 2 η ). In Appendix 6.9 we show that lim n→∞ ϕ n ω n = 0 .

(6.10)

This result and the previous inequality imply lim inf

n→∞ inf Ŝn ϕ n R 0 ( Ŝn ) ≥ ς(0) N * ε 2 γ * q (ε) Ω * (η, d) lim inf n→∞ 1 N N j=1 κ 2 j κ 2 j + 1 = ς(0) N * ε 2 γ * q (ε) Ω * (η, d) lim n→∞ 1 N N j=1 (1 -(j/N ) q ) = ς(0) N * ε 2 γ * q (ε) q q + 1 Ω * (η, d) = (1 -ε) 1/(2q+1) γ 0 q γ * q (ε) Ω * (η, d) , (6.11)
where

γ 0 q = γ q (0), Ω * (η, d) = lim inf N →∞ Ω N (η, d)
and The condition H 1 ) implies γ * q (ε) → γ 0 q as ε → 0. Therefore limiting ε → 0 , η → 0 , d → ∞ in (6.11) yields lim inf

Ω N (η, d) = N j=1 (α j (η) κ 2 j ) (β j (η) κ 2 j + i d ) -1 N j=1 κ 2 j (κ 2 j + 1) -1 . ( 6
n→∞ inf Ŝn ϕ n R0 ( Ŝn ) ≥ 1 .
Thus (6.8) implies the inequality (6.3) from which it follows (6.1) through Lemma 5.1.

Appendix

6.1

Properties of trigonometric basis Lemma 6.1. Let θ j ,θ j,n be coefficients from (3.3) and (4.2), respectively. Then for 1 ≤ j ≤ n and n ≥ 2 sup

S∈W q r |θ j,n -θ j | ≤ 2π √ r j n . (A.1)
Proof. Indeed, we have

|θ j,n -θ j | = n l=1 x l x l-1 (S(x l )φ j (x l ) -S(x)φ j (x)) dx ≤ 1 n n l=1 x l x l-1 | Ṡ(z)φ j (z)| + |S(z) φj (z)| dz = 1 n 1 0 | Ṡ(z)| |φ j (z)| + |S(z)| | φj (z)| dz .
By making use of the Bounyakovskii-Cauchy-Schwartz inequality we get

|θ j,n -θ j | ≤ 1 n Ṡ φ + φ S ≤ 1 n Ṡ + π j S .
The definition of class W q r in (2.2) implies (A.1). Hence Lemma 6.1.

Lemma 6.2. For any m ≥ 0, sup

N sup x∈[0,1] N -m N l=2 l m (φ 2 l (x) -1) ≤ 2 m . (A.2)
Proof. By the properties of the trigonometric functions we get

N l=2 l m (φ 2 l (x) -1) = 1≤l≤N/2 (2l) m [2 cos 2 (2πlx) -1] + 1≤l≤(N -1)/2 (2l + 1) m (2 sin 2 (2πlx) -1) = 1≤l≤N/2 (2l) m cos(4πlx) - 1≤l≤(N -1)/2 (2l + 1) m cos(4πlx) .
This yields

N l=2 l m (φ 2 l (x) -1) ≤ 1≤l≤(N -1)/2 ((2l + 1) m -(2l) m ) cos(4πlx) + N m ≤ 1≤l≤(N -1)/2 |(2l + 1) m -(2l) m | + N m = 1≤l≤(N -1)/2 m-1 j=0 m j (2l) j + N m .
This implies (A.2).

Lemma 6.3. For any function S ∈ W q r , sup

n≥1 sup 1≤m≤n-1 m 2q n j=m+1 θ 2 j,n φ j 2 n ≤ 2r π 2(q-1) , (A.3)
where ν n is defined by (3.2).

Proof. First, note that any function S from W q r we can be represented by its Fourier series, i.e. S = ∞ j=1 θ j φ j with the coefficients defined by (4.2). By denoting the residual term for S as

∆ m (x) = S - m j=1 θ j φ j = ∞ j=m+1 θ j φ j (x) , we obtain that n j=m+1 θ 2 j,n φ j 2 n = inf α 1 ,...,αm S - m j=1 α j φ j 2 n ≤ ∆ m 2 n . Moreover ∆ m 2 n = 1 n n k=1 ∆ 2 m (x k ) = n k=1 x k x k-1 ∆ 2 m (x k )dx ≤ 2 1 0 ∆ 2 m (x)dx + 2 n k=1 x k x k-1 (∆ m (x k ) -∆ m (x)) 2 dx .
The last term in this inequality we estimate as

(∆ m (x k ) -∆ m (x)) 2 = x k x ∆m (z)dz 2 ≤ 1 n x k x k-1 ( ∆m (z)) 2 dz . Therefore ∆ m 2 n ≤ 2 ∆ m 2 + 2 1 n 2 ∆m 2 = 2 ∞ j=m+1 θ 2 j + 2 1 n 2 ∞ j=m+1 θ 2 j φj 2 .
Taking here into account the inequality (4.3) we obtain (A.3). Hence Lemma 6.3.

Lemma 6.4. Let ξ j,n be defined in (3.3) for the model (1.1)-(2.1). Then, for any real numbers v 1 , . . . , v n ,

E n j=1 v j ξ j,n 4 ≤ ξ * E n j=1 v j ξ j,n 2 2
,

E n j=1 v j ξ j,n 2 ≤ σ * ν n n j=1 v 2 j , (A.4)
where σ * = max 1≤j≤n σ 2 j and ν n is defined in (3.2). Proof. By the definition of ξ j,n we obtain that n j=1 v j ξ j,n = n l=1 σ l ṽl ξ l with ṽl = 1 √ n n j=1 v j φ j (x l ). Therefore, this implies immediately the first inequality in (A.4). Moreover

E n j=1 v j ξ j,n 2 = n l=1 σ 2 l ṽ2 l ≤ σ * n l=1 ṽ2 l = σ * n i,j=1 v i v j (φ i , φ j ) n .
The orthogonal properties of the basis (φ j ) and (3.2) imply the second inequality in (A.4). Hence Lemma 6.4.

Proof of (3.21)

Indeed, we have

E S V * n (τ ) ≤ 3 i=1 τ =τ 1 E S |N i,n (τ ) -N i,n (τ 1 )| ≤ υ ε 3 i=1 sup τ ∈Aε E S |N i,n (τ ) -N i,n (τ 1 )| . (A.5)
We start with the first term in the right-hand part. We have

E S |N 1,n (τ ) -N 1,n (τ 1 )| = 1 n E S n j=1 (ρ τ (j) -ρ τ 1 (j)) ξj,n ≤ 1 n n j=1 (ρ τ (j) + ρ τ 1 (j)) (E ξ2 j,n ) 1/2 ,
where ξj,n is defined in (3.13). Moreover,

E ξ2 j,n = E ξ 2 j,n -m j,n 2 = Eξ 4 j,n -m 2 j,n = 1 n 2 n r,s=1 σ 2 r σ 2 s φ 2 j (x r )φ 2 j (x s ) E ξ 2 r ξ 2 s -m 2 j,n = (ξ * -1) 1 n 2 n r=1 σ 4 r φ 4 j (x r ) ≤ 2(ξ * -1)σ 2 * ν n n . (A.6)
Thus, for any fixed τ and τ 1 from A ε ,

E S |N 1,n (τ ) -N 1,n (τ 1 )| ≤ 2(ξ * -1)ν n σ * n √ n (w τ + w τ 1 ) ,
where

w τ = n 1/(2β+1) c τ ≤ c * n 1/3 since β ≥ 1. Therefore sup τ ∈Aε E S |N 1,n (τ ) -N 1,n (τ 1 )| ≤ 2 2(ξ * -1)ν n σ * c * n 7/6 . (A.7)
To estimate the second term in the right-hand part of (A.5), note that

N 2,n (τ ) = 1 n 2 n d=1 σ 2 d n j=1 ρ τ (j)φ j (x d ) - 1 n 2 n d=1 σ 2 d ρ τ (n)(1 - 1 ν n )φ n (x d ) .
Let us show that, for any n ≥ 2,

sup x | n j=1 ρ τ (j)φ j (x)| ≤ 3 • 2 β . (A.8)
Indeed, by definition (3.4), ρ τ (j) = 1 for j ≤ k 0 (τ ). Therefore the inequality (A.8) follows immediately from Lemma 6.2, for n ≤ k 0 (τ ). Let us show (A.8), for n > k 0 (τ ). In this case we can represent the sum in the left-hand part of (A.8) as

n j=1 ρ τ (j)φ j (x) = n j=2 ρ τ (j)φ j (x) = 2≤j≤wτ ∧n φ j (x) - 1 (w τ ) β 2≤j≤wτ ∧n j β φ j (x) + 1 (w τ ) β 2≤j≤k 0 (τ ) j β φ j (x) ,
where a ∧ b = min(a, b). Now it is easy to see that inequality (A.8) follows directly from Lemma 6.2. Therefore by (A.8)

sup τ ∈Aε E |N 2,n (τ ) -N 2,n (τ 1 )| ≤ 6 • 2 β * n 2 n d=1 σ 2 d + 2 n 2 n d=1 σ 2 d (1 - 1 ν n ) ≤ 7 σ * 2 β * n .
For the last addend in (A.5), in view of (3.4), we get

E |N 3,n (τ ) -N 3,n (τ 1 )| ≤ w τ + w τ 1 n E |ς n -ς n | ≤ c * (n 1/(2β+1) + n 1/(2β 1 +1 ) n E |ς n -ς n | ≤ 2 c * n 2/3 E |ς n -ς n | .

Proof of Proposition 3.2

First notice that taking into account (3.13) in (3.11) we obtain that

ςn -ς n = n j=l n +1 1 φ j 2 n θ 2 j,n + 2 √ n n j=l n +1 1 φ j 2 n θ j,n ξ j,n + 1 n n j=l n +1 1 φ j 2 n ξj,n + 1 n n j=l n +1 1 φ j 2 n mj,n - (l n + 1)ν n -1 nν n ς n .
Therefore by applying here (A.6) and Lemma 6.4 we come to the following upper bound

E S |ς n -ς n | ≤ n j=l n +1 1 φ j 2 n θ 2 j,n + 2 √ σ * ν n √ n   n j=l n +1 1 φ j 4 n θ 2 j,n   1/2 + 2(ξ * -1)ν n σ * √ n + (l n + 1)σ * n + 1 n n j=l n +1 1 φ j 2 n mj,n .
Moreover, by Lemma 6.2 we have

n j=l n +1 1 φ j 2 n mj,n = 1 n n d=1 σ 2 d n-1 j=l n +1 φ j (x d ) + 1 ν n mn,n ≤ σ * n d=1 n-1 j=l n +1 φ j (x d ) + 1 ν n | mn,n | ≤ 3 σ * .
This proposition follows now directly from Lemma 6.3

Mean forecast bound

Lemma 6.5. Let α, ξ be two integrable random variables. Let β be a positive real number and {Γ x , x > 0} be a family of events such that, for any x > 0,

P (ξ > α + βx , Γ x ) = 0 .
Assume also that there exists some positive integrable on R + function M (x) for which P(Γ c x ) ≤ M (x), for all x ∈ R + . Then

E ξ ≤ E α + βM * , where M * = ∞ 0 M (x)dx. Proof. Denote η = ξ -α. Then ξ = α + η ≤ α + η + and E ξ ≤ E α + E η + .
Moreover,

E η + = ∞ 0 P(η + > z)dz = β ∞ 0 P(η + > βx)dx = β ∞ 0 (P(η + > βx, Γ x ) + P(η + > βx, Γ c x )) dx = β ∞ 0 (P(η > βx, Γ x ) + P(η + > βx, Γ c x )) dx ≤ β ∞ 0 P(Γ c x )dx ≤ β ∞ 0 M (x)dx ≤ βM * .
6.5 Proof of Lemma 5.1

First notice that, for any S ∈ W q r , one has

Ŝn -S 2 n = 1 n n j=1
( Ŝn (x j ) -S(x j )) 2 = n j=1

x j

x j-1

( Ŝn (x j ) -S(x j )) 2 dx = n j=1

x j

x j-1

( Ŝn (x j ) -S(x)) 2 dx + 2 n j=1

x j

x j-1

( Ŝn (x j ) -S(x))(S(x) -S(x j ))dx

+ n j=1 x j x j-1 (S(x) -S(x j )) 2 dx = T ( Ŝ) n -S 2 + D n + ∆ n ,
where

D n = 2 n j=1
x j

x j-1

( Ŝn (x j ) -S(x))(S(x) -S(x j ))dx ,

∆ n = n j=1
x j

x j-1

(S(x) -S(x j )) 2 dx .

For any 0 < ε < 1, by making use of the elementary inequality 2ab ≤ εa 2 + ε -1 b 2 , one gets

D n ≤ ε T ( Ŝ) n -S 2 + ε -1 ∆ n .
Moreover, for any S ∈ W q r with q ≥ 1, by the Bounyakovskii-Cauchy-Schwartz inequality we obtain that

∆ n = n j=1 x j x j-1 x j x Ṡ(t)dt 2 dx ≤ 1 n n j=1 x j x j-1 ( Ṡ(t)) 2 dt = 1 n 2 Ṡ 2 ≤ r n 2 .
Hence Lemma 5.1.

Construction of the density ρ d

We set

ρ d (t) = 1 ς d f d t ς d , ς 2 d = d -d z 2 f d (z)dz -1 , (A.9)
where f d is a truncated Gaussian density which is defined by the following way. First we need to choose the smooth indicator function. We choose this function for d ≥ 3 as

Λ d (t) = R 1 (|z|≤d-2) G(z -t)dz ,
where

G(t) = G * exp{-1/(1 -t 2 )} I(t) , G * = 1 -1 exp{-1/(1 -t 2 )} dt -1
and I(t) = 1 {|t|≤1} . Notice that this smoothing density G is infinitely times differentiable. Now we set

f d (t) = f (t)Λ d (t) f * d , f (t) = e -t 2 /2 √ 2π and f * d = R f (t)Λ d (t)dt .
The properties of the density ρ d are given in the following Lemma Lemma 6.6. The density ρ d satisfies the following properties:

1. For any d ≥ 3, R tρ d (t)dt = 0, R t 2 ρ d (t)dt = 1 .
2. The Ficher information i d for ρ d goes to 1 as d → ∞.

Proof. Notice that the first part of this Lemma follows immediately from the definition (A.9). Let us shows the second one. Due to properties of Λ d ,

f d (z) = f (z)1 (|z|≤d-3) + f (z)Λ d (z)1 (|z|≥d-3) .
Therefore

f * d = R f (z)1 (|z|≤d-3) dz + R f (z)Λ d (z)1 (|z|≥d-3) dz → R f (z)dz = 1 , since sup x∈R |Λ d (x)| ≤ 1. Similarly ς 2 d → 1 as d → ∞.
Consider the Fisher information i d . We have

i d = d -d ( ρd ) 2 (z) ρ d (z) dz = 1 f * d ς 2 d d/ς d -d/ς d ( ḟd (u)) 2 f d (u) du = 1 f * d ς 2 d d/ς d -d/ς d ( ḟ (u)) 2 f (u) Λ d (u) du + 2 1 f * d ς 2 d d/ς d -d/ς d ḟ (u) Λd (u) du + 1 f * d ς 2 d d/ς d -d/ς d ( Λd (u)) 2 Λ d (u) f (u) du .
From the definition of Λ d it is easy to deduce that, for any integrable on R function f ,

lim d→∞ R f (u) Λ d (u) du = R f (u) du , lim d→∞ R f (u) Λd (u) du = 0
and Λd (u) = 0 for u ≤ d -3. Therefore we obtain the following assymptotic relationship for i d as d → ∞

i d = 1 + o(1) + d/ς d -d/ς d ( Λd (u)) 2 Λ d (u) f (u) 1 {|u|≥d-3} du . (A.10)
To estimate the last integral, notice that by the Bounyakovskii-Caushy-Schwartz inequality

( Λd (u)) 2 = R 1 (|z|≤d-2) Ġ(u -z)dz 2 ≤ R 1 (|z|≤d-2) dz R 1 (|z|≤d-2) Ġ(u -z) 2 dz ≤ 2d R 1 (|z|≤d-2) Ġ(u -z) 2 dz ,
Let now λn be an estimator of λ = λ(ϑ) based on observations (y j ) 1≤j≤n . For any

B(R n × R ι ) -mesurable integrable function ζ = ζ(x, ϑ), x ∈ R n , ϑ ∈ R ι , we set Ẽ ζ(Y, ϑ) = R ι E ϑ ζ(Y, ϑ) Φ(ϑ) dϑ ,
where E ϑ is the expectation with respect to the distribution P ϑ of the vector Y = (y 1 , . . . , y n ). Note that in this case

E ϑ ζ(Y, ϑ) = R n ζ(v, ϑ) f (v, ϑ) dv , where f (v, ϑ) = n j=1 1 √ 2πσ j (ϑ) exp - (v j -S ϑ (x j )) 2 2σ 2 j (ϑ)
.

We prove the following result (cp. [START_REF] Gill | Application of the van Trees inequality: a Bayesian Cramér-Rao bound[END_REF]).

Lemma 6.7. Assume that the function g(•, •) satisfies the condition H 1 ) -H 2 ) and the unknown function S ϑ (•) is differentiable with respect to ϑ l , for any 1 ≤ l ≤ ι, moreover,

∂S ϑ (•) ∂ϑ l ∈ L 1 [0, 1] .
Then for any square integrable estimator λn of λ and any 1 ≤ l ≤ ι,

Ẽ( λn -λ) 2 ≥ Λ 2 l A l + B l + I l ,
where

A l = n j=1 R ι 1 σ 2 j (ϑ) ∂S ϑ (x j ) ∂ϑ l 2 Φ(ϑ) dϑ , B l = 2 n j=1 R ι Ll (x j , S ϑ ) g(x j , S ϑ ) 2 Φ(ϑ) dϑ , Ll (x, S) = L x,S ∂S ϑ ∂ϑ l , the operator L x,S is defined in the condition H 2 ).
Proof. We put

l (v, ϑ) = ∂ ∂ϑ l (f (v, ϑ)Φ(ϑ)) f (v, ϑ)Φ(ϑ) .
Integrating by parts and taking into account that the density Φ l has a finite compact support yields

Ẽ ( λn -λ) l = R n R ι ( λn (v) -λ(ϑ)) ∂ ∂ϑ l (f (v, ϑ)Φ(ϑ)) dϑ dv = R n R ι-1 R ( λn (v) -λ(ϑ)) ∂ ∂ϑ l (f (v, ϑ)Φ(ϑ)) dϑ l ι j =l dϑ j dv = R n R ι-1 R ∂ ∂ϑ l λ(ϑ) f (v, ϑ)Φ(ϑ) dϑ l ι j =l dϑ j dv = R ι ∂ ∂ϑ l λ(ϑ) Φ(ϑ) R n f (v, ϑ) dv dϑ = Λ l .
Therefore by the Bouniakovskii-Cauchy-Schwartz inequality

Ẽ( λn -λ(ϑ)) 2 ≥ Λ 2 l Ẽ 2 l .
To evaluate the denominator in the last ratio, note that

l (v, ϑ) = ∂ ∂ϑ l (f (v, ϑ)) f (v, ϑ) + Φl (ϑ l ) Φ l (ϑ l ) = ∂ ∂ϑ l ln f (v, ϑ) + Φl (ϑ l ) Φ l (ϑ l ) .
By (A.11) we find that

∂ ∂ϑ l ln f (Y, ϑ) = n j=1 (ξ 2 j -1) 1 σ j (ϑ) ∂ ∂ϑ l σ j (ϑ) + n j=1 ξ j 1 σ j (ϑ) ∂ ∂ϑ l S ϑ (x j ) . Thus Ẽ ∂ ∂ϑ l ln f (Y, ϑ) = 0 .
Moreover, the condition H 2 ) implies the relationship

∂ ∂ϑ l σ j (ϑ) = ∂ ∂ϑ l g(x j , S ϑ ) = Ll (x j , S ϑ ) from which it follows Ẽ ∂ ∂ϑ l ln f (Y, ϑ) 2 = A l + B l .
Hence Lemma 6.7.

6.8 Proof of (6.13)

Notice that to prove (6.13) it suffices to show that lim sup η→0 , d→∞ sup N ≥1

|Ω N (η, d) -1| = 0 , (A.12)

where Ω N (η, d) is defined in (6.12). Indeed, by the direct calculation we find that, for any N ≥ 1,

|Ω N (η, d) -1| ≤ max j≥1 (|α j (η) -1| + |β j (η) -1|) + |i d -1| min(β * (η), i d ) ,
where β * (η) = min j≥1 β j (η). Therefore the definitions of α j (η) and β j (η) and the properties (6.2) and (6.7) imply (A.12).

6.9 Proof of (6.10)

In this proof we use the following Lemmas.

Lemma 6.8. Let S ϑ be defined in (6.4) and S

ϑ be its derivative of order l, 1 ≤ l ≤ q. Then, for sufficiently large n and any 0 < ε < 1,

q l=1 S (l) ϑ 2 ≤ (1 -ε/2)r + (1 + µ) q l=1 ζl + (1 + µ -1 ) L -2 n ,
where µ = ε/(2 -2ε) and

ζl = M m=1 g 2 0 (a m ) N j=1 κ 2 j 1 n jπ 2 h 2l ζ m,j , ζ m,j = ξ2 m,j -1 . (A.13)
Proof. From the definition of S ϑ in (6.4) we obtain that

S (l) ϑ (x) = M m=1 N j=1 ϑ m,j D (l) m,j (x) , where
D (l) m,j (x) = 1 h l Q (l) 0,j (v m (x)) + 1 h l Q (l)
1,j (v m (x)) , Q (l) 0,j (u) = e ϑ m,j Q (l) i,j (u) , i = 0, 1 , we find that

S (l) ϑ 2 = 1 h 2l-1 M m=1 1 -1 Q (l) 0,j (v) + Q (l) 1,j (v) 2 dv = 1 h 2l-1 M m=1 Q (l) m,0 2 + 2(Q (l) m,0 , Q (l) m,1 ) + Q m,1 2 .
Therefore, for any µ > 0, we get that ≤ (1 + o(1)) N 2q+1 π 2l q (q + 1)(2q + 1)n(2h) 2l+1 ς 0 . This yields that

S (l) ϑ 2 ≤ 1 + µ h 2l-1 M m=1 Q (l) m,0 2 + 1 + µ -1 h 2l-1 M m=1 Q (l) m,1 2 . (A.14) Moreover Q (l) m,0 2 = 1 -1 N j=1 ϑ m,j Q 0,j (u)
H l =    (1 + o(1))(1 -ε)r , if l = q , o n -1/(2q+1) , if l < q .
Let us consider now the second term in the right-hand part of (A. ϑ 2 m,j (πj) 2(q-i)

≤ C 1 (η) N j=1
ϑ 2 m,j j 2(q-1) ,

where C(η) = max 1≤l≤q max -1≤u≤1 l i=1 l i 2 (I (i) η (u)) 2 and C 1 (η) = qC(η) π 2q . From this it follows that

1 h 2l-1 M m=1 Q (l) m,1 2 ≤ C 1 (η) h 2l-1 M m=1 N j=1
ϑ 2 m,j j 2(q-1)

≤ d 2 C 1 (η) nh 2l M m=1
g(a m ) N j=1 κ 2 j j 2(q-1)

≤ C N 2q L 2l+1 n n 2(q-l)/(2q+1) , where C is a constant. Notice that, if l = q then 1 h 2q-1

M m=1 Q (l) m,1 2 ≤ C 1 L 2q+1 n N q N j=1 j q-2 ≤ C N 2q-1 L 2q+1 n ≤ C 1 L 2 n . If l < q then 1 h 2l-1 M m=1 Q (l) m,1 2 = O(L 2q n n -2/(2q+1) ) = O(L -1 n ) .
Therefore, putting in (A.14) µ = ε/2(1ε), we come to the assertion of this Lemma.

Lemma 6.9. Let ζl be defined in (A.13). Then, for any p ≥ 1/2,

Remark 3 . 1 .

 31 Note that the principal term in the right-hand side of (3.12) is best in the class of estimators ( Ŝτ , τ ∈ A ε ). Usually inequalities of such type are called the Oracle inequalities. Proof of Theorem 3.1. Fisrt, by (3.6)-(3.7) we get the following equality Ŝτ

  )uniformly over all functions bounded by any fixed constant c > 0, i.e. sup f sup -1≤x≤1 |f (x)| ≤ c.We choose the trigonometric basis in L 2 [-1, 1] as follows e 1 (x) = 1/ √ 2, e 2 (x) = cos(π x), e 3 (x) = sin(π x), . . . , e 2i (x) = cos(iπ x), e 2i+1 (x) = sin(iπ x), . . . .(6.3)For any array z = {z m,j , 1 ≤ m ≤ M, 1 ≤ j ≤ N } with M = [1/(2h)] -1 and N a positive integer, we denote

1 0S

 1 ẽ m,j (x)dx and λ m,j (z) = z (x)ẽ m,j (x) dx .

  (πj/2) 2l .Taking into account the definition of ϑ m,j yields the following bound for the first sum in the right-hand term of (A.[START_REF] Ibragimov | Statistical Estimation: Asymptotic Theory[END_REF]

  I (i) η (u)ψ m,i (u).Applying the Bouniakovskii-Cauchy-Schwartz inequality yields Q

.+ 1 ) 2 π 2q h 2q N 2q+1 ≤ C L 2q+1 n 1 h

 11 , d) L p n n p/(2q+1) , (A.15) where C(p, d) is a constant depending on p and d. Proof. From (A.13) it follows that The process ( ζm , m ≥ 1) is a square-integrable martingale-difference. Applying the Burkholder-Davis-Gundy inequality to the right-hand part of (A.16) yields, for p ≥ 1, p) is a constant depending on p. Moreover, | ζm | ≤ (d + 1) 2q = C L n n 2q/(2q+1) .

The second author is partially supported by the RFFI-Grant 04-01-00855.

where

From here we estimate the last integral in (A.10) by

Hence we get Lemma 6.6.

The van Trees inequality for regression models

) be a statistical model relative the observations (y j ) 1≤j≤n governed by the regression equation

where ξ 1 , . . . , ξ n are i.i.d. N (0, 1) random variables, ϑ = (ϑ 1 , . . . , ϑ ι ) is an unknown vector parameter and σ j (ϑ) = g(x j , S ϑ ), where the function g(x, S) is defined in the condition H 1 ). Assume that a prior distribution µ of the parameter ϑ in R ι is defined by the density Φ(ϑ) of the following form

where Φ j is a continuous density on R with a compact support such that

Let λ(•) be a real-valued function, λ : R ι → R. We assume that this function is differentiable with respect to ϑ l , for any 1 ≤ l ≤ ι, such that the partial derivatives are integrable with respect to the density Φ(•) and we put

Thus (A.17) implies

n p/(2q+1) and therefore (A.15).

Let us prove now (6.10). To end this, note that

Therefore, for sufficiently large n,

) . To obtain (6.10), it suffices to prove the relationship lim n→∞ ϕ n P q l=1 S (l) ϑ 2 > r = 0 . Indeed, Lemma 6.8 implies that, for sufficiently large n,

where µ = ε 2(1-ε) and ζl is defined by (A.13). Thus, for sufficiently large n,

where r 1 = εr/(2(1 + µ)). By making use of Inequality (A.15) with p > 2q we obtain that Hence (6.10).