Giovanni Gómez Gras

J Antonio Travieso-Rodríguez

Hernán González-Rojas

Amelia Nápoles-Alberro

Francisco Carrillo

Gilles Dessein

BALL-BURNISHING VIBRATION-ASSISTED PROCESS

Keywords: Ball-burnishing, plastic deformation, vibrations, manufacturing, acousto-plasticity, surface roughness

First of all, this work refers to the study of the ball-burnishing process assisted by a vibration. It starts by considering that this vibration helps to make the development of this finishing process easier, because it helps to deform the workpiece material more easily. Since there is no similar tool on the market, in order to conduct the study, it has been necessary to design, characterize and manufacture a tool that can perform the process, taking into account the critical components that are involved in the design, and the physical model characterizing the operation. Under these criteria, the characterization of the tool operation is also done by evaluating the surface roughness that remains after the process occurs. For experimental validation have been used work pieces of aluminium and steel. These results are compared to those obtained by the same tool without using vibration. Roughness results obtained using the ball-burnishing vibration-assisted process improves with respect to those obtained using the process without assistance, in both materials tested.

INTRODUCTION

Currently, industry features many mechanical components, which should have a good surface roughness, a geometric tolerance level, a surface hardness with a high degree and significant mechanical strength values to be able to work, even in hard conditions. Many traditional processes are being studied to improve and adapt them to new difficulties found the industry. Furthermore, parallel studies have provided novel solutions for longer life cycles of many components that undergo, for example, daily high rates of wear or are subject to the effect of cyclical forces. In recent years, the amount of research associated with different processes aiming to produce these pieces with appropriate features that allow them to meet these benefits, has markedly grown. The ballburnishing process meets all of these requirements. Ball-burnishing is defined as a technological operation that consists of plastically deforming surface irregularities, by the action of the force exerted by a ball [1]. This process could be performed with a conventional tool [2], or be assisted by vibration, as presented in this study. The vibration-assisted process affords a greater dislocation movement of the material work piece, which allows us to work on harder materials with less force, and even improve the results of the process [3]. The current market does not offer a tool that uses the assistance of vibration, the ball-burnishing process. Therefore, a prototype tool capable of performing the above process has been designed and manufactured. Therefore, this paper aims to study the behaviour of a ball-burnishing vibration-assisted tool. To this end, we first describe the tool design via the study of its fundamental parts. An analysis of the elements that provide the vibration via a physical model is then performed, which reflects that they constitute the essential elements for the tool operation. Subsequently, the model was experimentally validated. Finally, practical tests were performed to determine the surface roughness values obtained with the developed tool. The practical tests were carried out aluminium A92017 and steel G10380. Vibration is widely used to assist the conventional manufacturing process modern industry and it has been referenced by several authors [4], as a method to improve the surface quality of work pieces.

STUDY OF THE BALL-BURNISHING PROCESS 2.1. Schematic functional description

Before developing the tool components that affect the study of the ball-burnishing process, the method to introduce the vibrations must be considered. Vibrations are generated by an electromagnetic transducer that is used to convert alternating current into a variable magnetic field. This field produces an attraction cyclical force on metal plates M1 and M2 with thickness h1 and h2, deforms them and causes a vibration, whose frequency is determined by the magnetic field (Figure 1).

Figure 1 shows how the tool can be easily installed using an ISO cone in the same CNC machine where the piece has been previously machined. The thread cone is fixed to the tool body, which consists of a cylinder that contains a spring housed inside. This spring is deformed by the effect of the vibration, but it ensures the maintenance of a constant force throughout the process. The spring is joined to a rod that is in contact with the coil core. The coil is inside the cylinder closed on both faces using two plates, M1 and M2. These plates are coupled and vibrate at a predetermined working frequency. They transmit this vibration to the bottom part of the tool, where the ball burnishing is placed. The ball is in contact with a bearing, which facilitates its free rotational movement. Several spheres that are 2 mm in diameter form the bearing, which is responsible for transmitting the vibration generated by the coil. The effect of a hard ball that is 10 mm in diameter plastically deforms the work piece. The interaction between the hard ball and the work piece under the action of a constant and normal force is strong enough to modify the surface topography of the treated material.

Significant design criteria

Starting from these premises, monitoring certain parameters that will determine the correct functioning of the tool is important from the standpoint of design, as specified in the scheme of Figure 1. These elements of special interest are: 1. the thickness of plates M 1 and M 2 , 2. the interrelationship that is established between them, 3. the J gap which is defined as the distance between the coil core and the centre of the M 1 plate. Conceptually, this distance must be greater than the maximum relative plate deflection between M 1 -M 2 that is experienced during operation of the tool. Therefore, this paper seeks the maximum relative deflection between M 1 -M 2 plates to estimate the optimum value of the gap. At this point, studying the behaviour of the deflection when both plates are working (M 1 and M 2 ) is useful. Moreover, this deflection value will also depend on the thickness of the two plates. Therefore, various thicknesses were evaluated throughout the study to determine the appropriate thickness. The plate considered in this study is a rigid thin plate [7] because the plate diameter, D, is 53 mm, the plate thickness, h, is less than 5 mm and the maximum plate deflection, w, is less than 0.08 mm. The model developed in this study is based on the classical theory of Kirchhoff [8]. Therefore, assuming that the deformations in solids to be considered are infinitesimal, the relationship between the stress components and the strain components depends on the solid compound material. For an elastic and isotropic solid, the constitutive equations take the form of Hooke's generalised law [7]. The assumption of an isotropic material is suitable to define the behaviour of the C-45K steel (according to standard EN 10083-2), which was used to manufacture the vibrating plates and whose characterisation is specified in Table 1. The equations that relate the stress components to the deformation components are often called constitutive equations. Hooke's law is used for a reference system in polar coordinates with axial symmetry [7].

Model of the coupled plates

In addition to the influence of the vibration generator, other elements with equal importance influence the global behaviour of the plates of the tool under consideration. When the tool begins to apply a force on the burnishing tool, the M 1 and M 2 plates are deformed depending on the direction of the force, but in opposite directions to each other. The deformation produces a deflection between the two plates. The coupling of the deformation of the plates must be studied. Differential equations 1 and 2 are obtained by deducing the equations of all theories considered. These equations predict the deflection, w, of each of the circular plates depending on the radius, r, and time, t. These equations are the basis of the model to be developed. …( 1)

…(2)
The boundary conditions are defined in equation 3.

…(3)

P is the vertical load distribution and is defined by Equation 4. The value of the forcing function is constant. …(4) P 0 is the oscillatory forcing function produced by the magnetic field. is the initial value of the power system load, which is constant and results from spring compression. f is the driving frequency of the coil that generates the varying magnetic field, which also generates the deflection variable force. Furthermore, the flexural modulus of the rigidity plate, K, defined in [7], is shown in equation 5.

…(5)

Experimental check of the plate deflection

To verify the actual relative deflection, the M 1 plate was studied when the tool was subjected to the influence of the sinusoidal driver. A displacement digital electronic display unit produced by MARPOSS was used. The sensor was placed on the central point of the tool shank, which is the maximum displacement area, to take the appropriate reading when M 1 is vibrating at 2600 Hz (Figure 2). This frequency was determined from the simulation results of the deflection for M 1 =2 mm. According to the terms of the theoretical study, the maximum relative deflection of the plate designed for the tool was determined 0.0012 mm at 2600Hz, as seen in Figure 3. This value corresponds to what is shown in the graph, which shows a maximum deflection (0.0013 mm) at 2637Hz. These three points are indicated on the graph in Figure 2.To obtain various experimental values, the deflection mode resonance measurements were taken at various frequencies. A reading of 0.0010 mm could be obtained at 2200Hz, and another could be obtained from 0.0005 mm at 4800Hz. Both results corroborate the behaviour of the curve obtained from the simulation. Figure 3 shows the prototype of the manufactured tool, taking into account all the design criteria described.

Preliminary experiments to characterise the tool operation

To evaluate the behaviour of the prototype tool that was designed, burnishing tests on two pieces of aluminium A92017 and two pieces of steel G10380. The dimensions of said work pieces were 54x 44x15.5mm. The pieces were milled using a hard metal spherical cutter of 8 mm diameter, rotating at 3000min -1 with a feed rate of 330mm/min, a depth of pass of 0.5mm and a lateral swath width of 0.5mm. Conventional burnishing and vibration-assisted burnishing were compared for the same prototype tool. The process parameters used (Table 2) were defined in relation to those obtained by Travieso-Rodriguez, J.A. et al. [1]. To obtain the vibration, the tool was coupled to a vibration generator with coupled plates vibrating at a frequency of 2500Hz, which is defined in the simulation for the thicknesses h, in this study (Figure 2). After performing both burnishing processes on the work pieces, the resulting surface roughnesses of the burnished specimens were measured. Indicators measured the average surface roughness (Ra) and the maximum roughness total evaluation length (Rz) in two directions that were parallel and perpendicular to the advancement direction of the prior milling operation. Finally, the surface roughness values obtained with the vibration-assisted tool on the burnished surfaces are better than those of the tool without vibration. For the aluminium work pieces, Ra// was improved by 37%, Rz// was improved by 21%, Ra was improved by 21% and Rz was improved 24% on average. For the steel workpieces, Ra// was improved by 38%, Rz// was improved by 35%, Ra was improved by 33% and Rz was improved by 42% on average.

CONCLUSIONS

The changes in the surface roughness obtained when using the vibrations to assist the process were analysed compared to when the process was performed without the assistance of vibration. A method to characterise the plates involved in the design of a tool to be used in the ball-burnishing vibrationassisted process was successfully developed. Reliable numerical solutions were obtained to predict the plate deflection values that can be used to optimise the different basic design parameters: the thickness of the plates, the gap and the working frequency of the vibration generator. A completely new tool for the market was designed, characterised and manufactured.

[1] Travieso-Rodríguez, J. A.; Dessein, G.; González-Rojas, H. A., 2011. Improving the Surface Finish of Concave and Convex Surfaces Using a Ball-burnishing Process. Materials and Manufacturing Processes, Vol.26, Iss.12, pp 1494-1502.

[2] Travieso-Rodríguez, J. A.; 2010. Study to improve the surface finish of complex surfaces, applying a plastic deformation process (Ball-burnishing). PhD thesis, Universitat Politècnica de Catalunya.

  [3] B Guo, Q-L Zhao, M J Jackson. Ultrasonic vibration-assisted grinding of microstructured surfaces on silicon carbide ceramic materials. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture March 2012 vol. 226 pp 3553-559. [4] M R Razfar, P Sarvi, M M Abootorabi Zarchi. Experimental investigation of the surface roughness in ultrasonic-assisted milling. Proceedings of the Institution of Mechanical Engineers, Part B: Journal of Engineering Manufacture September 2011 vol. 225 no. 9 pp 1615-1620.

Figure 2 .

 2 Figure 2. Sweeping the sinusoidal excitation frequency of the M 1 plate, contrasted with the experimentally measured values at 2600Hz.

Figure 3 .

 3 Figure 3. Prototype of the ball-burnishing vibration-assisted tool.

Table 1 .

 1 Characterisation criteria for C-45K steel.[2]

	Material designation	C-45K Steel
	Density,	0.00784 gr/mm 3
	Young's modulus, E	211795 N/mm 2
	Poisson's ratio, v	0.2866
	Self-hardening coefficient, n	0.190
	Tensile Strength Yield, y	416 N/mm 2

Table 2 .

 2 Burnishing parameters used

	Parameter	Valour 1	Valour 2
	Feed-rate (mm/min), f	500	750
	Ball penetration depth (mm), p	0.75	1
	Side step width (mm), b	0.080	0.115


MOTS CLÉS

Télémédecine ; Santé mobile ; Médecine complémentaire ; Éducation thérapeutique ; Centre de santé intégré ; Pays en développement Résumé Les outils de la télémédecine et de la santé mobile offrent de grandes promesses pour ce qui est d'élargir l'accès à une santé de qualité, de soutenir la communication de l'information médicale et d'assister les processus de pharmacovigilance, dans les pays en voie de développement. Dans cet article, nous fournissons des arguments sur le potentiel d'applications de la télémédecine et de la santé mobile (mHealth) pour l'amélioration de la prestation des soins de santé en milieu rural en Afrique. Particulièrement, les développements des moyens mobiles de la télémédecine pourraient permettre de jeter les fondements d'une approche de la santé intégrant les connaissances médicales modernes avec les pratiques médicales ancestrales sur le continent africain. L'accès aux technologies de l'information et de la communication (TIC) dans les pays en développement est de plus en plus répandu ; notamment via l'utilisation de dispositifs techniques ou multimédias portables. Ceci peut favoriser une approche complémentaire de soins, en l'occurrence dans les maisons de santé communautaires (MSC), dans lesquels la pratique de la médecine conventionnelle se déroule dans un environnement où la croyance à la médecine traditionnelle est forte. Dans les MSC, on trouve notamment des volontaires communautaires qui offrent des premiers soins et font le lien entre les patients et le personnel médical qualifié exerc ¸ant dans les cliniques et les hôpitaux rapprochés. Ces volontaires ont des connaissances qui sont à la frontière de la médecine moderne et de la médecine traditionnelle qui a une forte influence dans leur approche pratique des soins de santé. L'article propose un regard holistique intéressant sur des applications potentielles de la télémédecine dans ce contexte et examine particulièrement l'éducation thérapeutique et préventive avec l'aspect toxicologique des plantes médicinales et la communication sur les potentiels effets secondaires de ces plantes.

Introduction

A recent study explains the prominence of environmental resources for health service delivery in general [START_REF] Pouliot | Relying on nature's pharmacy in rural Burkina Faso: empirical evidence of the determinants of traditional medicine consumption[END_REF]. The scale of medicinal plant use can be related to the advantages inherent in the practice of outcome-contingent contracts by African herbal healers [START_REF] Leonard | African traditional healers and outcomecontingent contracts in health care[END_REF]. Medicinal plant research has sometimes revealed certain fundamental properties (antimicrobial, anti-inflammatory, antioxidant, anticancer, and anti-diabetic activities) that have led to some progress in medical research and development [START_REF] Fokunang | Traditional medicine: past, present and future research and development prospect and integration in the national health system of Cameroon[END_REF]. We can cite the following examples: the use of phytosterols and phytostanols (for diminishing low density lipoprotein and total cholesterol), black cohosh (for relieving menopausal symptoms) and phytoestrogen extracts (e.g. isoflavones, lignans and coumestans in reducing plasma lipid levels and bone loss) [START_REF] Borrelli | Alternative and complementary therapies for the menopause[END_REF]; the medical use of African potato (Hypoxis hemerocallidea or Hypoxis rooperi) for a wide variety of diseases (e.g. intestinal parasites, cough, heart weakness and childhood convulsions) and biomedical evidence has revealed that hypoxis extracts (e.g. glucosides, sterols and sterolins tested in several in vitro and in vivo models as well as in a couple of clinical trials) possess certain pharmacological properties (antimicrobial, antiviral, anti-inflammatory, anti-diabetic, antioxidant, anticancer, cardiovascular and anticonvulsant) [START_REF] Ncube | Hypoxis (Hypoxidaceae) in African traditional medicine[END_REF]. However, there is a lack of sound studies on the comprehensive toxicity of the most used herbal products in African traditional medicine. It is therefore imperative to improve the means for informing people about toxicology aspects and adverse drug reactions of different medicinal plants used in traditional healthcare systems [START_REF] Edwards | Adverse drug reactions: definitions, diagnosis, and management[END_REF].

The challenges presented by the low number of health professionals and unequal distribution of medical infrastructures, within the context of rapid population growth, as well as existing budgetary constraints drive the need for new patient-management models and health care organizations across developing countries.

These new organizations can encourage cooperation and remove barriers between programs conducted within conventional medical frameworks and traditional medicine settings and nurture local competencies, a source of sustainable development. This would provide opportunities to extend knowledge, taking into account the expectations of patients more involved in their healthcare. We live in a pervasive (or ubiquitous) environment where widespread computing allows smart devices to recognize and automatically locate each other. This major breakthrough has been achieved thanks to recent progress in communication, information processing and human -computer interaction technologies. In fact, in both developed and developing countries, healthcare improvements must focus besides care, on prevention, promotion of good health habits and develop community medicine. This is why telemedicine must address these issues, among others, with the following objectives:

• improve access to quality health care services to all individuals regardless of their physical location (remote areas, territories with few medical professionals and dependent persons with multiple chronic conditions); • optimize the management of scarce medical resources (experts or specialists and infrastructures or devices for the prevention, monitoring and management of complex diseases); • improve collaboration between health professionals for the implementation of telemedicine procedures, whether in outpatient practices, or in medical or medico-social institutions (e.g. collaborative activities for the management of critical situations); • make the most appropriate coordinated healthcare pathways (e.g. greater regulation of healthcare channels or more standardized care pathways in healthcare provisions) by optimizing health programs within a secure and optimized medical framework.

Telemedicine is a fundamental lever for the development of these new organizations from cost-sharing arrangements to collaborative networks for data and information and knowledge exchanges. It enables organizations' socio-economic constraints to be related with national/international technological-medical advancements. Socio-medical impact includes not only conventional service delivery organizations, but also alternative medical organizations that identify complementary approaches to address contextual native problems. Collaborative management contributes to expressing a coordinated vision and harmonized direction, bonding and stimulating every level of the healthcare organization: mapping primary care (between community-based practice and conventional healthcare activity) with specialized care and rehabilitation services to which complex situations are referred. Telemedicine activities engage in emerging technologies to provide continuity of care in remote regions with an enhancement of professional capabilities through knowledge sharing. Particularly, it provides innovative methodologies to reinforce interdisciplinary practices that continuously improve the skills of medical workers in the fields of conventional and traditional medicines. As a result, there are new opportunities to improve quality of care by expediting disease prevention, management of chronic conditions and continuing medical education of people living and working in geographically remote locations (isolated regions, rural areas or islands).

Telemedicine and mobile health provide valuable tools for bridging between different cultures, traditional medical knowledge and medical information systems, and enables rich and complex African knowledge of diseases and traditional treatment approaches to be shared. We provide compelling reasons that telemedicine and mobile health (mHealth) will enhance rural healthcare delivery in Africa. Access to information and communication technologies (ICT) in developing countries is growing. We make the valid point that African home-based care (AHC) increases the number of people or patients to treat. We also link AHC to African Traditional Medicine (ATM). In Sub-Saharan Africa, AHC refers to community volunteers who provide home support and report back to medical and nursing staff at clinics and hospitals. They are involved in modern medicine and have a traditional medical background. The paper provides an interesting holistic perspective on telemedicine applications and also examines efforts in developing countries to help bridge the digital divide with the modern age of medical technology.

The paper is divided into four sections. The first section outlines various types of telemedicine and mobile health initiatives with informative characteristics. The second section describes some challenges of herbal medicine practices in an African context. The third section outlines at a high level various policy and practice issues with implementation. The fourth section describes the use of telemedicine in rural community centres. The fifth section integrates aspects concerning the relationships between information technologies and usability considerations. The conclusion delivers remarkable information on healthcare information technology (IT)/policy and provides directions for future research and development of healthcare IT applications.

Telemedicine and mobile health in Africa

Sharing knowledge between Western health practitioners and traditional practitioners is sometimes difficult, but telemedicine actually opens doors and creates avenues for the traditional medical community to work with modern medical service providers. Viable projects for rural communities create opportunities to integrate traditional and allopathic healthcare issues, for example by collaborating with formal or informal caregiver organisations and by increasing the adoption of ICT frameworks.

Intelligent decision support systems are particularly helpful, since they are able to integrate different components from emergent sensors for data capture through hardware, software, case-based reasoning technologies, communication and suggested recommendations.

For instance, in the Republic of South Africa, a recent project has implemented a remote and sustainable support system for rural healthcare delivery [START_REF] Barjis | A sustainable and affordable support system for rural healthcare delivery[END_REF]. More traditionally, primary healthcare in South Africa is home-based, established naturally within rural communities in order to provide basic nursing care by formal or informal caregivers to persons in their own homes. The proposed decision support system includes two advanced components: a home-based healthcare delivery model and a telemonitoring patient system. The former component simplifies patient information flow from home-based care workers to a local clinic or hospital while the latter component assists medical staff (nurses and doctors) in deciding the course of intervention or further treatment. Comments from participants and early evaluation results suggest that the proposed system has a positive impact on the quality of healthcare, error elimination, decision assistance and accuracy of alerting of critical cases.

In situations where there are programs that support interactive activities for information sharing between health professionals and traditional practitioners, the advances of communication technology in health care are beneficial. Realization of the telemedicine paradigm should facilitate information sharing amongst health professionals and primary caregivers so that they may adapt their decision support systems to address the specific issues of their patients. The decision support systems can provide advice based on a combination of experienced knowledge and patient numerical data. As such, the local health actors (e.g. disease generalists or specialists, social workers, nutritionists and psychologists) may provide referrals to specialized hospitals and agencies providing home-based care.

Based on the comprehensive characteristics of employed terminology and their meanings, the World Health Organization (WHO) has proposed a classification of key research areas in mHealth [START_REF] Who | New horizons for health through mobile technologies: second global survey on eHealth[END_REF]:

• healthcare call centers/health care telephone help line: provision of triage services and health care advice by trained professionals, by telephone; • toll-free emergency: often used for rapid access to health professionals or staff trained to deliver guidance during medical emergencies; • public health emergencies: can be defined as the use of mobile devices to react to urgent situations; • mobile telemedicine: can be defined as the use of functions of a mobile device (e.g., voice, text, data, imaging, or video) for different situations, such as teleconsultation or tele-expertise; • appointment reminders: comprise services that rely on voice or SMS (short message service) messages sent to patients; • Community mobilization and health promotion: defined as the use of text messaging for health promotion or alerting target groups of health campaigns; • patient records: the use of mobile devices to support the treatment of patients, including collecting and displaying patient records; • information initiatives: comprise services that offer access to health science publications or databases at the point-of-care, by means of portable devices; • patient monitoring: defined as using technology to manage, monitor, and treat a patient's illness from a distance (e.g., patients suffering from diabetes or cardiac conditions); • health surveys: the use of mobile devices for collecting and reporting health-related data; • surveillance: defined as the use of mobile devices for inputting and transmitting data that will be used by surveillance programs to track diseases; • awareness-raising: comprises the use of health information products, games, or quiz programs to instruct people on relevant health topic; • decision support systems: defined as software algorithms that help health providers to make their clinical diagnoses at the point-of-care or health managers to take actions based on data collected from health surveys.

The telemedicine system with ATM is described as an integrative medicine system, technically correct because clinical staff use the information sent by volunteers in the field to make clinical decisions. This integrative framework can be applied to health interventions across health domains to explore how and whether available eHealth technologies can support delivery of the associated types of assistance or interventions with the target African populations, e.g. the surveillance strategy (usually considered an eHealth activity). But it can also be used in telemedicine and mobile health tools. The use of surveillance procedures is helpful in emphasizing the fact that it is important for primary caregivers to be involved in analyzing the consequences of public health policy on AHC practices and intelligence reports with regard to drug safety strategy.

Challenges of herbal medicine in Africa

Nowadays, quality control of herbal medicine and good practices are indispensable for the advancement of the herbal medicine system [START_REF] Sen | Challenges and opportunities in the advancement of herbal medicine: India's position and role in a global context[END_REF]. Quality problems of herbal medicines can be categorized into: external factors (contamination, adulteration and misidentification) and internal factors (complexity and non-uniformity of the ingredients) [START_REF] Zhang | Quality of herbal medicines: Challenges and solutions[END_REF]. More generally, ensuring and improving patients' safety in integrative healthcare involves the following five research priorities (listed in order of importance) [START_REF] White | Reducing the risk of complementary and alternative medicine (CAM): Challenges and priorities[END_REF]:

• active surveillance projects including vulnerable patients and concomitant use of conventional care; • attitude to safety among ATM practitioners (i.e. the extent to which safety is integral to clinical practice) and attitude to safety in ATM professional organizations (i.e. the extent to which safety is considered integral in all thinking and decisions); • influences on, and changes in, public and patient beliefs and attitudes to ATM safety; • procedures (and their effectiveness) that ATM professional organizations use to ensure continued safe practice by their members;

There is particularly a requirement for the safety monitoring of herbal remedies, and there is a growing necessity to correctly inform consumers about the medicinal products they use. The WHO Drug Dictionary (WHO-DD), the herbal anatomic-therapeutic-chemical (HATC) classification and the system checklist for cross-referencing botanical and vernacular names are particularly valuable tools for improving the operational performance of the safety monitoring programs of herbal medicines for national pharmacovigilance centres [START_REF] Shetti | Pharmacovigilance of herbal medicines: Current state and future directions[END_REF].

Regulation is fundamental to professionalizing and protecting the image of traditional medicine practice. It will help to eliminate harmful practices and promote the positive aspects of traditional medicine [START_REF] Ncube | Hypoxis (Hypoxidaceae) in African traditional medicine[END_REF]. This will involve the adoption of common diagnostic nomenclature, therapeutic methods, or curriculum [START_REF] Peltzer | Traditional health practitioners in South Africa[END_REF]. Moreover, in situations where traditional medicine is deemed to be essential, research and development including an evaluation of therapeutic potential and toxicological profile are crucial in fostering patient confidence [START_REF] Kamsu-Foguem | Adverse drug reactions in certain African herbal medicine: literature review and stakeholders' interview[END_REF].

Potential herb-drug interactions can cause haemorrhaging, mild serotonin syndrome, induction of mania, exacerbation of extrapyramidal effects, increased risk of hypertension, and decreased or increased blood concentrations or bioavailability of certain blood substances (enzymes, hormones,. . .) or others drugs [START_REF] Fugh-Berman | Herb-drug interactions[END_REF], glycemic disorders or more severe side effects such as convulsions, medullar aplasia with leucopoenia, and therefore risk of severe infections. Governments have a responsibility to sensitize their traditional medicine practitioners about adverse drug reactions and provide them with the necessary knowledge and training that will promote rational traditional medicine practices [START_REF] Shaw | Pharmacovigilance of herbal medicine[END_REF]. An effective system of surveillance for adverse drug effects, from both a reflexive reporting and a dynamic surveillance viewpoint, must also be established [START_REF] Walji | Governance of natural health products regulation: an iterative process[END_REF]. Computational methods (e.g. surveillance methods) for adverse drug reactions or semantic formalization techniques to remove interoperability barriers, can contribute to ensuring that safe, effective and good-quality traditional medicines are available [START_REF] Kamsu-Foguem | Conceptual graphbased knowledge representation for supporting reasoning in african traditional medicine[END_REF][START_REF] Doumbouya | Telemedicine using mobile telecommunication: towards syntactic interoperability in teleexpertise[END_REF].

Telemedicine in rural community centres

Generally, rural community development incorporates the improvement of healthcare service through the provision of basic health care for the largest possible number of people. Particularly, the use of telemedicine in rural community centres implies the incorporation of modern information and communication technologies for improving medical services of persons living in rural regions. Rural telemedicine development consists primarily of teleconsultation and telemonitoring activities [START_REF] Simon | Le décret franc ¸ais de télémédecine: une garantie pour les médecins[END_REF] for community-based health services:

• teleconsultation: the medical community worker and patient are based at a rural site, while the medical professional delivers interactive consultation services from a remote urban location; • telemonitoring: the medical community worker and patient are based at a rural site, while the medical professional must correctly interpret medical data (clinical, radiological or biological), necessary for a given medical monitoring procedure in the remote workplace.

In both cases, the conditions for deployment are [START_REF] Simon | L'an III de la télémédecine en France[END_REF]:

• patient identification;

• the requirement to directly or indirectly collect a patient's free and informed consent; • the authentication of each health personnel involved in the remote consultation; • the training or preparation of the medical community worker in the use of the telemedicine device; • noting in the patient record and in the observation record (a summary of the medical act with context, diagnosis and prescription or recommendations).

A telemedicine project is based on dedicated organization of care and professional practices for the target medical environment [START_REF] Simon | Tribute to Professor Louis Lareng[END_REF][START_REF] Lareng | Telemedicine in Europe[END_REF][START_REF] Lareng | La télémédecine: grandeurs et tristesses[END_REF]. Clarifications and details on the conditions under which the telemedicine activity is performed must take into account the specificities for the provision of care in the considered region of the world [START_REF] Bernard | Systemic modeling in telemedicine[END_REF][START_REF] Kamsu-Foguem | Ontological view in telemedicine[END_REF]. Particularly, the implementation of telemedicine programs in developing countries needs to ensure the good understanding of specific social, cultural and economic characteristics. For instance, the African reality clearly shows that natural products are perceived as safe and secure, differentiating them from other pharmaceutical products. This means that patients would tend to want to increase the doses, but when they do so, they might increase risks to their health. Telemedicine activities (e.g. teleconsultation and telemonitoring) may harmonize safety monitoring organization and the effective dissemination of safety rules and procedures. The roles and responsibilities of each actor involved in the telemedicine activity must have formal written terms of description and should entail functional locations, workplace sheets and behavior guidelines with update options. Therefore, each medical or paramedical actor has to act in accordance with the required professional conditions (in force in the country) in order to participate in a telemedicine activity. Furthermore, telemedicine activities must meet some technical requirements for assuring compliance with personal health data protection and the guarantee of fundamental rights and freedoms in practice.

Community health centres play a pivotal role in rural areas in preventive and therapeutic support for local populations. They ensure the social and medical links between conventional hospitals and patients. The medical worker of a community health centre assisting with a telemedicine activity plays an essential role in highlighting real living conditions, actual experiences and expectations of the patient in consultation. As previously mentioned, at an initial stage, the two main telemedicine activities are both the teleconsultation and telemonitoring.

A framework for support in teleconsultation: the community health centre's actor assisting the patient becomes the mediator of the consultation, the relationship between the hospital doctor and the patient, and guarantees the transmission of different exchanges [START_REF] Kamsu-Foguem | User-centered visual analysis using a hybrid reasoning architecture for intensive care units[END_REF].

A framework for support in telemonitoring: the principle of the telemonitoring of patients with chronic diseases (e.g. heart disease, chronic kidney disease and diabetes) is increasingly recognized as essential in meeting the important needs of these patients [START_REF] Kamsu-Foguem | Conceptual graph operations for formal visual reasoning in the medical domain[END_REF][START_REF] Kamsu-Foguem | Using conceptual graphs for clinical guidelines representation and knowledge visualization[END_REF].

These professional actors involved in community health centres describe an improved relationship with the patient because of this support of telemedicine activities. As a result, knowledge sharing occurs between the health professionals and the patient and his family. In fact, the family members of patients can provide useful information for the planning and management of health care; in turn they benefit from the interesting care counselling provided by health professionals.

The situation of developing countries requires the deployment of telemedicine projects with low operating costs. mHealth provides the means to increase the use of digital technologies and Internet access around the world. The definition of purchase requirements for a given telemedicine project should also develop guidelines to ensure good levels of access for large numbers of African people. Besides, coherent choices with sustainable considerations can be made for specific operation and maintenance of the medical devices and information systems used in rural conditions. Some direct or indirect funding by the African governments can reduce costs for community health centres and make mHealth devices prices more affordable. In addition, non-governmental organizations and other charitable foundations may provide supplementary budgetary resources for the acquisition of critical materials, devices, and information technology equipment. Hence, the rural African population can have access to a certain number of healthcare services at more reasonable prices.

Discussion

There is an expectation that mobile phones will facilitate a range of telemedicine activities (cardiology, radiology, nephrology, dermatology, obstetrics, psychiatry and ophthalmology) in Africa [START_REF] Mars | Telemedicine and advances in urban and rural healthcare delivery in africa[END_REF]. Nevertheless, there may be certain cases or circumstances in which health workers do not necessarily accept mHealth in their routine activities. Furthermore, due to network disruptions that affect the quality of mobile phone and Internet services in Africa, the guarantee of services is not permanent. There are also healthcare professionals' organisational barriers to health information technologies [START_REF] Lluch | Healthcare professionals' organisational barriers to health information technologies -A literature review[END_REF]:

• structure of healthcare organisations; • tasks; • people policies; • incentives; • information and decision processes.

Usability issues are a great obstacle to health information technology (IT) acceptance. The health IT usability evaluation model (Health-ITUEM) was designed and used in a number of policy exercises focusing on interactions between the user and tasks in a web-based communication system [START_REF] Yen | Health information technology usability evaluation: methods? models and measures[END_REF]. Health-ITUEM concepts include:

• perceived ease of use: error prevention, other outcomes, information requirements and memorability; • perceived usefulness: learnability, competency, perform speed, flexibility/customizability; • efficiency and effectiveness.

The Health-ITUEM framework has recently been tested and evaluated to provide appropriate and useful understanding and knowledge of adjusted management in the usability issues related to mHealth technology [START_REF] Brown | Assessment of the Health IT Usability Evaluation Model (Health-ITUEM) for evaluating mobile health (mHealth) technology[END_REF]. As the potential inherent in this technology is broadly recognized by global health agencies and industrial practitioners working in the field, evidence-based strategies are needed to provide guidance to implementing organizations, help legitimize mHealth and promote related national government policies [START_REF] Labrique | H pe for mHealth: More ''y'' or ''o'' on the horizon?[END_REF]. In this respect, improvement measures, which optimize meaningful mHealth utilization and health systems, are required to assist developing countries less advanced in the field. In practice, the strengths and disadvantages associated with each mHealth category can be of a different nature, depending on the level of health workers' involvement, the degree of formality of the organisational structure, the availability of resources, initial arrangements with mobile operators and requirements regarding the application's capabilities [START_REF] Sanner | From pilot to scale: towards an mHealth typology for low-resource contexts[END_REF].

Conclusion

Telemedicine and mHealth can play an important role in forging strategic links between the different stakeholders working in research issues of traditional, complementary and alternative medicines. Particularly, the effective communication of information on toxicological evaluations of the most frequently used medicinal plants can be important in the safety and monitoring of the African traditional medicine. The promising potential of telemedicine and mHealth applications for an integrated approach in the African healthcare system can significantly contribute to the holistic dimension of traditional societies and local communities with cultural values and ancestral knowledge.

There is evidence to demonstrate that the ''informal mobile telecenter'' besides the formal one, denotes a booming informal business in Africa's mobile telephone sector. This informal business, which is practiced by millions of Africans both in urban and rural areas, involves obtaining a mobile phone and a table in an open area on a street. In addition to buying airtime, the customers can also make calls and cover the costs of communications. It is also possible to borrow mobile phones from relatives or friends and reimburse them the associated costs of communicating the findings to medical members and other healthcare organizations. As a consequence, it is realistic to consider that mobile phones can play an important role (e.g. the educational advantages of telemedicine) in reaching people living on the rural areas.

However, more research is required to learn lessons from socio-cultural and economic contexts for continuous improvement that effectively and efficiently supports primary healthcare services (evaluated from a sustainable user-centred perspective) in remote regions of Africa or others regions of the world [START_REF] Kamsu-Foguem | Telemedicine and mobile health with integrative medicine in developing countries[END_REF][START_REF] Kamsu-Foguem | Verifying a medical protocol with temporal graphs: The case of a nosocomial disease[END_REF][START_REF] Tafin-Kampé | Acute osteomyelitis due to Staphylococcus aureus in children: What is the status of treatment today?[END_REF][START_REF] Tafin-Kampé | Vitamine D et personnes âgées: enquête auprès de 192 médecins généralistes dans les Hautes-Pyrénées[END_REF]. Organizational, legal, ethical, conceptual and technological issues are crucial in allowing mHealth services to become widespread throughout the emerging area and particularly with some imperatives for their adoption in Africa [START_REF] Adebesin | A Review of Interoperability Standards in E-health and Imperatives for their Adoption in Africa[END_REF].

Disclosure of interest

The authors declare that they have no conflicts of interest concerning this article.


C Dion

V Genon-Catalot

BIDIMENSIONAL RANDOM EFFECT ESTIMATION IN MIXED STOCHASTIC DIFFERENTIAL MODEL

Keywords: AMS Subject Classification 62M05, 62G07, 60J60 Adaptive bandwidth, density estimation, kernel estimator, mean integrated squared error, mixed-effects models, stochastic differential equations

HAL is

Introduction

Mixed-effects models are used to analyse repeated measurements with similar functional form but with some variability between experiments (see e.g. [START_REF] Davidian | Nonlinear models for repeated measurement data[END_REF][START_REF] Pinheiro | Mixed-effect models in S and Splus[END_REF][START_REF] Diggle | Analysis of longitudinal data[END_REF]. The advantage is that a single estimation procedure is used to fit the overall data simultaneously. In the present work the model of interest is a mixed-effects stochastic differential equation (SDE). Each equation represents the behaviour of one subject and corresponds to one realization of the random effects. Hence the random effects represent the particularity of each process. Mixed-effects SDEs have various applications such as neuronal or pharmacokinetic modelling (see e.g. [START_REF] Picchini | Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal[END_REF][START_REF] Donnet | A review on estimation of stochastic differential equations for pharmacokinetic -pharmacodynamic models[END_REF]. Estimation methods in SDEs with random effects have been proposed in literature. The main purposes are the estimation of the common distribution of the random effects in a parametric or nonparametric way. The estimation of the common density of the random effects is mainly parametric. Most methods assume normality of the random effects and estimate the population parameters (see e.g. [START_REF] Picchini | Parameters of the diffusion leaky integrate-and-fire neuronal model for a slowly fluctuating signal[END_REF]Delattre et al., 2013;[START_REF] Delattre | Coupling the SAEM algorithm and the extended Kalman filter for maximum likelihood estimation in mixed-effects diffusion models[END_REF]). However, one can ask if this assumption is reasonable in some application contexts. Nonparametric estimation can allow us to get around this problem. To the best of our knowledge, the first reference in this context is Comte et al. (2013) who propose different nonparametric estimators and then [START_REF] Dion | Nonparametric estimation in a mixed-effect Ornstein-Uhlenbeck model[END_REF] who develops two adaptive nonparametric estimators for the Orsntein-Uhlenbeck model with an application to a neuronal database. But these two references focus on a one-dimensional random effect.

In the present work we study the case of two random effects or, in other words, of one bidimensional random effect. We want to deal with two points: the estimation of the random effects and the nonparametric estimation of their common density. This bivariate context makes the study more complex. In fact the estimation of the random effects is done using matrix norm and operator, and anisotropy appears in the density estimation part. We consider N trajectories, observed on the interval [0, T ] where T is given. For j = 1, ..., N the dynamics of each process is described by the stochastic differential equation dX j (t) = b t (X j (t))φ j dt + σ(X j (t))dW j (t) X j (0) = γ j

(1)

where φ j = (φ j,1 , φ j,2 ) t ∈ R 2 is the bidimensional random effect, b(.) = (b 1 (.), b 2 (.)) t , σ(.) are known functions defined on R, (W j ) 1≤j≤N are N independent Wiener processes and γ j is a real valued random variable. The random variables ((φ 1 , γ 1 ), . . . , (φ N , γ N )) are i.i.d. and the sequences ((φ 1 , γ 1 ), . . . , (φ N , γ N )) and (W 1 , . . . , W N ) are independent. The two random effects are not assumed independent. This model is more general than the ones investigated in Comte et al. (2013) where the drift has the form b(.)φ or b(.) + φ with φ a real valued random effect. Notice that the N trajectories (X j (t), 0 ≤ t ≤ T ) 1≤j≤N are i.i.d.. We assume that the φ j = (φ j,1 , φ j,2 ) t 's have a common bivariate density f . Our goal is twofold: first estimate the random effects φ j 's and then their density f , from the observations (X j (t), 0 ≤ t ≤ T ) 1≤j≤N , with large T and N . The estimation of the random variables φ j 's follows the steps of [START_REF] Genon-Catalot | Estimation for stochastic differential equations with mixed effects[END_REF] where only one multiplicative random effect in the drift is considered. We build an estimator A j (T ) of φ j based on the trajectory (X j (t), 0 ≤ t ≤ T ) and study its L 2 -risk. This leads to a bound of order 1/T . Then we propose a kernel estimator of the density f which uses the sample ( A j (T )) j . When f is in a Nikol'ski space a bound of the mean integrated squared error is established and the rate of convergence is evaluated. Finally a data-driven choice of the bandwidth based on a Goldenshluger and Lepski's criterion for anisotropic multi-index is proposed (see e.g. [START_REF] Kerkyacharian | Nonlinear estimation in anisotropic multiindex denoising[END_REF][START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] and leads to an adaptive estimator. Section 2 is dedicated to assumptions and definitions of some useful quantities for the estimation of φ j . In Section 3 the estimator of the random effects is built and its L 2 -risk is bounded. In particular we deal with two main examples: the Ornstein-Uhlenbeck model and the Cox-Ingersoll-Ross model.

In Section 4 the estimator of the density f is studied. Finally, Section 6 is devoted to numerical simulations to illustrate estimators. Proofs are relegated in Section 8.

Notation and assumptions

2.1. General assumptions on the model. Consider real valued processes (X j (t)) j=1,...,N given by (1). We assume that (W j ) j=1...,N and (φ j , γ j ) j=1,...,N are defined on a common probability space (Ω, F, P). Consider the following assumptions (see e.g. [START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF]Kessler et al., 2012, for details). (A2) There exists an open set Φ of R 2 and an interval (l, r) ⊂ R such that σ 2 (x) > 0 for x ∈ (l, r), and for all ϕ ∈ Φ the function s ϕ : x → exp(-2

(A1) The real valued functions x → b 1 (x), x → b 2 (x)
x x 0 b t (u)ϕ σ 2 (u) du), x 0 ∈ (l, r) satisfies l s ϕ (y)dy = +∞, r s ϕ (y)dy = +∞.
The function

m ϕ : x → 1/(σ 2 (x)s ϕ (x)) satisfies M (ϕ) = r l m ϕ (x)dx < +∞. We set π ϕ (x) := 1 (l,r) (x) m ϕ (x) M (ϕ) . ( 2
) (A3) φ j takes values in Φ, has distribution f (ϕ)dϕ and (φ j , γ j ) has distribution on Φ × (l, r) π(dϕ, dx) = f (ϕ)dϕ ⊗ π ϕ (x)dx.
Assumption (A1) ensures the existence and uniqueness of a strong solution X j (.) to (1) for all random variables (φ j , γ j ) ∈ R 3 , adapted to the filtration (F t = σ((φ j , γ j ) j=1,...,N , W j (s), s ≤ t, j = 1 . . . , N ), t ≥ 0). Then (φ j , γ j ) is F 0 -measurable and W j is a (F t , t ≥ 0) Brownian motion.

In the following, for ϕ = (ϕ 1 , ϕ 2 ) t a fixed value in Φ we denote by X ϕ the process solution of the stochastic differential equation with fixed ϕ:

dX ϕ j (t) = b t (X ϕ j (t))ϕdt + σ(X ϕ j (t))dW j (t) X ϕ j (0) ∼ π ϕ , X ϕ j (0) independent of W j .
(3)

Under (A1)-(A2), for ϕ ∈ Φ, the process defined by ( 3) is strictly stationary and ergodic with scale density s ϕ , speed density m ϕ and marginal distribution π ϕ (x)dx. Under (A1)-(A3), according to Genon-Catalot and Larédo (2014), the conditional distribution of X j given φ j = ϕ is identical to the distribution of (3), the process ((φ j , X j (t)), t ≥ 0) is strictly stationary with marginal distribution π and P(X j (t) ∈ (l, r), ∀t > 0) = 1.

Finally under (A1)-(A3) as for j = 1, . . . , N , for all t ≥ 0, X j (t) ∈ (l, r), ∀T > 0: Example.

T 0 b 2 1 σ 2 (X j (s))ds < +∞, T 0 b 2 2 σ 2 (X j (s))ds < +∞, a.s.
[1] The Ornstein Uhlenbeck model (O-U) with two random effects is defined as

dX j (t) = (φ j,1 -φ j,2 X j (t))dt + σdW j (t) X j (0) = γ j (4)
with σ(x) = σ > 0, (l, r) = R, and b(x) = (1, -x) t . Assumption (A2) requires Φ = R × (0, +∞) and leads to π ϕ = N ϕ 1 /ϕ 2 , σ 2 /(2ϕ 2 ) as invariant distribution for fixed ϕ.

Example.

[2] The Cox-Ingersoll-Ross model (C-I-R) with two random effects is defined as

dX j (t) = (φ j,1 -φ j,2 X j (t))dt + σ X j (t)dW j (t) X j (0) = γ j (5) with σ(x) = σ √
x with σ > 0, and b(x) = (1, -x) t . Assumption (A2) requires 2ϕ 1 /σ 2 ≥ 1 and 2ϕ 2 /σ 2 > 0 (in particular the process is always positive). This leads to consider Φ = (σ 2 /2, +∞) × (0, +∞). The invariant distribution, for fixed ϕ, π ϕ = Γ(2ϕ 1 /σ 2 , σ 2 /(2ϕ 2 )) is the Gamma distribution with shape parameter 2ϕ 1 /σ 2 and scale parameter σ 2 /(2ϕ 2 ).

2.3. Specific assumptions and notations for estimation. Our estimation is based on the following quantities. As in Comte et al. (2013) we define for j = 1, ..., N,

U j (T ) := T 0 b σ 2 (X j (s))dX j (s), (6)
which is a column vector with size 2 × 1 and the 2 × 2 symmetric matrix:

V j (T ) := T 0 b b t σ 2 (X j (s))ds = T 0 b 2 1 σ 2 (X j (s))ds T 0 b 1 b 2 σ 2 (X j (s))ds T 0 b 1 b 2 σ 2 (X j (s))ds T 0 b 2 2 σ 2 (X j (s))ds . (7)
Using (1), U j (T ) can be decomposed as follows

U j (T ) = V j (T )φ j + M j (T ) with M j (T ) := T 0 b σ (X j (s))dW j (s). (8)
Note that

V j (T ) =< M j > T (9)
where < . > is the quadratic variation of a continuous local martingale. For all measurable function h : (l, r) → R π ϕ -integrable for all ϕ ∈ Φ, we define the random variable π φ j (h) := r l h(x)π φ j (x)dx. (A4) For ϕ ∈ Φ and i ∈ {1, 2}, we assume

π ϕ b 2 i σ 2 = r l b 2 i σ 2 (x)π ϕ (x)dx < +∞.
We define the random matrix

L j := π φ j b b t σ 2 .
(A5) For j = 1, . . . , N , P (L j invertible) = 1.

Under (A1)-(A4), Theorem 3.1. of Genon-Catalot and Larédo (2014) gives:

< M j > t t = V j (t) t → t→+∞ L j , a.s. ( 10
)
One can notice that assumption (A5) implies that V j (t) is invertible for t large enough.

In the study we denote . the L 2 (R)-norm, . p the L p (R)-norm when p = 2, 2 the euclidean norm of R 2 and F the Frobenius norm of matrices defined for A ∈ M 2 (R) by A 2 F = i,j A 2 i,j = Tr( A t A). We denote S 2 (R) the subset of symmetric matrices of M 2 (R).

Random effect estimation

We define first for each j an estimator of the random variable φ j based on the trajectory X j (t), t ∈ [0, T ]. Then these estimators are used to build an estimator of the density f . 3.1. Definition of the estimator of the random effects. Let us define N bidimensional random variables

A j (T ) := (A j,1 (T ), A j,2 (T )) t = V j (T ) -1 U j (T ), (11)
which corresponds to the maximum likelihood estimator of φ j when φ j = ϕ is deterministic. Note that

A j (T ) = φ j + V j (T ) -1 M j (T ). (12)
Thus A j (T ) is a consistent estimator of φ j as T tends to infinity according to convergence (10).

Because of the presence of the inverse matrix V j (T ) -1 in formulae ( 11)-( 12), it is difficult to prove that A j (T ) has finite moments and to compute any of them. To overcome this theoretical difficulty, we consider a truncated estimator of φ j . Let

A j (T ) := A j (T )1 B j (T ) , B j (T ) := {V j (T ) ≥ κ √ T I 2 } = {min(λ 1,j (T ), λ 2,j (T )) ≥ κ √ T } (13)
where λ i,j (T ), i = 1, 2 are the eigenvalues of V j (T ) and I 2 is the identity matrix of M 2 (R). The inequality in the definition of B j (T ) has a matrix sense: for two matrices (A, B) ∈ S 2 (R), A ≤ B if and only if B -A is a non negative matrix (see Appendix 8.5). Relations ( 7) and (9) show that V j (T ) is a non negative symmetric matrix, thus its eigenvalues are non negative. We are able to bound the L 2 -risk of the estimator A j (T ) of φ j . This bound is needed to evaluate the mean integrated squared error of the nonparametric estimator of the density f . 3.2. Main result. We denote L ϕ the infinitesimal generator of the process (3), given for F ∈ C 2 ((l, r)), by L ϕ F (x) := (σ 2 (x)/2)F (x) + ( b t (x)ϕ)F (x). Its domain is included in L 2 πϕ which is the space of functions f such that r l f 2 dπ ϕ < ∞ (for details see e.g. [START_REF] Genon-Catalot | Stochastic volatility models as hidden markov models and statistical applications[END_REF]). When F is a matrix, the notation L ϕ F indicates that we apply the operator on each coefficient of the matrix. We also define analogously F as the matrix of derivatives. For all g = (g i,k ) 1≤i,k≤2 ∈ M 2 (R) such that π ϕ (g 2 i,k ) < +∞ for all ϕ ∈ Φ, we associate the matrix

F g ϕ = F g 1,1 ϕ F g 1,2 ϕ F g 2,1 ϕ F g 2,2 ϕ
satisfying -L ϕ F g ϕ = g -π ϕ g for all ϕ ∈ Φ . We denote for simplicity

H φ j := F b b t /σ 2 φ j . ( 14
)
Examples below show how H φ j can be constructed. We are now able to announce the main result on the estimator on the random effects.

Proposition 1. Consider the processes (X j (t), j = 1, . . . , N ) given by ( 1) under (A1)-(A5). Assume that for j = 1, . . . , N ,

E L j 2 F [det(L j )] 2 Tr (L j ) < ∞, E L j 4 F [det(L j )] 4 Tr π φ j b b t σ 2 2 < ∞, E φ j 2 2 1 L 2 j,1,1 + 1 L 2 j,2,2 π φ j ( H φ j 2 F ) + E φ j 2 2 1 L 2 j,1,1 + 1 L 2 j,2,2 π φ j ( H φ j 2 F σ 2 ) < ∞ and E π φ j ( H φ j 4 F ) + E π φ j ( H φ j 4 F σ 4 ) < ∞.
Then, there exists a constant C > 0 such that

E A j (T ) -φ j 2 2 ≤ C T . ( 15
)
The two last assumptions of Proposition 1 correspond to the application of Proposition 7 given in Section 8.1 with g = b b t /σ 2 , p = 1 and p = 2. This proposition is the key of the nonparametric estimation procedure set up in Section 4.

Key examples continued. Let us investigate the two examples given in Section

2.2. Example. [1](continued) In this case b b t σ 2 (x) = 1 σ 2 1 -x -x x 2 , V j (T ) = 1 σ 2 T - T 0 X j (s)ds - T 0 X j (s)ds T 0 X j (s) 2 ds .
For simplicity we set here φ j = φ = (φ 1 , φ 2 ) t . The limit matrix L of V (T )/T is given by

L = π φ b b t σ 2 = 1 σ 2 1 -φ 1 φ 2 -φ 1 φ 2 σ 2 2φ 2 + φ 2 1 φ 2 2 , det(L) = σ 2 2φ 2 > 0.
We can check that

H φ (x) =   1 -1 φ 2 σ 2 (x -φ 1 φ 2 ) -1 φ 2 σ 2 (x -φ 1 φ 2 ) 2φ 1 σ 2 φ 2 2 x -φ 1 φ 2 + 1 2σ 2 φ 2 x -φ 1 φ 2 2 -σ 2 2φ 2   .
Details are in Appendix. The assumptions of Proposition 1 are fulfilled if

E[φ 6 1 + φ -10 2 ] < ∞. Example. [2](continued) In this case, b b t σ 2 (x) = 1 σ 2 1 x -1 -1 x , V j (T ) = 1 σ 2 T 0 1 X j (s) ds -T -T T 0 X j (s)ds . For Φ = (σ 2 , +∞) × (0, +∞), π ϕ (x → 1/x 2 )
< ∞ for all ϕ ∈ Φ and the limit matrix L is given by:

L = π φ b b t σ 2 = 1 σ 2 2φ 2 2φ 1 -σ 2 -1 -1 φ 1 φ 2 . , det(L) = σ 2 2φ 1 -σ 2 > 0.
One can verify (see Appendix) that

H φ (x) = 2 σ 2 (2φ 1 -σ 2 ) log(x) 1 1 1 φ 2 σ 2 x -φ 1 φ 2 .
The details are relegated in Appendix. For the assumptions of Proposition 1 we must have

π φ (x → | log(x)|), π φ (x → 1/x 2 ) and π φ (x → 1/x 4 ) finite. The last condition imposes a reduction of Φ to Φ = (2σ 2 , +∞) × (0, +∞). Moreover the other assumptions are fulfilled if E[log 2 (φ 2 ) + φ 5 1 + φ 8 2 + φ -4 2 + ψ 2 (2φ 1 /σ 2 -1)] < ∞, where ψ(x) := Γ (x)/Γ(x) is the digamma function.

Nonparametric estimation

In this section we assume that f ∈ L 2 (R 2 ) and set up a nonparametric estimation procedure based on a kernel estimator. We study the obtained estimator and determinate its rate of convergence. 4.1. Nonparametric estimator of the density of the random effects. By Proposition 1, A j (T ) is a consistent estimator of φ j when T tends to infinity. It is therefore natural to define a kernel estimator based on the A j (T )'s. Let us denote by K a kernel in C 1 (R 2 ) such that the partial derivatives

∂K ∂u and ∂K ∂v are in ∈ L 2 (R 2 ), K is integrable, K(u, v)dudv = 1 and K 2 = K 2 (u, v)dudv < +∞. For all h = (h 1 , h 2 ), h 1 > 0, h 2 > 0, for all (u, v) ∈ R 2 , we denote K h (u, v) = 1 h 1 h 2 K u h 1 , v h 2 .
For example one can consider the Gaussian kernel

K(u, v) = K 1 (u)×K 1 (v), with K 1 (u) = (1/ √ 2π) exp(-u 2 /2).
We define the estimator of the density f for

x = (x 1 , x 2 ) ∈ R 2 by f h (x) = 1 N N j=1 K h (x -A j (T )). ( 16
) Denoting f h (x) := K h f (x) = f (y 1 , y 2 )K h (x 1 -y 1 , x 2 -y 2
)dy 1 dy 2 , the following result holds.

Proposition 2. Under (A1)-(A5) and under the assumptions of Proposition 1,

E[ f h -f 2 ] ≤ 2 f -f h 2 + 2 max 1 h 3 1 h 2 , 1 h 1 h 3 2 ∂K ∂u 2 + ∂K ∂v 2 C T + K 2 N h 1 h 2 (17)
with C the constant which appears in Proposition 1.

This bound comes from the bias-variance decomposition:

E[ f h -f 2 ] ≤ 2 f -f h 2 + 2 E[ f h ] -f h 2 + E[ f h -E[ f h ] 2 ].
The first term is a bias term due to the approximation of f by f h , it decreases when h 1 , h 2 decrease. The third term is a variance term which increases when h 1 , h 2 decrease. Finally, the middle term is an error term due to the approximation of the φ j 's by the A j (T )'s also increasing when h 1 , h 2 decrease. Note that the orders are consistent with the result of Comte et al. (2013), Proposition 1 for the multiplicative model. In fact in the case of a single random effect, multiplicative in the drift, the second term has the order 1/(T h 3 ) and the third term: 1/(N h).

To choose the best h, a compromise must be done between between the bias term and the variance and middle terms. 4.2. Rates of convergence. We consider anisotropic Nikol'ski classes of functions which are well fitted to evaluate the order of the bias term (see [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF] and [START_REF] Comte | Anisotropic adaptive kernel deconvolution[END_REF] for example).

Definition 3. A function f is in the Nikol'ski class N (β, R) if f : R 2 → R admits partial derivatives of order β i , i ∈ {1, 2} such that, with y 1 = (y, 0), y 2 = (0, y), for all y ∈ R, for i ∈ {1, 2}   ∂ β i f (∂x i ) β i ((x 1 , x 2 ) + y i ) - ∂ β i f (∂x i ) β i (x 1 , x 2 ) 2 dx   1/2 ≤ R|y| β i -β i , ∂ β i f (∂x i ) β i ≤ R
(with β denotes the largest integer strictly less than β).

Recall that kernel K : R → R is of order l ∈ N * if for j = 1, . . . , l, |x| j |K(x)|dx < +∞ and x j K(x)dx = 0 (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]). In this context we are now able to bound the bias term

f -f h 2 . Proposition 4. If f ∈ N (β, R) and K(x) = K(x 1 , x 2 ) = K 1 (x 1 )K 2 (x 2 ) where K 1 is a kernel of order l 1 := β 1 , K 2 a kernel of order l 2 := β 2 , |K 1 (u)u β 1 |du < ∞ and |K 2 (u)u β 2 |du < ∞, then there exist two constants c 1 , c 2 > 0 such that f -f h 2 ≤ c 1 h 2β 1 1 + c 2 h 2β 2 2 . ( 18
)
In order to minimize the MISE, if we only consider the first and the last term of the right hand side of (17), we have to minimize the following function of two variables:

(h 1 , h 2 ) → ξ(h 1 , h 2 ) = 1 N h 1 h 2 + h 2β 1 1 + h 2β 2 2 .
With 1/ β = 1/(2β 1 ) + 1/(2β 2 ), we get the minimizing values

h * 1 = O N -1 2β 1 β β+1 , h * 2 = O N -1 2β 2 β β+1 (19)
and this implies

ξ(h * 1 , h * 2 ) = O N - 4β 1 β 2 (2β 1 +1)(2β 2 +1)-1 = O N -2 β 2+2 β .
Therefore, we require (see ( 17))

1 T max 1 h * 3 1 h * 2 , 1 h * 1 h * 3 2 ≤ 1 T h * 1 h * 2 1 h * 2 1 + 1 h * 2 2 ≤ N - 4β 1 β 2 (2β 1 +1)(2β 2 +1)-1 .
This holds under the following condition

T ≥ N 1+ 4β 1 (2β 1 +1)(2β 2 +1)-1 + N 1+ 4β 2 (2β 1 +1)(2β 2 +1)-1 . ( 20
)
This condition is implied by T ≥ N 3 , and if β i > 1/2 by T ≥ N 2 . We are now able to state the order of the risk, for large N and T .

Corollary 5. Under (A1)-(A5), under the assumptions of Proposition 4, under conditions ( 19) and (20), we have 2β+2) . The classical rate of convergence for a kernel density estimator, with one bandwidth, is N -2β/(2β+1) . We observe here a 2 on the denominator which comes from the number of bandwidths.

E[ f h * -f 2 ] = O N -2 β 2 β+2 with 1/ β = 1/(2β 1 ) + 1/(2β 2 ). Notice that if β 1 = β 2 = β, thus β = β and N -2β/(2β+2) = N -2β/(
In practice the regularity parameters β and R are unknown. Thus we propose a data-driven way of choosing h based on the [START_REF] Goldenshluger | Bandwidth selection in kernel density estimation: oracle inequalities and adaptive minimax optimality[END_REF]) method. 4.3. Data-driven bandwidth and adaptation. The idea is to replace the unknown term f h -f 2 by an estimator. For this we introduce the iterated auxiliary kernel estimators

f h,h (x) := K h f h (x) = 1 N N j=1 K h K h (x -A j (T )).
Let H N,T be a finite set of bandwidths h = (h 1 , h 2 ). Following Goldenshluger and Lepski's approach, we define the estimator of the bias term by

B(h) = sup h ∈H N,T f h,h -f h 2 -v(h ) + ( 21
)
where

v(h) := κ 1 K 2 K 2 1 N h 1 h 2 + κ 2 K 2 1 1 h 3 1 h 2 + 1 h 1 h 3 2 ∂K ∂u 2 + ∂K ∂v 2 C T , (22)
is a term which has the same order as the variance terms in (17), C is the constant from Proposition 1 and κ 1 , κ 2 are numerical constants. The bandwidth is finally selected as follows

h = argmin h∈H N,T (B(h) + v(h)). ( 23
)
Then we obtain the following Theorem.

Theorem 6. We assume that the elements of H N,T satisfy 0

< h i < 1, i = 1, 2, 1/(N h 1 h 2 ) ≤ 1, 1/(h 3 1 h 2 ) ≤ T, 1/(h 1 h 3 2 ) ≤ T, and that ∀c > 0, ∃Σ(c) < ∞, h∈H N,T (h 1 h 2 ) -1/2 e -c/ √ h 1 h 2 ≤ Σ(c).
Under (A1)-(A5), under the assumptions of Proposition 1 and if K 4/3 < +∞, there exist numerical constants κ 1 , κ 2 such that

E[ f h -f 2 ] ≤ C 1 inf h∈H N,T { f -f h 2 + v(h)} + C 2 N (24) with C 1 a positive constant depending on K 1 , C 2 is a positive constant depending on f , ∂K ∂u 2 + ∂K ∂v 2 , K 1 , K 4/3 and C (the constant from Proposition 1).
This is a non-asymptotic result proving that the bias variance compromise is automatically realized by the final estimator f h which is therefore adaptive. Theorem 6 is proved in [START_REF] Dion | Nonparametric estimation in a mixed-effect Ornstein-Uhlenbeck model[END_REF] for a single random effect. As the proof is analogous we do not repeat it here. The key is to prove that

E[B(h)] ≤ f -f h 2 + c/N
. The theoretical study gives κ 1 ≥ max(40/ K 2 1 , 40) and κ 2 ≥ max(10/ K 2 1 , 10). Nevertheless, the constants κ 1 , κ 2 obtained in the proof are not optimal and thus it is standard to calibrate them from a simulation study. The conditions of Theorem 6 are fulfilled for example with the set

H N,T = {(1/k 1 2 , 1/k 2 4 ), k 1 = 1 . . . , N 1/4 , k 2 = 1 . . . , N 1/8 , N 2 ≤ T }.
In practice, note that in v(h), the constant C is unknown, and must be replaced by a rough estimator.

From the proofs of Propositions 1 and 7 it appears that C = E[C(φ j )] for an explicit function C(φ j ), thus C can be estimated by (1/N ) N j=1 C( A j (T )).

Discrete data.

In practice, only discrete observations are available. The estimation method must take into account this fact. Let us assume that we observe synchronously (X j (t)), j = 1, . . . , N at times t k = k∆, k = 1, . . . , n and set T = n∆. As it is classically done, we replace the stochastic and ordinary integrals by their discretized versions. The estimator of φ j is thus defined by

A j,n := A j,n 1 B j,n , with A j,n = V -1 j,n U j,n , (25)
and

U j,n := n-1 k=0 b σ 2 (X j (k∆))(X j ((k + 1)∆) -X j (k∆)), V j,n := n-1 k=0 ∆ b b t σ 2 (X j (k∆)), B j,n := {V j,n ≥ κ √ n∆I 2 }.
To extend the present work to discrete data, it is enough to find conditions ensuring that

E[ A j,n -φ j 2 2 ] ≤ C n∆
. The proof follows the steps of Proposition 1, with the discretized integrals instead of the continuous ones. We must study

E 1 n∆ (U j,n -U j (T )) 2p 2 , E 1 n∆ (V j,n -V j (T )) 2p F
for p = 1, 2 and extend Proposition 7 (see Section 7) to the discretized version of (1/T ) T 0 g(X j (s))ds. This is relatively standard as we assume that the processes (φ j , X j (t)) t≥0 are in stationary regime. The discretizations induce a new bias term which of order O(∆). To make this bias negligible, a classical condition on the sampling interval is : ∆ 1/(n∆), i.e. n∆ 2 1 (see [START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF]; [START_REF] Kessler | Statistical methods for stochastic differential equations[END_REF] for example). Note that other approximation of stochastic or ordinary integrals based on discrete data are available and can improve the constraint n∆ 2 1 (see e.g. [START_REF] Kloeden | Numerical solution of stochastic differential equations[END_REF] [START_REF] Iacus | Simulation and Inference for stochastic differential equation[END_REF]).

Moreover, due to the presence of random effects, additional moment conditions on the distribution of the initial condition (φ j , γ j ) are required. To avoid technical developments, we do not give more details.

Simulation study

In this section, we illustrate on simulated data the performances of our nonparametric estimation procedure on the two examples of Sections 2.2 and 3.3. For simplicity we denote f := f h and

f 1 (x 1 ) = f (x 1 , x 2 )dx 2 , f 2 (x 2 ) = f (x 1 , x 2 )dx 1 . ( 26
)
For the two examples, we simulate N independent discretized sample paths (X j (k∆), k = 1, . . . , n, j = 1, . . . , N ). This is done by simulating independently for each j first the random variable φ j , then, given φ j = ϕ, the initial variable γ j which is simulated according to the invariant distribution π ϕ corresponding to each model. Lastly, the discretized sample path is simulated. For the OU model, we use exact simulations of the discretized path. For the CIR model, we use a discretization scheme given in [START_REF] Alfonsi | On the discretization schemes for the cir (and bessel squared) processes[END_REF].

For simplicity we simulate independent components for φ j = (φ j,1 , φ j,2 ), thus the density f has the form

f (x 1 , x 2 ) = f 1 (x 1 ) × f 2 (x 2 ).
The computation of the estimators A j (T ) given by ( 13) requires to choose a value for the cut-off parameter κ. To choose κ, different functions f have been investigated with different values of ∆ and a large number of repetitions. We compared the MISEs of estimators as functions of the constant κ and selected the value κ = 0.125 as satisfactory. For comparison, we also computed the estimators for κ = 0 (no cut-off). For κ = 0 the estimator is denoted f and its marginals f 1 and f 2 .

There remains the choice of the adaptive bandwidth. In [START_REF] Dion | Nonparametric estimation in a mixed-effect Ornstein-Uhlenbeck model[END_REF] the kernel estimator with datadriven bandwidth selected by the Goldenshluger and Lepski criterion is implemented for one random effect. It is also implemented using the R-function density choosing the bandwidth by cross-validation or with the rule of thumb. The Goldenshluger and Lepski method performs better for mixture densities.

Here we have chosen to use the R-function kde2d for the choice of the bandwidths with a standard bivariate Gaussian kernel.

Example.

[1] The simulation scheme of each sample path given ϕ is as follows:

X ϕ (0) = γ ∼ N ϕ 1 ϕ 2 , σ 2 2ϕ 2
and G 1 , . . . , G n i.i.d. N (0, 1) independent of X ϕ (0),

X ϕ ((k + 1)∆) = X ϕ (k∆)e -∆ϕ 2 + ϕ 1 ϕ 2 (1 -e -∆ϕ 2 ) + σ 2 2ϕ 2 (1 -exp(-2∆ϕ 2 ))G k+1 . ( 27
)
The following distributions are chosen for the components of the random effects. We drop the index j:

• φ 1 ∼ N (1, 0.5), φ 2 ∼ Γ(10.1, 0.25), • φ 1 ∼ N (1, 0.5), φ 2 ∼ 1/Γ(3, 0.25) • φ 1 ∼ Γ(1.8, 0.8), φ 2 ∼ Γ(2, 1)
• φ 1 ∼ Γ(1.8, 0.8), φ 2 ∼ 1/Γ(3, 0.25) Note that, for the third case, the distribution of φ does not satisfy all assumptions as E[φ -10 2 ] does not exist. However the simulations show that the estimation procedure works even when theoretical assumptions do not hold.

Example.

[2] The simulation scheme of each sample path given ϕ is the one given in [START_REF] Alfonsi | On the discretization schemes for the cir (and bessel squared) processes[END_REF] which is based on the following approximation for small δ:

X ϕ (k+1)δ ∼ = X ϕ kδ + (ϕ 1 -σ 2 /2 -ϕ 2 X ϕ (k+1)δ )δ + σ X ϕ (k+1)δ (W (k+1)δ -W kδ )
. Solving for X ϕ ((k + 1)δ) yields the explicit relation:

X ϕ ((k + 1)δ) =   √ δσG k+1 + σ 2 δG 2 k+1 + 4(δ(ϕ 1 -σ 2 /2) + X ϕ (kδ))(1 + ϕ 2 δ) 2 + 2ϕ 2 δ   2 ( 28
)
where

G k+1 = (W (k+1)δ -W kδ )/ √ δ and X ϕ (0) = γ ∼ Γ(2 ϕ 1 σ 2 , σ 2 2ϕ 2 ).
The values δ = T /20000, ∆ = 10×δ and n = T /∆ = 2000 are chosen. The two random effects must be positive random variables and the first component must satisfy φ 1 > 2σ 2 . We have chosen to simulate:

•

φ 1 ∼ 2σ 2 + Γ(5, 0.5), φ 2 ∼ 1 + Γ(1.8, 0.8) • φ 1 ∼ 2σ 2 + Γ(5, 0.5), φ 2 ∼ 1/Γ(8.1, 0.05). The chosen distribution for φ 1 satisfies the assumption E[ψ 2 (2φ 1 /σ 2 -1)] < ∞. Indeed, when φ 1 ∼ 2σ 2 + Γ(k, θ), E[ψ 2 (2φ 1 /σ 2 -1)] = +∞ 2σ 2 ψ 2 (2x/σ 2 -1)e -(x-2σ 2 )/θ (x -2σ 2 ) k-1 dx.
One can use the asymptotic equivalent for the di-gamma function ψ(x) ∼ +∞ log(x) to see that the integral is convergent.

Here, the first case does not satisfy E[φ -4 2 ] < ∞. But the estimation results are satisfactory. Figures 1 and4 illustrate the influence of the cut-off κ on the estimators for the two examples. On top, the estimators A j (T ) (κ = 0) are plotted as function of the simulated variables φ j . Top left is for the first component, top right for the second. The bottom figures represent the estimators A j (T ) (κ = 0.125) as function of the simulated φ j . The random variables φ j are well estimated and On Figures 2 (OU example) and 5 (CIR example), estimators of the two marginal densities, i.e. the densities of the two components of the random effects, are shown. The true marginals are in bold black. Three density estimators f 1 (left) and f 1 (right), built using the A j (T )'s (κ = 0), are in dotted (blue) lines. Three density estimators f 1 (left), f 2 (right), built using the A j (T )'s (κ = 0.125), are in continuous (red) lines. The cut-off for estimating the φ j 's improves the density estimators: especially, the first marginal f 1 fits better the true density than f 1 .

Figures 3 and6 show the true bivariate density (top left) and the estimator f (top right). The bottom graphs correspond to sections x 1 → f (x 1 , x 2 ) (bold left) and x 1 → f (x 1 , x 2 ) (dotted left) for a fixed x 2 , and x 2 → f (x 1 , x 2 ) (bold right) and x 2 → f (x 1 , x 2 ) (dotted right) for a fixed x 1 . The estimators are close to the true density.

In order to evaluate with more acuracy the performances of our estimator on the two examples, we have computed their empirical MISE from 100 simulated data sets, for different values of σ and T and for the chosen distributions of (φ 1 , φ 2 ). Results are presented in Tables 1 and2.

On Tables 1 and2 the influence of the cut-off is often visible when N = 20: the MISEs of f are larger than the MISEs of f . It almost disappears on the Tables when N = 200. For a given value of N , when T increases from 10 to 100 the MISE decreases (twice smaller sometimes). The best results are obtained when T = 100 and N = 200. When φ 1 is Gaussian the estimation seems more difficult than when it is Gamma. The influence of σ may be seen when T is not very large. For example for the OU-model, the quantity σ 2 /T appears is the theoretical bound of the MISE and thus it must not T δ ∆ T /∆ f f 10 0.005 0.005 × 2 1000 0.038 0.037 100 0.05 0.05 × 2 1000 0.021 0.025 100 0.005 0.005 × 2 10000 0.019 0.020 T δ ∆ T /∆ f f 10 0.005 0.005 × 2 1000 0.029 0.025 100 0.05 0.05 × 2 1000 0.006 0.007 100 0.005 0.005 × 2 10000 0.006 0.007 be too large. When T is large (T = 100) there is very little difference between the MISE's values for σ = 0.1 and σ = 1.

σ T N f f f 1 f 2 φ 1 ∼ Γ, φ 2 ∼ Γ
Table 3 highlights the different roles of T and ∆. When T increases (from 10 to 100) with the same ∆, the MISEs are divided by 2. When both T and ∆ (from 0.005 × 2 to 0.05 × 2) increase, we still note a slight improvement. However, T /∆ has clearly no influence and this is why in the previous tables we fixed the number of observations per trajectory at T /∆ = 2000 (∆ = 10 × δ). The right table highlights again the role of N . 1

Concluding remarks

In this work, we first provide an estimator of a bivariate random effect in the drift of a stochastic differential equation based on the observation of one trajectory given by (1). The definition of the estimator uses a cut-off parameter which allows us to study and bound its L 2 -risk. Then, using the estimators of the random effects, we build a kernel estimator of the common bivariate density of the random effects, from N i.i.d. observed trajectories and propose a data-driven selection of the bandwidth. The Orsntein-Uhlenbeck and the Cox-Ingersoll-Ross models both with two random effects can be studied with our estimation procedure. Illustrations on simulated data are done and show the good performances of our estimator.

As these two models have many applications in finance and in neuroscience, it would be interesting to investigate our nonparametric method on real data.

The method developed here for two random effects can be easily developped for d random effects. The extension to more general drift forms would require further work and other tools to accommodate the estimation procedure, especially for estimating the random effects. Models including a linear random effect in the diffusion coefficient could be investigated, for instance using the ideas developed in [START_REF] Delattre | Estimation of population parameters in stochastic differential equations with random effects in the diffusion coefficient[END_REF].

1 Program codes are available on the web page: https://owncloud.math-info.univ-paris5.fr/public.php?service= files&t=bc269d7843f844a74b7cff211a751d26

The nonparametric estimation of the drift function in SDEs with no random effect has been largely investigated in the literature (see e.g. [START_REF] Kutoyants | Statistical Inference for Ergodic Diffusion Processes[END_REF][START_REF] Comte | Penalized nonparametric mean square estimation of the coefficients of diffusion processes[END_REF][START_REF] Schmisser | Penalized nonparametric drift estimation for a multidimensional diffusion process[END_REF]. In the case of SDEs with random effects and a general drift b(x, ϕ 1 , ϕ 2 ), the nonparametric estimation of the function b is open and of interest. 8. Proofs 8.1. An additional result and its proof. We denote L ϕ the generator of the process given by (1) when φ j = ϕ, given for all F ∈ C 2 ((l, r)), by L ϕ F (x) := (σ 2 (x)/2)F "(x) + ( b t (x)ϕ)F (x). Then when F is a matrix, the notation L ϕ F only indicated that we apply the operator on each coefficient of the matrix. We also define in this sense F as the derivative matrix coefficients by coefficients.

Proposition 7. Consider the processes (X j (t)) given by ( 1) under (A1)-(A4). Let g = (g i,k ) 1≤i,k≤2 a matrix of S 2 (R), such that π ϕ g 2 i,k < +∞ for all ϕ ∈ Φ. Assume that

F g ϕ = F g 1,1 ϕ F g 1,2 ϕ F g 1,2 ϕ F g 2,2 ϕ
satisfies -L ϕ F g ϕ = g -π ϕ g for all ϕ ∈ Φ . Let J(φ j ) be a non negative measurable function of φ j such that for all p ≥ 1

E J(φ j ) 1 T p π φ j F g φ j 2p F + π φ j (F g φ j ) 2p F σ 2p < ∞.
Then for all p ≥ 1, there exists a constant C p > 0 depending on p such that

E J(φ j ) √ T 1 T T 0 g(X j (s))ds -π φ j (g) 2p F ≤ C p 1 T p E J(φ j )π φ j F g φ j 2p F +E J(φ j )π φ j (F g φ j ) 2p F σ 2p .
We demonstrate the result without the function J but it can be added all along. The subscript j is omitted for simplicity. Denote g ϕ := g -π ϕ g. Ito's formula applied to F g ϕ term by term, and the equality L ϕ F g ϕ = -g ϕ , lead to

F g ϕ (X ϕ (T )) = F g ϕ (X ϕ (0)) + T 0 L ϕ F g ϕ (X ϕ (s))ds + T 0 (F g ϕ ) (X ϕ )σ(X ϕ (s))dW (s) = F g ϕ (X ϕ (0)) - T 0 g ϕ (X ϕ (s))ds + T 0 (F g ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s). Thus, T 0 g ϕ (X ϕ (s))ds = T 0 (F g ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s) + F g ϕ (X ϕ (0)) -F g ϕ (X ϕ (T )) .
Then with Hölder's inequality it yields

E T 0 g ϕ (X ϕ (s))ds 2p F ≤ 2 2p-1 (T a + T b )
with

T a := E T 0 (F g ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s) 2p F , T b := E F g ϕ (X ϕ (0)) -F g ϕ (X ϕ (T )) 2p F . ( 29
)
We study first the term T a given by (29). Again, Hölder's inequality gives

T a = E     1≤i,k≤2 T 0 (F g i,k ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s) 2   p   ≤ 4 p-1 E   1≤i,k≤2 T 0 (F g i,k ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s) 2p   .
The first term of the sum can by studied with the Burkholder-Davis-Gundy (B.D.G.) inequality (see e.g. [START_REF] Revuz | Continuous martingales and Brownian motion[END_REF][START_REF] Gall | Calcul stochastique et processus de markov[END_REF]. Thus there exists a constant c p > 0 such that

E T 0 (F g 1,1 ϕ ) (X ϕ (s))σ(X ϕ (s))dW (s) 2p ≤ c p E T 0 ((F g 1,1 ϕ ) (X ϕ (s))σ(X ϕ (s))) 2 ds p ≤ c p T p-1 E T 0 ((F g 1,1 ϕ ) (X ϕ (s))σ(X ϕ (s))) 2p ds = c p T p-1 T π ϕ (((F g 1,1 ϕ ) σ) 2p
). Finally there is a constant C p := 4 p-1 c p verifying

T a ≤ C p T p π ϕ     1≤i,k≤2 (F g i,k ϕ ) 2p   σ 2p   ≤ C p T p π ϕ (F g ϕ ) 2p F σ 2p .
Furthermore, we study term T b (29) and Cauchy-Schwarz's inequality leads to

T b ≤ E ( F g ϕ (X ϕ (0)) 2p F + F g ϕ (X ϕ (T )) 2p F ) 1/(2p) 2 1-1/(2p) 2p ≤ 2 2p-1 E F g ϕ (X ϕ (0)) 2p F + F g ϕ (X ϕ (T )) 2p F = 2 2p π ϕ ( F g ϕ 2p F ).
Finally, with C p := max(2 2p , C p ) it yields

E 1 T p T 0 g ϕ (X ϕ (s))ds 2p F ≤ C p π ϕ (F g ϕ ) 2p F σ 2p + 1 T p π ϕ ( F g ϕ 2p F ) .
8.2. Proof of Proposition 1. The subscript j is omitted for simplicity.

Proof. First note that A -φ = (A -φ)1 B(T ) -φ1 B(T ) c . Thus:

T A -φ 2 2 = T (A -φ)1 B(T ) -φ1 B(T ) c 2 2 ≤ 2 (T 1 + T 2 ) with T 1 := T (A -φ) 2 2 1 B(T ) , T 2 := T φ 2 2 1 B(T ) c . ( 30
)
• Study of T 1 . We have

T 1 = 1 B(T ) T V (T ) -1 M (T ) 2 2 = 1 B(T ) T V (T ) -1 M (T ) √ T 2 2 = 1 B(T ) (T V (T ) -1 -L -1 + L -1 ) M (T ) √ T 2 2 ≤ 2T 1 + 2T 1 "
with

T 1 := 1 B(T ) (T V (T ) -1 -L -1 ) M (T ) √ T 2 2 , T 1 " := 1 B(T ) L -1 M (T ) √ T 2 2 (31)
Let us study this two terms separately. Note that

E V (T ) T φ = 1 T T 0 E b b t σ 2 (X(s)) φ ds, E b b t σ 2 (X(s)) φ = ϕ = E b b t σ 2 (X ϕ (s)) = E b b t σ 2 (X ϕ (0)) = π ϕ b b t σ 2 . Thus L := π φ b b t σ 2 = E V (T ) T φ . • Study of T 1 ". As L is F 0 measurable, E[T 1 "] ≤ E L -1 2 F M (T ) √ T 2 2 = E L -1 2 F E M 2 1 (T ) T + M 2 2 (T ) T F 0 = E L -1 2 F < M 1 > T T + < M 2 > T T = E L -1 2 F Tr V (T )
T Thus we obtain:

E[T 1 "] ≤ E L -1 2 F E Tr V (T ) T φ = E L -1 2 F Tr(L) .
Using Property 9 (see Appendix), we must assume :

E L 2 F det(L) 2 Tr (L) < ∞. (32)
• Study of T 1 given by ( 31). This term will use the cut-off B(T ). We have:

T V (T ) -1 -L -1 = (T V (T ) -1 L -I)L -1 = V (T ) -1 (T L -V (T ))L -1 . ( 33
)
Then we obtain:

T V (T ) -1 -L -1 M (T ) √ T = V (T ) -1 (T L -V (T ))L -1 M (T ) √ T = T V (T ) -1 L - V (T ) T matrix L -1 M (T ) √ T vector and E[T 1 ] ≤ E T V -1 (T ) 2 F 1 B(T ) L - V (T ) T 2 F L -1 M (T ) √ T 2 2 . But V (T ) -1 2 F = det(V (T ) -2 ) V (T ) 2 F = λ 1 (T ) -2 + λ 2 (T ) -2 .
And the definition of the set B(T ) implies:

T V (T ) -1 2 F ≤ 2T κ 2 . ( 34
)
Thus the Hölder inequality yields

E[T 1 ] ≤ E 2T κ 2 L - V (T ) T 2 F L -1 M (T ) √ T 2 2 ≤ 2 κ 2 (E[T c ]E[T d ]) 1/2 with T c := √ T L - V (T ) T 4 F , T d := L -1 M (T ) √ T 4 2 . ( 35
)
• Study of term T c . We apply here Proposition 7 with p = 2 and g = b t b/σ 2 and get

E [T c ] ≤ C 1 T 2 E H φ 4 F + E H φ 4 F σ 4 < ∞. (36)
• Study of term T d . We notice that:

E[T d ] ≤ E L -1 4 F M (T ) √ T 4 2 .
Proceeding as for T 1 " and using the B.D.G. inequality implies that for some constant c > 0,

E L -1 4 F M (T ) √ T 4 2 = E L -1 4 F M 2 1 (T ) + M 2 2 (T ) T 2 ≤ c E L -1 4 F E Tr V (T ) T 2 φ .
And by the Hölder inequality,

E Tr V (T ) T 2 φ = ϕ = E 1 T T 0 b 2 1 σ 2 (X ϕ (s)) + b 2 2 σ 2 (X ϕ (s)) ds 2 ≤ E 1 T T 0 b 2 1 σ 2 (X ϕ (s)) + b 2 2 σ 2 (X ϕ (s)) 2 ds = Tr π ϕ b b t σ 2 2 .
Thus,

E [T d ] ≤ E L -1 4 F Tr π φ b b t σ 2 2 .
Finally we must assume that

E L 4 F det(L) 4 Tr π φ b b t σ 2 2 < ∞. (37)
• Study of term T 2 . Term T 2 is given by (30). Note that

E[T 2 ] = T E[E[ φ 2 2 1 {B(T ) c } φ]] = T E[ φ 2 2 E[ 1 {B(T ) c } φ]].
We need to compute the quantity P(B(T ) c |φ = ϕ)] for all ϕ. We have:

B(T ) c = V (T ) T ≤ κ √ T I 2 = L - V (T ) T ≥ L - κ √ T I 2 . If we denote l := π ϕ b b t σ 2 , l - κ √ T I 2 ≥ l 2 ⇔ l - l 2 ≥ κ √ T I 2 ⇔ l ≥ 2κ √ T I 2 . Furthermore, P V (T ) T ≤ κ √ T I 2 φ = ϕ ≤ 1 l≥ 2κ √ T I 2 P V (T ) T ≤ κ √ T I 2 φ = ϕ + 1 l≤ 2κ √ T I 2 ≤ T α + T β with T α := P l - V (T ) T ≥ l 2 φ = ϕ , T β := 1 l≤ 2κ √ T I 2 . ( 38
)
We study the two terms separately. First:

T β = 1 l≤ 2κ √ T I 2 = 1 l -1 ≥ √ T 2κ I 2 = 1 min{ 1 λ ϕ,1 , 1 λ ϕ,2 }≥ √ T 2κ ≤ 4κ 2 T min 1 λ ϕ,1 , 1 λ ϕ,2
Secondly, for the term T α given by formula (38), we use Property 11 (Appendix)

l - V (T ) T ≥ l 2 ⇒ l i,i - V i,i (T ) T ≥ l i,i 2 , i = 1, 2 then T α ≤ P l 1,1 - V 1,1 (T ) T ≥ l 1,1 2 φ = ϕ + P l 2,2 - V 2,2 (T ) T ≥ l 2,2 2 φ = ϕ .
For the two terms of the right hand side we are able now to use Markov inequality:

T α ≤ 2 l 1,1 2 E l 1,1 - V 1,1 (T ) T 2 φ = ϕ + 2 l 2,2 2 E l 2,2 - V 2,2 (T ) T 2 φ = ϕ .
Finally we obtain

E[T 2 ] ≤ T E φ 2 2 2 L 1,1 2 L 1,1 - V 1,1 (T ) T 2 + 2 L 2,2 2 L 2,2 - V 2,2 (T ) T 2 (39) + 4κ 2 T min 1 λ 1 , 1 λ 2 ≤ E 4 φ 2 2 1 L 2 1,1 + 1 L 2 2,2 √ T L - V ( T ) T 2 + E 4κ 2 φ 2 2 min 1 λ 1 , 1 λ 2 (40)
We are able to conclude this proof using the Proposition 7 with J

(φ) = φ 2 2 1 L 2 1,1 + 1 L 2 2,2
. Indeed, as

T E[ A -φ 2 ] ≤ 4E[T 1 ]4E[T 1 "] + 2E[T 2 ],
gathering the bounds (32), ( 36), (37),and (39) we obtain that there is a constant

C > 0 such that E[ A j (T ) -φ j 2 2 ] ≤ C/T . 8.3. Proof of proposition 2.
Proof. With the Cauchy-Schwarz inequality, we obtain the following decomposition

E[ f h -f 2 ] = f -E[ f h ] 2 + E[ E[ f h ] -f h 2 ] ≤ 2 f -f h 2 + 2 f h -E[ f h ] 2 + E[ f h -E[ f h ] 2 ].
Let us study the second term of the bound. For all

x = (x 1 , x 2 ) ∈ R 2 , E[ f h (x)] = 1 N N j=1 E[K h (x -A j (T ))] = E[K h (x -A 1 (T ))], thus f h = E[ f h ].
The subscript j (or 1) is omitted in the following for simplicity. Note that

f h (x) = f h (x 1 , x 2 ) = f K h (x 1 , x 2 ) = 1 h 1 h 2 R 2 f (u, v)K x 1 -u h 1 , x 2 -v h 2 dudv = E[K h (x -φ)]. Then to study f h -E[ f h ] 2 = R 2 E[K h (x -A(T )) -K h (x -φ)] 2 dx we denote DK h = ( ∂K h ∂u , ∂K h ∂v ) t , with ∂K h ∂u (u, v) = 1 h 2 1 h 2 ∂K ∂u u h 1 , v h 2 , ∂K h ∂v (u, v) = 1 h 1 h 2 2 ∂K ∂v u h 1 , v h 2
and applying Taylor's formula with U (x, t) := (U 1 (x, t), U 2 (x, t)) := x -φ + t(φ -A(T ), it yields

K h (x -A(T )) -K h (x -φ) = 1 0 DK h (U (x, t)).(φ -A(T ))dt. Thus f h -E[ f h ] 2 ≤ E (K h (x -φ) -K h (x -A(T ))) 2 dx = E (K h (x -φ) -K h (x -A(T ))) 2 dx ≤ E 2 R 2 1 0 2 h 2 1 h 2 ∂K ∂u U 1 (x, t) h 1 , U 2 (x, t) h 2 dt 2 dx 1 dx 2 (φ 1 -A 1 (T )) 2 +E 2 R 2 1 0 2 h 1 h 2 2 ∂K ∂v U 1 (x, t) h 1 , U 2 (x, t) h 2 dt 2 dx 1 dx 2 (φ 2 -A 2 (T )) 2 .
The first term of the previous sum is bounded according to Proposition 1:

E R 2 1 0 1 h 2 1 h 2 ∂K ∂u U 1,1 h 1 , U 1,2 h 2 dt 2 dx 1 dx 2 (φ 1,1 -A 1,1 (T )) 2 ≤ E 1 h 4 1 h 2 2 1 0 dt R 2 ∂K ∂u (y 1 , y 2 ) 2 dy 1 dy 2 h 1 h 2 (φ 1,1 -A 1,1 (T )) 2 ≤ 1 h 3 1 h 2 R 2 ∂K ∂u (y 1 , y 2 ) 2 dy 1 dy 2 E[(φ 1,1 -A 1,1 (T )) 2 ] ≤ C h 3 1 h 2 T ∂K ∂u 2
with C the constant from Equation (15). The same arguments works for the second term and finally it yields

f h -E[ f h ] 2 ≤ C max 1 h 3 1 h 2 , 1 h 1 h 3 2 ∂K ∂u 2 + ∂K ∂v 2 T .
Finally, the last term is bounded by

E[ f h -E[ f h ] 2 ] ≤ K 2 N h 1 h 2 .
8.4. Proof of Proposition 4. For all x = (x 1 , x 2 ) ∈ R 2 ,

f h (x) -f (x) = f K h (x) -f (x) = R 2 K(v)[f (x -vh) -f (x)]dv.
Then,

f (x 1 -v 1 h 1 , x 2 -v 2 h 2 )-f (x 1 , x 2 ) = f (x 1 -v 1 h 1 , x 2 -v 2 h 2 )-f (x 1 -v 1 h 1 , x 2 )+f (x 1 -v 1 h 1 , x 2 )-f (x 1 , x 2 ).
As in [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF] we apply Taylor's formula to the two partial functions: t → f (x 1 -v 1 h 1 , t) at the order l 2 and t → f (t, x 2 ) at the order l 1 . Using the orders of the two kernels yields for the first term,

B 1 (x) := R 2 K 1 (v 1 )K 2 (v 2 )[f (x 1 -v 1 h 1 , x 2 -v 2 h 2 ) -f (x 1 -v 1 h 1 , x 2 )]dv 1 dv 2 (41) = - R 2 (h 2 v 2 ) l2 (l 2 -1)! 1 0 (1 -t) l2-1 ∂ l2 ∂x l2 2 f (x 1 -v 1 h 1 , x 2 -v 2 h 2 + th 2 v 2 )dt K 1 (v 1 )K 2 (v 2 )dv 1 dv 2 = - R 2 (h 2 v 2 ) l2 (l 2 -1)! 1 0 (1 -t) l2-1 ∂ l2 ∂x l2 2 f (x 1 -v 1 h 1 , x 2 -v 2 h 2 + th 2 v 2 ) - ∂ l2 ∂x l2 2 f (x 1 -v 1 h 1 , x 2 -v 2 h 2 ) dtK 1 (v 1 )K 2 (v 2 )dv 1 dv 2
and the analogue term B 2 (x). To evaluate f -f h 2 = (f h (x) -f (x)) 2 dx we remind the generalized Minkowski's inequality (see [START_REF] Tsybakov | Introduction to nonparametric estimation[END_REF]: for any measurable function f on R 2 , we have

f (u, x)du 2 dx ≤ f 2 (u, x)dx 1/2 du 2 .
Looking at the first term (41), and applying twice the above inequality implies for f ∈ N (β, R):

R 2 B 2 1 (x)dx ≤ R 2 |K 1 (v 1 )K 2 (v 2 )| h 2 |v 2 | l2 (l 2 -1)! 1 0 (1 -t) l2-1 R 2 ∂ l2 ∂x l2 2 f (x 1 -v 1 h 1 , x 2 -v 2 h 2 + t h 2 v 2 ) - ∂ l2 ∂x l2 2 f (x 1 -v 1 h 1 , x 2 -v 2 h 2 ) 2 dx 1 dx 2   1/2 dt dv 1 dv 2    2 ≤ R 2 |K 1 (v 1 )K 2 (v 2 )|h 2 β2 |v 2 | β2 l 2 ! R dv 1 dv 2
8.6. Details on examples. We refer to [START_REF] Genon-Catalot | Stochastic volatility models as hidden markov models and statistical applications[END_REF] and Genon-Catalot and Larédo (2014) for details and properties on the generator infinitesimal. Nevertheless we recall one of them. For f ∈ C 2 ((l, r)),

L ϕ f (x) = 1 2 σ 2 (x)f (x) + ( t b(x)ϕ)f (x) = 1 2m ϕ (x) f s ϕ (x). (42)
Thus for g ∈ L 2 πϕ when we are looking for the associated F g ϕ such that L ϕ F g ϕ = -(g -π ϕ (g)) we can use the relation (F g ϕ ) (x) = -2s ϕ (x)

x l (g(u) -π ϕ (g))m ϕ (u)du.

For the O-U model.

We want to explicit

H φ = F b b t /σ 2 φ
. Let us denote: g = b b t /σ 2 = (g i,k ) 1≤i,k≤2 . First: g 1,1 (x) -π ϕ g 1,1 = 0 = -L ϕ F g 1,1

ϕ (x). For example F g 1,1

ϕ (x) = 1 is suitable. Then, g 1,2 (x) -π ϕ g 1,2 = -1 σ 2 x -ϕ 1 ϕ 2 . We look for F g 1,2 ϕ (x) = -a σ 2 x -ϕ 1 ϕ 2 and we obtain

F g 1,2 ϕ (x) = - 1 ϕ 2 σ 2 x - ϕ 1 ϕ 2 . Finally g 2,2 (x) -π ϕ g 2,2 = x 2 σ 2 - 1 2ϕ 2 + ϕ 2 1 σ 2 ϕ 2 2 = 1 σ 2 x - ϕ 1 ϕ 2 2 + 2 ϕ 1 ϕ 2 x - ϕ 1 ϕ 2 - 1 2ϕ 2 .
We look for F g 2,2 ϕ with the same functional form and we obtain

F g 2,2 ϕ (x) = 2ϕ 1 σ 2 ϕ 2 2 x - ϕ 1 ϕ 2 + 1 2σ 2 ϕ 2 x - ϕ 1 ϕ 2 2 - σ 2 2ϕ 2 .
For the C-I-R model.

We want to explicit H φ = F b b t /σ 2 φ

. First: g 1,1 (x) -π ϕ g 1,1 = 1/(σ 2 x) -2ϕ 2 /((2ϕ 1 -σ 2 )σ 2 ). Here we use formula (42). We have

(F g 1,1 ϕ ) (x) = -2s ϕ (x) x 0 1 σ 2 u - 2ϕ 2 σ 2 (2ϕ 1 -σ 2 )
m ϕ (u)du with s ϕ (x) = e 2ϕ 2 x/σ 2 x -2ϕ 1 /σ 2 . This is equivalent to

(F g 1,1
ϕ ) (x) = -2 σ 4 e 2ϕ 2 x/σ 2 x -2ϕ 1 /σ 2

x 0 e -2ϕ 2 u/σ 2 u 2ϕ 1 /σ 2 -2 du -x 0 2ϕ 2 2ϕ 1 -σ 2 e -2ϕ 2 u/σ 2 u 2ϕ 1 /σ 2 -1 du and an integration by part in the first integral gives

(F g 1,1 ϕ ) (x) = - 2 σ 4 e 2ϕ 2 x/σ 2 x -2ϕ 1 /σ 2 - e -2ϕ 2 x/σ 2 x 2ϕ 1 /σ 2 -1 2ϕ 1 /σ 2 -1 = 2 σ 2 (2ϕ 1 -σ 2 )x .
We finally set

F g 1,1 ϕ (x) = 2 σ 2 (2φ 1 -σ 2 ) log(x).
Furthermore g 1,2 (x) -π ϕ g 1,2 = 0 = -L ϕ F g 1,2

ϕ (x), and then for example F g 1,2

ϕ (x) = 1. At last g 2,2 (x) -π ϕ g 2,2 = 1 σ 2 (x -ϕ 1 ϕ 2 ) and we obtain

F g 2,2 ϕ (x) = 1 ϕ 2 σ 2 x - ϕ 1 ϕ 2 .

  and x → σ(x) are continuous on R and b 1 , b 2 Lipschitz and σ Hölder with exponent belonging to [1/2, 1].

Figure 1 .

 1 Figure 1. O-U example. Top left: A j,1 (T )'s as a function of φ j,1 , top right: A j,2 (T )'s as a function of φ j,2 . Bottom left: A j,1 (T )'s as a function of φ j,1 , bottom right: A j,2 (T )'s as a function of φ j,2 . With φ 1 ∼ N (1, 0.5), φ 2 ∼ Γ(10.1, 0.25), T = 10, N = 200, σ = 0.1

Figure 3 .

 3 Figure 3. O-U example. Top left: true density f , top right: estimator f . Bottom: sections. Bottom left: dark line for x 1 → f (x 1 , x 2 ) versus the estimator x 1 → f (x 1 , x 2 ) light line (red) for x 2 fixed. Bottom right: dark line for x 2 → f (x 1 , x 2 ) versus the estimator x 2 → f (x 1 , x 2 ) light line (red), for x 1 fixed. Simulation with φ 1 ∼ N (1, 0.5), φ 2 ∼ Γ(10.1, 0.25), T = 100, N = 200, σ = 0.1.

Figure 4 .

 4 Figure 4. C-I-R example. Top left: A j,1 (T )'s as a function of φ j,1 , top right A j,2 (T )'s as a function of φ j,2 . Bottom left: A j,1 (T )'s as a function of φ j,1 , top right: A j,2 (T )'s as a function of φ j,2 . With φ 1 ∼ 2σ 2 + Γ(5, 0.5), φ 2 ∼ 1 + Γ(1.8, 0.8), T = 10, N = 200, σ = 0.1.

Figure 5 .

 5 Figure5. C-I-R example. On the left: dark bold line f 1 , 3 estimators f 1 (x 1 ) in dotted grey (green) and f 1 (x 1 ) in grey (red), on the right: dark bold line f 2 , 3 estimators f 2 (x 2 ) in dotted grey (green) and f 2 (x 2 ) in grey (red). Simulation with φ 1 ∼ 2σ 2 + Γ(5, 0.5), φ 2 ∼ Γ(1.8, 0.8), T = 100, N = 200, σ = 0.1.

Figure 6 .

 6 Figure 6. C-I-R example. Top left: true density f , top right: estimator f . Bottom: sections. Bottom left: dark line for x 1 → f (x 1 , x 2 ) versus the estimator x 1 → f (x 1 , x 2 ) light line (red) for x 2 fixed. Bottom right: dark line for x 2 → f (x 1 , x 2 ) versus the estimator x 2 → f (x 1 , x 2 ) light line (red), for x 1 fixed. Simulation with φ 1 ∼ 2σ 2 + Γ(5, 0.5), φ 2 ∼ 1/Γ(8.1, 0.05), T = 100, N = 200, σ = 0.1.

  CIR example. MISE computed from 100 simulations, with N = 20 on the left and N = 200 on the right, σ = 0.1, Gamma distributions for the random effects.

Table 1 .

 1 O-U example. MISE computed from 100 simulations

	Case	σ	T	N	f	f	f 1	f 2
	φ 1 ∼ N , φ 2 ∼ Γ 0.1 10 20 0.064 0.064 0.067 0.073
				200 0.039 0.038 0.038 0.047
			100 20 0.046 0.046 0.047 0.030
				200 0.010 0.010 0.009 0.006
		1 100 20 0.046 0.046 0.044 0.033
				200 0.012 0.012 0.011 0.007
	φ 1 ∼ N , φ 2 ∼ 1/Γ 0.1 10 20 0.077 0.075 0.113 0.081
				200 0.050 0.051 0.008 0.060
			100 20 0.068 0.068 0.083 0.040
				200 0.021 0.027 0.035 0.010
		1 100 20 0.061 0.061 0.061 0.044
				200 0.021 0.021 0.035 0.011
	φ 1 ∼ Γ, φ 2 ∼ Γ 0.1 10 20 0.028 0.024 0.056 0.047
				200 0.015 0.013 0.030 0.028
			100 20 0.025 0.025 0.029 0.027
				200 0.009 0.009 0.007 0.011
		1 100 20 0.022 0.022 0.025 0.025
				200 0.008 0.008 0.007 0.010
	φ 1 ∼ Γ, φ 2 ∼ 1/Γ 0.1 10 20 0.045 0.042 0.061 0.074
				200 0.031 0.030 0.030 0.076
			100 20 0.030 0.030 0.028 0.035
				200 0.008 0.008 0.007 0.009
		1 100 20 0.029 0.029 0.027 0.034
				200 0.008 0.008 0.007 0.008

the cut-off seems to have the adequate effect: it sets to zero the values which are too far from the diagonal.

Table 2 .

 2 C-I-R example. MISE computed from 100 simulations Case

Appendix 8.5. Useful results of algebra.