Monique Grandbastien

Suzana Loskovska
email: suzana.loshkovska@finki.ukim.mk

Samuel Nowakowski

Jelena Jovanovic

Using online presence data for recommending human resources in the OP4L project

Keywords: Web-based learning, social presence, online presence, social semantic web, ontology based resource recommendation

In order to help and support learning practices, the development of web-based Personal Learning Environments (PLE) is widely adopted. A PLE is a set of services customized by the student. Among these services, resource (either digital or human) recommendation is a crucial one. The paper briefly reviews existing approaches for recommending resources in PLE. Then it describes a novel approach implemented in the OP4L prototype. OP4L is combining Social Web presence data and semantic web technologies, based on an intensive use of ontological models to represent the learning context. Then the paper reports about qualitative studies that were conducted with students on the currently available version of the OP4L prototype. The aim of the study was to get students' feedbacks about new online presence services offered in a LMS.

Introduction

In order to help and support learning practices, the development of web-based Personal Learning Environments (PLE) is widely adopted in the TEL research community [START_REF] Atwell | Personal Learning Environments -the future of eLearning? eLearning Papers[END_REF]. From the technical perspective, a PLE is a customizable set of services aimed at enhancing the learning experience and learning outcomes for a student. Among these services, resource (either digital or human) recommendation is a crucial one, given the number and the diversity of available resources on the web. In this context, various approaches have been proposed to customize the recommendation of resources to the learners' needs. They all rely on the use of a learner profile and include a more or less rich description of the learning context, often through ontologies.

The recent increase in the use of social software tools by learners leads to the inclusion of novel learning practices, such as collaborative tagging, into PLEs. The combination of Semantic Web and Social Web technologies allows to take these practices into account in an online learning environment. Moreover access to on line people is offered through chats and other synchronized means.

This paper reports on qualitative studies that were conducted with students using a PLE prototype developed in the scope of the OP4L project. This prototype offers recommendation services based on dynamic integration of online presence data and an intensive use of ontologies and web technologies. In section 2 we briefly summarize some results from previous studies that explored social semantic web and PLEs. Then, in section 3, we present the OP4L framework with a focus on the available social presence features. In section 4 we describe how we organized data collection in order to get initial feedbacks from master degree students and discuss the obtained results that have to be considered for improving the prototype. This is a work in progress, so we conclude about the planned forthcoming steps.

Social semantic web and learning resources

In her "vision" paper [START_REF] Vassileva | Towards Social Learning Environments[END_REF], Vassileva states that the design of learning technologies needs to focus on supporting social learning in context. She defines three main roles that should be performed by PLEs: (1) support the learner in finding the right content (right for the context, particular learner, specific purpose of the learner and pedagogically), [START_REF] Vassileva | Towards Social Learning Environments[END_REF] support learner to connect with the right people (…) and ( 3) motivate/incentivize people to learn.

To achieve these goals, researchers and developers build on experiences gained from several domains. The discovery and retrieval of learning resources is one of them and has been widely investigated, beginning with the work on metadata interoperability, then going on with the use of ontologies to better match the learners' needs and context. As social web applications, such as resources tagging, became available, solutions mixing both ontology-and tagging-based approaches were proposed. Meanwhile the recommender systems community developed powerful algorithms for the e-commerce sector and PLE developers tried to adapt them to e-learning purposes [START_REF] Duval | Special Issue on Social Information Retrieval for Technology Enhanced Learning[END_REF].

Social presence is another relevant research topic. It has been identified as a crucial success factor in e-learning for many years [START_REF] Aragon | Creating social presence in online environments[END_REF], [START_REF] Cob | Social Presence and Online Learning: A current view from a Research Perspective[END_REF], [START_REF] Lowenthal | Social Presence, Social Computing : Concepts, methodologies, tools and applications[END_REF]. At the beginning social presence was mostly implemented through online forums and Instant Messaging tools that allowed for establishing and maintaining social presence in online learning settings. The wide adoption of social web applications, such as Facebook, resulted in the inclusion of online social networks and connections established in these networks into online learning environments. Though in theory students can interact with their entire social network, in practice they do not get any indicator about who is really available in the given moment and who is really competent for helping in the current task. Although recommending knowledgeable people for performing a given task is not new, it has been mostly investigated in company settings such as reported, for instance, in [START_REF] Beham | Recommending knowledgeable people in a work-integrated learning system, 1 st RecSysTEL workshop[END_REF]. OP4L project brings solutions for these last two questions as described below.

OP4L framework

Background and objectives

OP4L (Online Presence For Learning) [8] is a European SEE-ERANET project bringing together partners from 6 countries from 2010 to 2012. The project aims at exploring the use of web tools and services for supporting social presence in online learning environments and thus contributing to an improved learning experience. In this paper, we use OP4L to name both the project and the developed prototype. OP4L defines online presence as a temporary description of a user's presence in the online world. It can be considered as an image that a person projects about him/herself into the online world. In this project we explored online presence in the context of the DEPTHS PLE [START_REF] Jovanovic | Ontologies for Effective Use of Context in e-Learning Settings[END_REF]. DEPTHS (DEsign Patterns Teaching Help System) is designed for a Design Pattern course in Software Engineering -Computer Science -master level. It makes use of ontologies as a common base for the integration of different systems, services and tools in a common environment for collaborative learning of software design patterns. OP4L extends DEPTHS services by online presence data at the semantic level.

Functional description

A complete technical description of the OP4L framework can be found in [START_REF] Jeremic | OP4L : Online Presence Enabled Personal Learning Environments[END_REF], [START_REF] Milikic | Infrastructure for Exchanging Online Presence Data in Learning Applications[END_REF] as well as in deliverables available on the project's web site. However, it is necessary to draw an overall picture of the system and the ontologies that make possible the provided services. From a functional perspective, the primary aim of the project was the development of a context-aware PLE by integrating learning context data from different learning systems/tools/services, using a flexible ontology-based model [START_REF]OP4L Models[END_REF]. We define learning context, i.e., the context of a given learning situation as an integration of the following main components:

• the learning activity that was performed or the learning-related even that occurred, • the content that was used and/or produced during the learning activity,

• the individual(s) involved (e.g., learners, teachers, experts),

• the (online) environment where the learning activity took place, • the time when the learning activity took place.

The notion of learning context is formally modeled through an interlined set of ontologies collectively named LOCO (Learning Object Context Ontologies) framework [START_REF] Jovanovic | Ontologies for Effective Use of Context in e-Learning Settings[END_REF]. These ontologies served as the foundation for the development of the DEPTHS PLE with the following main features:

• Integration of data and resources from diverse learning applications that students interact with; • Context-aware recommendation of resources on software design patterns from online repositories, learning artifacts produced and shared by peers, software projects, discussion threads, chats, etc.; • Context-aware recommendation of other students, experts and/or teachers to offer help in the given situation.

Within the OP4L project, the notion of learning context is extended to include the notion of Online Presence. Accordingly, links have been established between the LOCO ontologies and the Online Presence Ontology (OPO) [START_REF] Jovanovi | Online Presence in Adaptive Learning on the Social Semantic Web[END_REF] to allow for explicitly defining the semantic of this extended notion of learning context. The integration of LOCO and OPO ontologies served as a foundation for the development of onlinepresence-aware educational services within the DEPTHS PLE. These services make use of users' online presence data when providing learners with recommendations about whom to ask for help or collaborative work. These data are periodically "pushed" towards the PLE by specific software modules developed for that purpose. Within the online presence data, a key indicator is the "online status" [START_REF] Stankevic | Modeling Online Presence[END_REF] as declared by the user. For instance, a peer whose online status indicates that he/she is busy in the given moment will not be recommended; on the other hand, the system would recommend a face-to-face study session with a peer who has just checked in the same building and whose status indicates that he/she can be freely contacted.

For course designers, one of the main challenges is to adapt interactions to the students' state of presence and to provide services so that interactions can be established smoothly among the participants.

Main features of OP4L prototype

OP4L services are implemented within the existing DEPTH PLE. They are accessible through a dedicated Moodle platform. The services become available after a student selects a course to study (e.g., the Design Pattern course) and a learning activity (e.g., updating patient's data problem UML modeling tool). Fig. 1 shows the OP4L on line presence services as presented to the student in the user interface. Specifically, they appear in the form of an online presence box in the upper left side of the screen. The box indicates who is competent and available online for help or collaboration. It also indicates how to contact potential helper(s)/collaborator(s), either on the Moodle platform itself, or via Facebook or Twitter. Services linking Moodle to Facebook and Twitter have been developed so that each student remains using his/her current application, for instance, Moodle for the student looking for help/collaboration and Facebook for the student being contacted. Based on the online statuses declared by the peer students, the learner who is looking for help/collaboration will know in which manner he/she can communicate with the peers. In the case presented on Fig. 1, all of the peers can be contacted on Facebook and by email. The platform shows several other services as described below. The platform recommends appropriate contents related to the topic of the course (Fig. 2). Its originality is to augment the digital library with the resources brought and built by the students during the course.

Fig. 2. Interface illustrating recommendation of digital resources

For enhancing collaboration, students are also given a brainstorming tool where ideas can be annotated and rated. Finally, students can upload their work on the platform and benefit from a system of peer evaluation. They can assess other proposals only when they have uploaded their own solution as shown in Fig. 3.

First feedbacks from students

In this paper we report about two studies performed between January and May 2012 with the first versions of the OP4L prototype. The first study was performed with master students who were not studying Software Engineering at Université de Lorraine (France). The second one was run with Computer Science students (several levels) at University of Skopje (Republic of Macedonia).

Experiment at Université de Lorraine

Objectives and method.

As explained in section 2, previous research work related to the notion of social presence and its role in online learning did not consider the use of Online Presence educational services. So we do not have data about the expectations of students as they never used such a set of services. As Brooks et al. stated in [START_REF] Brooks | Lessons Learned using Social and Semantic Web Technologies for E-Learning in Semantic Web Technologies for e[END_REF], we believe that "there is no substitute for constantly trying to test techniques in the real world of students' learning". Our method was to test the OP4L environment with students in order to analyze how end-users use, like or dislike, and benefit from the newly provided features.

Experimental population and settings.

Our population included 15 students and the study was conducted in February 2012. Whereas the DEPTHS environment is designed for a Software Patterns course, namely a course for Software Engineering students, the students participating in the study came from several master courses on other subjects. Indeed it was important for the project to get a feedback about recommendation services not only from software engineering students, but also from students studying other subjects in several countries.

The students' profiles, organized in 5 categories, are: Cat. The data collection was run into three steps.

Step 1: A first questionnaire (see results below) was passed in order to get some descriptive data about students' understanding and current use of web based and social network services.

Step 2: We demonstrated the OP4L-DEPTHS prototype, especially the online presence services; then, the students were free to analyze more deeply the services. We remind that they were not students in Computer Science and consequently we were not able to assign the software engineering tasks available in OP4L-DEPTH.

Step 3: We passed the second questionnaire in order to get the students' feedback about the features of online services. In order to know more about their expectations concerning the inclusion of such services, they were also asked to describe a scenario including the kind of services they would dream about. Finally, they were invited to provide free suggestions.

Students' feedback

The participants' use of technology according to the first questionnaire is as follows:

Frequency of Moodle use: less than one time a week (except Master in Laws: about 3 times per week)

Frequency of connection to a personal social network: daily Frequency of using email to contact others students: never or less than one time a week Frequency of cell phone use to contact others students: two times per week.

An example of scenario created by students is provided hereafter (translated from French).

"We are students in Communication, Laws, Sciences and Medical sciences and Digital design, we have to realize a project on a specific domain, but we cannot understand some of the technical terms and Wikipedia cannot help us (the articles are not adapted to our problem).

We have to contact people not only in our curriculum but others in connected domains who will be able to answer to our questions with "students" words and/or adapted vocabulary (easier to understand). The different services (more specifically, brainstorming tool) could provide opportunities for exchanges, collaborations, details and advices given in peer-to-peer mode." Table 1 summarizes students' answers to the second questionnaire about several dimensions of social presence in online learning environments. Students' categories are as described above. Students were asked to rank each of the proposed dimensions from 1 (the most important) to 8 (the least important) Students' suggestions and comments.

We provide hereafter additional suggestions and comments written by the students.

• "To create and display detailed users' profiles including curricula, centers of interest and whom they helped and in which domains. This gives us a possibility to have a better knowledge of the people who could be contacted for help" • "To have in the list of recommended peers, not only our friends in our social networks but also people who have the appropriate profile to help us." • "This tool can be useful to help us identifying the appropriate additional contents, to collaborate on specific topics and to get advices on the already done work. There is a true social aspect (peer-to-peer) which could really help us". • "The overall ergonomic design is quite good, clear and well organized. It is really user friendly". • "This tool could give the possibility to ask questions and to collaborate with other students who have the capability and the availability to answer to questions. To have all contacts and friends in the platform will bring a gain of time. Three important remarks: to know who is available to know who cannot be disturbed to know who is online."

• "Thus, we can only discuss with people who are ready for discussion or to answer question. This gives the possibility to directly access to the appropriate person without using Facebook or Small Messages Services. Further more, if these persons show that they are ready to collaborate, they will answer quicker and in a better mood. The consequences will be: gain of time."

Preliminary conclusion.

We can already notice that the students immediately show an interest in contacting peer students for the kind of project they have to complete for their master degree.

The hypothesis was that providing students with online presence recommendation services in a LMS could significantly help them in performing their learning tasks. The analysis of the students' current use of communication technologies shows that they are not daily users of the university's Moodle environment. However, they join each other preferably through social networks. Given that context, their appreciation of the prototype is quite encouraging as they immediately understand the benefits that such a tool could bring to them in one of the tasks they have to achieve for their master degree, namely the project they described in the proposed scenario. Moreover we collected their appreciations (ranking) about the main dimensions of online presence. The results again show their interest. The rankings deserve additional studies, but knowing who is online and who is available are the most appreciated. Finally among the positive points they mention several times a gain of time in achieving their tasks, which is certainly an important criterion for students as well as for academic staff.

Evaluation at UKIM-FEEIT

Objectives and method.

Our objective was to provide evidence that the developed PLE can be successfully by students for collaborative problem solving activities. In that context, testing scenario considered project-based learning with collaborative learning support where a teacher defines a specific problem to be solved in a workshop-like manner. The scenario required from the learners first to present their ideas for solving the given problem, comment on each others' ideas, discuss the ideas with the peers and rate them, and then to provide solution and to evaluate others students solutions. To perform these collaborative learning activities, students had to use developed services: Recommending peer (on-demand and proactive suggestions regarding other students, teachers or experts as possible collaborators) and recommending reading (provide suggestions for relevant learning content to be consulted and/or used when working on the problem's solution).

Experimental population and settings.

The evaluation of the system was performed at the Faculty of Computer Science and Engineering, Ss. Cyril and Methodius University in Skopje in period between February and May 2012. Students from two faculties: Faculty of Computer Science and Computer Engineering and Faculty of Electrical Engineering and Information Technologies were involved in the testing process. The students were divided in two different groups who performed the same testing scenarios with different tasks. The groups profiles are:

Group. 1. Students enrolled at Software Engineering master study program. The group was formed by the 22 students enrolled to the course Design Patterns. The evaluation process was incorporated into assessment of the course. In that context, two of five course homeworks students were obliged to do using the system. The course lessons were already finished when the students received their assignments.

Group. 2. Undergraduate students on the course Human-Computer Interaction. 14 students from the second and the third year of Computer Science and Engineering undergraduate study program, voluntarily applied to participate in the evaluation. Again, the course marks are obtained as combination of results from exams, quizzes and independent project. In this case, participation in testing is treated as independent course projects.

The testing scenario was the same for both groups although the students from the second group do not have or have limited knowledge of Design Patterns. The difference between the groups was in the tasks they should solve. Evaluation scenario for both groups consisted of

Step 1. All students filled a motivation questionnaire (MSLQ).

Step 2. Working with the system. In this step, each student got two tasks from Design Patterns and for each task he/she has to do three actions:

Proposing and submitting the idea for solution, discussion and grading ideas proposed by the colleagues Proposing solutions that included development of an UML model using ArgoUML case tool

Assessing their own solutions and solutions of other students

Step 3. Each student filled questionnaire related to the tasks. Questions from the questionnaire were concerned with: the habits of using Facebook, its chat and Web sources, the experience in OP4L learning environment, the collaborative learning and using Facebook in favor of collaborative learning.

Students in regular situations can use the help of a teacher and/or other students, but this time they were encouraged to perform this communication using the system.

Preliminary results.

Students were positive about the system' services which include:

• Possibility to discuss with peers (more than 50% stated that this discussion helped them to solve the problem) • Recommending reading (more than 50% reported that reference materials helped them to solve the problem) But, the results showed that the students are not keen to use Facebook, its chat and applications for learning. On the question do they use DEPTHS' Facebook connection only eight answered positively. 18 students answered that they did not use this service. Students answers related to the use of Facebook can be grouped as follows:

• Seven students even don't have a Facebook account.

• Those who have Facebook's accounts, usually use the Facebook for fun or for "private matters" as one of students declared. • Some students didn't use this possibility because of testing scenarios. In fact, students had access to the solutions of others, so for them it was enough to solve the problem. • One student reported that the complete use of the system was too difficult and that was the reason why he/she did not try this option.

Preliminary conclusion.

Considering the answers in the questionnaire we can conclude that the developed PLE can be successfully used by students for collaborative problem solving activities. Generally the students as most positive assessed the service Recommending reading and "offline" discussion with peers (providing ideas for solution and commenting and assessing the ideas and the solutions as well).

The results have shown that the use of Facebook, its chat and applications did not increase after period of working with the system. It was for us not an expected result because the students usually use the Facebook chat to communicate with teaching staff (personal experience and experience of my colleagues) for matters connected with the courses. But the evaluation showed that majority used their Facebook accounts as they had used before the study, which mainly includes fun and communication.

5

Lessons learned and further work.

We have described the OP4L services as implemented in the DEPTH PLE. The main originality of the OP4L project is to recommend digital resources and human resources through the same ontology based process by dynamically integrating online presence data from other social web applications such as Facebook or Twitter. This study reports about the first use of the OP4L prototype with students. Those from France were not the specific targeted population whereas those from Macedonia were Computer Science students. Both do not belong to the implementing team.

Such a study only aims at providing a qualitative analysis of the provided functionalities. Other evaluations of OP4L learning services are currently under way by other partners. They adopt a software engineering perspective using the SUMI (Software Usability Measurement Inventory) approach.

We noticed a strong interest in the provided services as well as low real use of Facebook integration in Macedonia. Next step would be to improve the software developed in order to be able to deploy it in more universities and to provide authors with ways to add new lessons and new courses. We remind here that in OP4L advising human resources for a given task relies on the provision of task domain descriptions through ontologies. Providing the corresponding ontologies for a new domain as well as describing resources -digital and human -is a very time-consuming task. So intermediate solutions could be welcome. The students' ranking of the proposed dimen-sions of online presence, the services effectively used could help making intermediate choices easier to implement. Then further evaluation in larger learning settings (longer period in the academic year, lower dependency to exam ratings, etc.) should take place.

Fig. 1 .

 1 Fig. 1. Inside the prototype with recommended peers on the left

Fig. 3 .

 3 Fig. 3. Rating other students' work.

Table 1 .

 1 Students appreciations about dimensions of social presence in online learning environments

	Proposed ranking by students categories Cat1 Cat2 Cat3 Cat4 Cat5
	To know who is online on the platform	1	2	2	3	1
	To know what are the others doing?	5	4	1	5	3
	To know who is available for discussion	2	1	4	1	4
	To know who is not available and does not	6	3	3	2	2
	want to be disturbed
	To know where is someone	7	5	7	7	8
	To know what are the feelings of the others	8	7	8	8	7
	(stress, tired, etc.)
	To know what are their subjects of interest	4	6	5	4	5
	To know what they like or not	3	8	6	6	6

Acknowledgements:

This work was supported by the SEE-ERA Net Plus program, contract n° 115, from the European Union.


François Portier

Bernard Delyon

Bootstrap Testing of the Rank of a Matrix via Least Squared Constrained Estimation

Keywords: Rank estimation, Least squared constrained estimation, Bootstrap, Hypothesis testing

In order to test if an unknown matrix has a given rank (null hypothesis), we consider the family of statistics that are minimum squared distances between an estimator and the manifold of fixed-rank matrix. Under the null hypothesis, every statistic of this family converges to a weighted chi-squared distribution. In this paper, we introduce the constrained bootstrap to build bootstrap estimate of the law under the null hypothesis of such statistics. As a result, the constrained bootstrap is employed to estimate the quantile for testing the rank. We provide the consistency of the procedure and the simulations shed light one the accuracy of the constrained bootstrap with respect to the traditional asymptotic comparison. More generally, the results are extended to test if an unknown parameter belongs to a sub-manifold locally smooth. Finally, the constrained bootstrap is easy to compute, it handles a large family of tests and it works under mild assumptions.

Introduction

Let M 0 ∈ R p×H be an unknown matrix (arbitrarily p ≤ H). To infer about the rank of M 0 with hypothesis testing, the general framework usually considered is the following: there exists an estimator M ∈ R p×H of M 0 such that

n 1/2 ( M -M 0 ) d -→ W, with vec (W ) = N (0, Γ) (1)
where vec(•) vectorizes a matrix by stacking its columns. In the whole paper the hatted quantities are random sequences that depends on the sample number n, all the limit are taken with respect to n. Moreover there exists an estimator Γ such that

Γ P -→ Γ, (2)
and in some cases, one may ask that Γ is full rank.

(

) 3
Let d 0 be the rank of M 0 and m ∈ {1, ..., p}, we consider the set of hypotheses

H 0 : d 0 = m against H 1 : d 0 > m, (4)
Thus d 0 can be estimated the following way: we start by testing m = 0, if H 0 is rejected we go a step further m := m + 1, if not we stop the procedure and the estimated rank is d = m. In this paper, by considering the hypotheses (4) we focus on each step of this procedure. Many different statistical tests appeared in the literature for this purpose. For instance Cragg and Donald [START_REF] Cragg | On the asymptotic properties of LDU-based tests of the rank of a matrix[END_REF] introduced a statistic based on the LU decomposition of M , Kleibergen and Paap [START_REF] Kleibergen | Generalized reduced rank tests using the singular value decomposition[END_REF] studied the asymptotic behaviour of some transformation of the singular values of M , and Cragg and Donald [START_REF] Cragg | Inferring the rank of a matrix[END_REF] considered the minimum of a squared distance under rank constraint. In some other fields with similar issues, close ideas have been developed : Bura and Yang [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF] examined a Wald type statistic depending on the singular decomposition of M and Cook and Ni [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF] also considered the minimum of a squared distance under rank constraint. Although based on different considerations, each of the previous work relies on the test described by [START_REF] Bentler | Corrections to test statistics in principal hessian directions[END_REF]. For comprehensiveness, in this paper we consider the following three statistics. The first one is introduced by Li [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF] as

Λ 1 = n p k=m+1 λ 2 k (5)
where ( λ 1 , ..., λ p ) are the singular values of M arranged in descending order. Under H 0 and (1), this statistic converges in law to a weighted chi-squared distribution [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF]. The main drawback of such a test is that Λ 1 is not pivotal, i.e. its asymptotic law depends on unknown quantities that are M 0 and Γ. Accordingly the consistency of the associated test requires assumptions [START_REF] Arcones | On the bootstrap of M -estimators and other statistical functionals[END_REF] and [START_REF] Barbe | The weighted bootstrap[END_REF]. In [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF] a standardized version of Λ 1 is studied with

Λ 1 = n vec( Q 1 M Q 2 ) T [( Q 2 ⊗ Q 1 ) Γ( Q 2 ⊗ Q 1 )] + vec( Q1 M Q2) (6)
where M + stands for the Moore-Penrose inverse of M and Q 1 and Q 2 are respectively the orthogonal projectors on the left and right singular spaces associated with the pm smallest singular values of M . The authors proved that under H 0 , if (1) and ( 2) hold, the Waldtype statistic Λ 2 is asymptotically chi-squared distributed. Besides, [START_REF] Cragg | Inferring the rank of a matrix[END_REF] and [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF] proposed a constrained estimator by minimizing a squared distance under a fixed-rank constraint as

Λ 3 = n min rank(M )=m vec( M -M ) T Γ -1 vec( M -M ), (7)
which is also asymptotically chi-squared distributed under H 0 , assuming (1), ( 2) and (3). We will refer the minimum discrepancy approach. Although the statistics Λ 2 and Λ 3 have the convenience of being pivotal, they both require the inversion of a large matrix and this may cause robustness problems when the sample number is not large enough. For α ∈]0, 1[ and under the relevant assumptions, each of these statistics Λ 1 , Λ 2 and Λ 3 , is consistent at level α in testing (4), i.e. the level goes to 1α and the power goes to 1 as n goes to ∞.

Nevertheless the estimation of the quantile is difficult because either the asymptotic distribution depends on the data (non pivotality represented by Λ 1 ), or the true distribution may be quite different than the asymptotic one (slow rates of convergence represented by Λ 2 and Λ 3 ). The objective of the paper is to propose a bootstrap method for quantile estimation in this context.

An important remark which instigates the sketch of the paper is that all the previous statistics share the form

Λ = n B vec( M -M c ) 2 with M c = argmin rank(M )=m A vec( M -M ) 2 ( 8
)
where

• is the Euclidean norm, A ∈ R pH×pH , B ∈ R pH×pH .
The values of A and B corresponding to the statistics Λ 1 , Λ 2 and Λ 3 are summarized in the Table 1 (See Section 2 for the details). We refer to traditional testing (resp. bootstrap testing) when the statistic is compared to its asymptotic quantile (resp. bootstrap quantile). The bootstrap test is said to be consistent

Λ 1 Λ 2 Λ 3 A I I Γ -1/2 B I [( Q 2 ⊗ Q 1 ) Γ( Q 2 ⊗ Q 1 )] +1/2 Γ -1/2
P H 0 Λ > q(α) -→ 1 -α and P H 1 Λ > q(α) -→ 1, (9)
where q(α) is the quantile of level α calculated by bootstrap. The advantage of bootstrap testing is its high level of accuracy under H 0 with respect to traditional testing. This fact is emphasized by considering the two possibilities: when the statistic is pivotal and when the asymptotic law of the statistic depends on unknown quantities. First, as highlighted by Hall [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF], when the statistic is pivotal, under some conditions the gap between the distribution of the statistic and its bootstrap distribution is O P (n -1 ). Since the normal approximation leads to a difference O(n 1/2 ), the bootstrap enjoys a better level of accuracy. Secondly if the asymptotic law of the statistic is unknown, the bootstrap appears even more as a convenient alternative because it avoids its estimation. In [START_REF] Hall | Two guidelines for bootstrap hypothesis testing[END_REF], Hall and Wilson give two advices for the use of the bootstrap testing:

A) Whatever the sample is under H 0 or H 1 , the bootstrap estimates the law of the statistic under H 0 .

B) The statistic is pivotal.

The first guideline is the most crucial because if it fails it may lead to inconsistency of the test. The second guideline aims at improving the accuracy of the test by taking full advantage of the accuracy of the bootstrap. In this paper we propose a new procedure for bootstrap testing in least square constraint estimation (LSCE) (estimators as [START_REF] Bura | Extending sliced inverse regression: the weighted chi-squared test[END_REF] are particular cases), called constrained bootstrap (CS bootstrap). More precisely, the CS bootstrap aims at testing whether a parameter belongs or not to a submanifold and so generalised the test (4). Our main result is the consistency of the CS bootstrap under mild conditions. As a consequence we provide a consistent bootstrap testing procedure for testing (4) with the statistic Λ 1 , Λ 2 and Λ 3 . For the sake of clarity, we address the CS bootstrap in the next section. Section 3 is dedicated to rank estimation with special interest to the bootstrap of the statistic Λ 1 , Λ 2 and Λ 3 . Finally, the last section emphasizes the accuracy of the bootstrap in rank estimation by providing a simulation study in sufficient dimension reduction (SDR). Accordingly, the sketch of the paper is as follows:

• The CS bootstrap in LSCE • Bootstrap testing procedure for Λ 1 , Λ 2 and Λ 3

• Application to SDR

The constrained bootstrap for LSCE and hypothesis testing

Because of ( 8) LSCE has a central place in the paper. Moreover since LSCE intervenes in many statistical fields as M-estimation or hypothesis testing, this section is independent from the rest of the paper.

LSCE

Let θ 0 ∈ R p be called the parameter of interest, and let θ ∈ R p be an estimator of θ 0 . We define the constrained estimator of θ 0 as

θ c = argmin θ∈M ( θ -θ) T A( θ -θ), (10)
where M is a submanifold of R p with co-dimension q, and A ∈ R p×p . The constrained statistic is defined as

Λ = n( θ -θ c ) T B( θ -θ c ). ( 11
)
where B ∈ R p×p . Note that if A is full rank, the unique minimizer of (10) without constraint is θ, hence it could be understood as the unconstrained estimator. We introduce now the notion of nonsingular point in M. This one is needed to express the Lagrangian first order condition of the optimization [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF]. For any function g = (g 1 , . . . , g p ) : R p → R q , define its Jacobian as J g = (∇g 1 , ..., ∇g q ), where ∇ stands for the gradient operator.

Definition 1. We say that θ is M-nonsingular if θ ∈ M and if there exists a neighbourhood V and a function g : R p → R q continuously differentiable on V with J g (θ) full rank such that

V ∩ M = {g = 0}.
As a consequence any point of a submanifold locally smooth is nonsingular, e.g. any matrix with rank m is a nonsingular point in the submanifold rank(M ) = m. We prove in Proposition

2 that if θ 0 is M-nonsingular, √ n( θ -θ 0 ) d → N (0, ∆) and B = A P → A is full rank, then we have Λ d -→ p k=1 ν k W 2 k , (12)
where the W k 's are i.i.d. Gaussian random variables and the ν k 's are the eigenvalues of the matrix ∆ 1/2 J g (θ 0 ) T (J g (θ 0 )A -1 J g (θ 0 ) T ) -1 J g (θ 0 )∆ 1/2 . Especially, the case A = ∆ -1 is interesting because Λ is asymptotically chi-squared distributed with q degrees of freedom. Otherwise, if θ 0 / ∈ M, Λ goes to infinity in probability. Those facts shed light on a consistent testing procedure based on LSCE with the hypotheses

H 0 : θ 0 ∈ M against H 1 : θ 0 / ∈ M (13)
and the decision rule to reject H 0 if Λ is larger than a quantile of its asymptotic law. Accordingly the previous framework can be seen as an extension of the Wald test statistic which handles the simple hypothesis θ 0 = θ with the statistic ( θθ) T ∆ -1 ( θθ).

The bootstrap in LSCE

Since LSCE is a particular case of estimating equation, we review the bootstrap literature with two principal directions: estimating equation and hypothesis testing. For clarity we alleviate the framework in this section: let X 1 , • • • , X n be an i.i.d. sequence of real random variables with law P , define γ = var(X 1 ), γ = (X -X) 2 , we put θ 0 = E[X 1 ], θ = X, and A = B = γ -1 where • stands for the empirical mean. The original bootstrap was introduced in [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF] in the following way. Let X * 1 , . . . , X * n be an i.i.d. sequence of real random variables with law P = n -1 n i=1 δ X i , define θ * = X * , the distribution of √ n(θ *θ) conditionally on the sample, that we call the bootstrap distribution, is "close" to the distribution of √ n( θθ 0 ), that we call the true distribution (in the rest of the paper we just say "conditionally" instead of "conditionally on the sample"). For instance, it is shown in [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF] that the bootstrap distribution converges weakly to the true distribution almost surely. One says that √ n(θ *θ) bootstraps √ n( θθ 0 ) and we will write

L ∞ (n 1/2 (θ * -θ)| P ) = L ∞ (n 1/2 ( θ -θ 0 )) a. s.,
where L ∞ (•) and L ∞ (•| P ) both mean the asymptotic laws with the difference that the later is conditional on the sample. Equivalently, one has for every x ∈ R, P( √ n(θ *θ) ≤ x| P ) a.s.

→ P(

√ n( θθ 0 ) ≤ x), but the use of the bootstrap is legitimate by a more general results stated in [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF], which says that

|P(n 1/2 (θ * -θ)/γ * ≤ x| P ) -P(n 1/2 ( θ -θ 0 )/ γ ≤ x)| = O P (n -1 ) (14)
with γ * = (X * -X * ) 2 , provided that P is non-lattice. Besides, one has

|P(n 1/2 ( θ -θ 0 )/ γ ≤ x) -Φ(x)| = O P (n -1/2 ),
where Φ is the cumulative distribution function (c.d.f.) of the standard normal law. Variations of Efron's resampling plan are proposed in [START_REF] Barbe | The weighted bootstrap[END_REF] under the name of weighted bootstrap. For a complete introduction about the bootstrap we refer to [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF]. We now present three different bootstrap techniques related to LSCE 1 .

(i) The classical bootstrap (C bootstrap)

The literature about the bootstrap in Z and M-estimation, see respectively [START_REF] Chatterjee | Generalized bootstrap for estimating equations[END_REF] and [START_REF] Arcones | On the bootstrap of M -estimators and other statistical functionals[END_REF], is based on the following principle: if θ M = argmin

θ * M = argmin θ∈Θ n -1 n i=1 w i φ(X i , θ), (15)
where (w i ) is a sequence of random variables. The particular case where the vector (w 1 , . . . , w n ) is distributed as mult(n, (n -1 , . . . , n -1 )) leads to a direct application of original Efron's bootstrap to M-estimation. Since such a bootstrap has been extensively studied, we refer to the C bootstrap. To the knowledge of the authors, the C bootstrap when Θ has empty interior has not been studied yet. Nevertheless one may sight its bad behaviour for the test of equal mean H 0 : θ 0 = µ. The associated least squared constrained statistic

n γ -1 ( θ -µ) 2 ,
is indeed the score statistic associated to the M-estimator with φ(x, θ) = γ -1 (xθ) 2 and Θ = {µ}. Clearly the C bootstrap through nγ * -1 (θ *µ) 2 does not work because of its bad behaviour under H 1 for instance. In this case it is better to use

nγ * -1 (θ * -θ) 2 ,
but it cannot handle the cases of more involved hypotheses2 . Whereas the C bootstrap is not really connected with hypothesis testing, the two following bootstrap procedures are more related to the present work.

(ii) The biaised bootstrap (B bootstrap)

The B bootstrap is introduced in [START_REF] Hall | Intentionally biased bootstrap methods[END_REF] and is directly motivated by hypothesis testing. The original idea of their work is to re-sample with respect to the distribution P b = n i=1 ω i δ X i , where the ω i 's maximize n i=1 log(ω i ) under the constraints

1 n n i=1 ω i X i = µ n i=1 ω i = 1 . ( 16
)
Since the ω i 's minimize the Kulback-Leibler distance between P and P b , one can see the resulting distribution as the closest to the original one satisfying the mean constraint.

The authors presented interesting results for the test of equal mean

θ 0 = µ, essentially the bootstrap statistic nγ * -1 (θ * b -µ) 2 , with θ * b = X * b , X * b,i sampled from P b
, has a chi-squared limiting distribution either H 0 or H 1 is assumed. As a result both guidelines (A) and (B) are checked. They go further by showing that the B bootstrap outclasses the asymptotic normal approximation for quantile estimation in the sense that | q(α)q n (α

)| = O P (n -1 ) whereas |q n (α)-q ∞ (α)| = O(n -1/2
), where q ∞ , q n and q n are the quantile functions of the standard normal distribution, the statistic n γ -1 ( θµ) 2 under H 0 and the bootstrapped statistic, respectively. Although the B bootstrap matches the context of hypothesis testing, it has been designed to handle the particular test of equal mean. To the knowledge of the authors the study of the B bootstrap has not been extended to other tests. Facing ( 16), the main drawback of the B bootstrap deals with algorithmic difficulties. Indeed when the constraint becomes more involved, solving ( 16) is more difficult. As a result it is not sure that this method could handle other situations such as fixed-rank constraints.

(iii) The estimating function bootstrap (EF bootstrap) Now X i ∈ R p . Some other ideas about the bootstrap of the Z-estimators can be found in [START_REF] Lele | Resampling using estimating equations[END_REF] and [START_REF] Hu | The estimating function bootstrap[END_REF], and can be summarized as follows. Considering the score statistic S = √ n n i=1 ∂φ ∂θ (X i , θ 0 ), [START_REF] Hu | The estimating function bootstrap[END_REF] showed that it could be bootstrapped by

S * = n -1/2 n i=1 w i ∂φ ∂θ (X i , θ),
where (w i ) is a sequence of random variables. This bootstrap is called the EF bootstrap and revealed nice computational properties. Moreover the authors argued for its use in quantile estimation in order to test if g(θ 0 ) = 0, where g : R p → R q is the constraint function, by recommending essentially to use

S * T J g ( θ) T J g ( θ)γ * J g ( θ) T -1 J g ( θ)S * . Applying it to the least squared context φ(x, θ) = γ -1/2 (x -θ) 2 , the EF bootstrap is carried out by n(θ * -θ) T J g ( θ) T J g ( θ)γ * J g ( θ) T -1 J g ( θ)(θ * -θ).
of J g ( θ) than on the bootstrap of

√ n( θ -θ c ). Indeed √ n(θ * -θ) bootstraps the non constrained estimator √ n( θ -θ 0 )
. Then as the authors noticed, it is first of all a bootstrap of the Wald-type statistic n S T J g (θ 0 ) T J g (θ 0 ) γJ g (θ 0 ) T -1 J g (θ 0 ) S which has fortunately the same asymptotic law than the targeted one. This may induce some loss in accuracy. Moreover, it requires the knowledge of the function J g which is not the case for fixed rank constraints where the g depends on the limit M 0 (see Remark 1 for some details).

Essentially both (i) and (ii) provide a bootstrap for testing simple hypotheses. The EF bootstrap proposed in (iii) extends this limited scope by including tests of the form g(θ 0 ) = 0 where g is known. Nevertheless it does not handle the test (4) as it is highlighted by the following remark.

Remark 1. Testing (4) with Λ 3 results in an optimization with the constraint rank(M ) = m. Since the subspace of fixed rank matrices is a submanifold locally smooth with co-dimension (pd)(Hd), at every point M , there exists a neighbourhood V and a

C ∞ function g : V → R (p-d)(H-d) such that V ∩ {rank(M ) = m} = {g = 0} and J g (M ) has full rank. Moreover, we have Γ -1/2 vec( M c -M 0 ) ≤ 2 Γ -1/2 vec( M -M 0 ) .
If now (1) holds, the right-hand side term goes to 0 in probability and M c P → M 0 . As a consequence, if Γ is invertible, for any neighbourhood of M 0 , from a certain rank, M c belongs to it with probability 1. Then under H 0 since M 0 has rank m the constrained estimator has the expression

M c = argmin g(M )=0 Γ -1/2 vec( M c -M ) ,
with g depending on M 0 . Unfortunately we do not know neither g nor J g (M 0 ). This entails some problems relating to the later approach.

The constrained bootstrap

The CS bootstrap is introduced in order to solve all the issues we have raised through the previous little review which are essentially: computational difficulties and small scope of the existing methods. The CS bootstrap targets an estimation q(α) of the quantile under H 0 of Λ. The consistency of the procedure, i.e. [START_REF] Chatterjee | Generalized bootstrap for estimating equations[END_REF], forms the main result about the CS bootstrap. Another important issue which occurs beforehand in the section is the bootstrap of the law of

n 1/2 ( θ c -θ 0 ) under H 0 .
Basically, we show that a bootstrap of the unconstrained estimator √ n( θ-θ 0 ) allows a bootstrap of the constrained estimator √ n( θ c -θ 0 ) under H 0 . We point out that the CS bootstrap heuristic is rather different than the C and EF bootstrap. Otherwise it shares the idea to "reproduce" H 0 even if H 1 is realized with the B bootstrap. Assuming that we can bootstrap √ n( θθ 0 ), the CS bootstrap calculation of the statistic is realized as follows:

The CS bootstrap procedure Compute

θ * 0 = θ c + n -1/2 W * , with L ∞ (W * | P ) = L ∞ (n 1/2 ( θ -θ 0 )) a. s., (17)
where the simulation of W * can be done by a standard bootstrap procedure 3 . Calculate

θ * c = argmin θ∈M (θ * 0 -θ) T A * (θ * 0 -θ), and Λ * = n(θ * 0 -θ * c ) T B * (θ * 0 -θ * c ), (18)
where

A * ∈ R p×p and B * ∈ R p×p 4 .
Intuitively, this choice appears natural because θ * 0 equals θ c plus a small perturbation going to 0. Accordingly θ * 0 is somewhat reproducing the behaviour of θ under H 0 , especially because W * has the right asymptotic variance. As we should notice, A * and B * could be chosen as A and B but this is not the best choice in practice. As it is highlighted in [START_REF] Efron | Bootstrap methods: another look at the jackknife[END_REF], we should normalized by the associated bootstrap quantities (e.g. the variance computed on the bootstrap sample).

The following lemma gives a first order decomposition of the bootstrap law √ n(θ * cθ c ) under mild conditions. The following lemma is proved in the Appendix. Lemma 1. Let M be a submanifold. Assume there exists θ c ∈ M and θ c a M-nonsingular point such that θ c a.s.

→ θ c . If moreover L ∞ ( √ n(θ * 0 -θ c )| P ) exists a.
s. and conditionally a.s. A * P → A is full rank, then we have conditionally a.s.

n 1/2 (θ * c -θ c ) = (I -P )n 1/2 (θ * 0 -θ c ) + o P (1)
,

with P = A -1 J T g (θ c )(J g (θ c )A -1 J T g (θ c )) -1 J g (θ c ).
Note that if θ 0 is M-nonsingular and L ∞ ( √ n( θθ 0 )| P ) exists, we can apply Lemma 1 with θ c = θ c = θ 0 . This gives the following proposition:

Proposition 2. Let M be a submanifold. Assume that L ∞ ( √ n( θθ 0 )| P ) exists with θ 0 Mnonsingular. Assume also that A P → A is full rank, then we have

n 1/2 ( θ c -θ 0 ) = (I -P )n 1/2 ( θ -θ 0 ) + o P (1), with P = A -1 J T g (θ 0 )(J g (θ 0 )A -1 J T g (θ 0 )) -1 J g (θ 0 ).
Proposition 2 leads easily to [START_REF] Cragg | Inferring the rank of a matrix[END_REF] and extends classical results [START_REF] Dennis | On generalized score tests[END_REF] about constrained estimators with constraint {g = 0} to manifold type constraints. Besides statements of Lemma 1 and Proposition 2 together explain the preceding definition of θ * 0 in [START_REF] Hall | Two guidelines for bootstrap hypothesis testing[END_REF]. They also lead to the following theorem. → A is full rank, then we have

L ∞ (n 1/2 (θ * c -θ c )| P ) = L ∞ (n 1/2 ( θ c -θ 0 )) a. s. .
Essentially, Theorem 3 is an application of Lemma 1 under H 0 , indeed as we seen in the proof of Lemma 1, equation [START_REF] Andrew | An f approximation to the distribution of a linear combination of chi-squared variables[END_REF], the assumption θ a.s.

→ θ 0 ∈ M implies that θ c a.s. → θ c . Nevertheless under H 1 nothing guarantee such a convergence (see Example 1 below). Roughly speaking, asking for an equality in law under H 1 as in Theorem 3 may be too much to ask. However as stated in the following theorem we do not require that θ c converges a.s. to a constant to provide that the power of the corresponding test goes to 1. This leads to the consistency of the CS bootstrap for hypothesis testing. For the statement of the consistency theorem, we need to define the quantile function of the bootstrap statistic

q(α) = inf {x : F (x) ≥ 1 -α},
where F is the c.d.f. of Λ * conditionally on the sample. → θ 0 with θ 0 M-nonsingular under H 0 . We assume also that A

P → A is full rank, B P → B. If moreover L ∞ ( √ n(θ * 0 -θ c )| P )=L ∞ ( √ n( θ -θ 0 ))
a.s. has a density, and conditionally a.s. A * P → A, B * P → B, then we have

P H 0 ( Λ > q(α)) -→ 1 -α, and P H 1 ( Λ > q(α)) -→ 1.
In other words, the test described in [START_REF] Eaton | The asymptotic distribution of singular values with applications to canonical correlations and correspondence analysis[END_REF] with statistic Λ and CS bootstrap calculation of quantile is consistent.

We provide the following example under H 1 , where θ c does not converge to a constant in probability. Although we cannot get the conclusion of Theorem 3, the least squared constrained statistic still converges in distribution.

Example 1. Let (X i ) i∈N be a i.i.d. sequence such that X 1 d = N (0, 1). Define θ = X, and H 0 : θ 2 0 = 1. Clearly H 0 does not hold and naturally the statistic n min

θ 2 =1
θθ 2 goes to infinity in probability. One can find that θ c = sign(X) which does not converge. Since

θ * c = argmin θ 2 =1 θ * 0 -θ 2 and θ * 0 = θ c + n -1/2 W * ,
we get that θ * c = θ c a.s. and naturally, we do not have the asymptotic given by Theorem 3. Besides, the convergence to a chi-squared distribution holds for the quantity n min

θ 2 =1 θ * 0 -θ 2 .

Rank estimation with hypothesis testing

In this section through a review of the literature about rank estimation, we apply the results obtained in section 2.1 to provide a consistent bootstrap procedure for the test described by ( 4) associated with the statistics Λ 1 , Λ 2 and Λ 3 . We define q 0 = pd 0 the dimension of the kernel of M T 0 . We denote by (λ 1 , ..., λ p ) the singular values of M 0 arranged in descending order and we write the SVD of M 0 as

M 0 = (U 1 U 0 ) D 1 0 0 0 V T 1 V T 0 , with U 1 ∈ R p×d 0 , U 0 ∈ R p×q 0 , V 1 ∈ R H×d 0 , V 0 ∈ R H×q 0 , and D 1 = diag(λ 1 , ..., λ d 0 )
. For m ∈ {1, • • • , p}, we note q = pm and we write the SVD of M as

M = ( U 1 U 0 ) D 1 0 0 D 0 V T 1 V T 0 , with U 1 ∈ R p×m , U 0 ∈ R p×q , V 1 ∈ R H×m , V 0 ∈ R H×q , D 1 = diag( λ 1 , .
.., λ m ) and D 0 = diag( λ m+1 , ..., λ p ). We also introduce the orthogonal projectors

Q 1 = I -P 1 = U 0 U T 0 , Q 2 = I -P 2 = V 0 V T 0 , Q 1 = I -P 1 = U 0 U T 0 and Q 2 = I -P 2 = V 0 V T 0 .
Whereas the link between Λ 3 and LSCE is evident, the one conecting Λ 1 and Λ 2 to LSCE relies on the following classical lemma, whose proof is avoided.

Lemma 5. Let M ∈ R p×H , it holds that argmin rank(M )=m M -M 2 F = P 1 M P 2 , and
M -P 1 M P 2 2 F = p k=m+1 λ 2 k ,
where λ 1 , . . . , λ p are the singular values of M arranged in descending order, and P 1 and P 2 are orthogonal right and left singular projectors of M associated with λ 1 , . . . , λ m .

Note that in the previous lemma, P 1 and P 2 are uniquely determined if and only if λ m = λ m+1 .

Nonpivotal statistic

As stated in the introduction, the statistic Λ 1 = n p k=m+1 λ 2 k can be used to arbitrate between the hypotheses of (4). Basically, if H 0 : d 0 = m is realized, all the eigenvalues of the sum goes to 0 and Λ 1 has a weighted chi-squared limiting distribution. Otherwise, at least one eigenvalue converges in probability to a positive number and for any A > 0, P( Λ 1 > A) -→ 1. The following proposition describes the asymptotic behaviour of Λ 1 5 . It was stated in [START_REF] Bura | Extending sliced inverse regression: the weighted chi-squared test[END_REF] and some recent extension can be found in [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF]. Our statement goes further because we are also concerned about the estimation of the asymptotic law of Λ 1 , i.e. the estimation of the weights that intervenes in the weighted chi-squared asymptotic law. Besides, the proof we give in the Appendix is quite simple6 . Proposition 6. Under H 0 , if (1) holds we have

Λ 1 d -→ ν k W 2 k
where the ν k 's are the eigenvalues of the matrix where the ν k 's are the eigenvalues of the matrix

(Q 2 ⊗ Q 1 )Γ(Q 2 ⊗ Q 1 )
( Q 2 ⊗ Q 1 ) Γ( Q 2 ⊗ Q 1 ).
Remark 2. Unlike Theorem 1 in [START_REF] Bura | Extending sliced inverse regression: the weighted chi-squared test[END_REF] or Theorem 1 in [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF], we prefer to state this theorem with the quantities Q 1 and Q 2 rather than with U 0 and V 0 . Because we do not assume that the kernel of M has dimension 1, the vectors that form U 0 or V 0 are not unique because vector spaces with dimension larger than 2 have an infinite number of basis. As a consequence it does not make sense to estimate either U 0 or V 0 . To characterize convergence of spaces, a suitable object is their associated orthogonal projectors.

In general, we do not know the asymptotic distribution of Λ 1 because it depends on (Q 2 ⊗ Q 1 )Γ(Q 2 ⊗ Q 1 ). On the first hand, one can estimate consistently this matrix to get an approximation of the law of Λ 1 under H 0 . Some conditions providing the consistency of the estimation are stated in Proposition 6. On the other hand, one can apply the CS bootstrap to estimate the quantile of Λ 1 in order to test. The main advantage of such an approach is that we no longer need to have a consistent estimator of Γ so that (2) is not needed anymore. Following section 2.1 and by using Lemma 5, we define

M * 0 = P 1 M P 2 + n -1/2 W * with W * | P d → W a. s., (19)
with W defined in [START_REF] Arcones | On the bootstrap of M -estimators and other statistical functionals[END_REF]. Accordingly, we introduce the CS bootstrap statistic → M 0 hold, then the test described in (4) with the statistic Λ 1 and calculation of quantile with Λ * 1 is consistent.

Λ * 1 = n p k=m+1 λ *

Wald-type statistic

The Wald-type statistic Λ 2 = vec(

Q 1 M Q 2 ) T [( Q 2 ⊗ Q 1 ) Γ( Q 2 ⊗ Q 1 )] + vec( Q 1 M Q 2 )
has been introduced in [START_REF] Bura | Dimension estimation in sufficient dimension reduction: a unifying approach[END_REF] to get a pivotal statistic 7 . They obtained the following theorem for which we provide a different proof in the appendix. Following [START_REF] Hu | The estimating function bootstrap[END_REF], we define the associated bootstrap statistic by

Λ * 2 = vec(Q * 1 M * 0 Q * 2 ) T [(Q * 2 ⊗ Q * 1 )Γ * (Q * 2 ⊗ Q * 1 )] + vec(Q * 1 M * 0 Q * 2 ),
where → M 0 and Γ * P → Γ hold, then the test described in (4) with the statistic Λ 2 and calculation of quantile with Λ * 2 is consistent.

M * 0 is defined in (19), Γ * ∈ R pH×pH , Q * 1 ,

Minimum Discrepancy approach

Noting that {rank(M ) = m} has co-dimension (Hm)(pm) and applying [START_REF] Cragg | Inferring the rank of a matrix[END_REF] we get the following proposition8 .

Proposition 10. If (1), (2), and (3) hold, we have

Λ 3 d -→ χ 2 (H-m)(p-m) .
In general a minimizer

M c = argmin rank(M )=m vec( M -M ) T Γ -1 vec( M -M )
does not have an explicit form as it was for the constrained matrix associated with Λ 1 and Λ 2 . Therefore, we define

M * 0 = M c + n -1/2 W * with W * | P d → W a. s., ( 20
)
where W is defined in [START_REF] Arcones | On the bootstrap of M -estimators and other statistical functionals[END_REF]. We also define the associated CS bootstrap statistic

Λ * 3 = n min rank(M )=m vec(M * 0 -M ) T Γ * -1 vec(M * 0 -M ),
and applying Theorem 4 we have the following result.

Proposition 11. If ( 1), ( 2), ( 3), ( 20), Γ * P → Γ, and M a.s.

→ M 0 hold, then the test described in (4) with the statistic Λ 3 and calculation of quantiles with Λ * 3 is consistent.

Remark 3. The set of assumptions needed to obtain Proposition 10 is stronger than the ones stated in propositions 6 and 8 ensuring the convergence of Λ 1 and Λ 2 . As a consequence this is also true for Proposition 11 with respect to propositions 7 and 9. The main difference is that we add the assumption on Γ to be non deficient. This assumption cannot be alleviated in the statement but is not as restrictive in practice. On the first hand, if Γ is deficient the optimization under constraint has a free coordinate which implies the non-convergence of the minimizer. On the other hand, because of the semi-definite character of Γ the projection of M on the null space of Γ is null. Then one can apply the proposition to the restriction of M on the range of Γ. This is the case in the application to SDR in Section 4.

Remark 4. Unlike the situation of Λ 1 and Λ 2 , an optimization algorithm is needed to obtain Λ 3 and Λ * 3 , this point out an important issue of such a procedure. In [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF], the authors noticed that

Λ 3 = n min A∈H d ,B∈R d×l (vec( M ) -vec(AB)) T Γ -1 (vec( M ) -vec(AB))
where H d is the set of orthogonal basis lying in R p with dimension d. We follow their algorithm in the computation of Λ 3 (see [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF], Section 3.3 for the details).

3.4

The statistics Λ 1 , Λ 2 , Λ 3 through an example

In the introduction, we already mentioned several drawbacks and advantages of the use of Λ 1 , Λ 2 , or Λ 3 . The remark relied on both pivotality of the statistics and large matrix inversion.

Here we develop another point of view related to the algebraic nature of the statistics. Facing the representation provided by Table 1, each statistic Λ 1 and Λ 2 evaluates a different distance between M and M c . The first one is the distance that is optimized, but the second is another one. This has raised the issue we present here through the following example. For the sake of clarity, we consider

M = λ 1 0 0 λ 2 with λ k = 1 n n i=1 λ k,i , for k = 1, 2, and (λ k,i ) k,i i.i.d.,
and we test H 0 :

d 0 = 1 against H 1 : d 0 > 1. We assume that λ 1 > λ 2 , we have Λ 1 = n λ 2 2 . Otherwise, one can show that Λ 2 = n λ 2 2 v 2 + o P (1), with v k = (λ k -λ k ) 2 .
For Λ 3 it is clear that the minimization can be done over the diagonal matrix diag(λ 1 , λ 2 ) and one has

Λ 3 = n argmin λ 1 λ 2 =0 λ 1 -λ 1 v 1 + λ 2 -λ 2 v 2 + o P (1) = n min λ 2 1 v 1 , λ 2 2 v 2 + o P (1).
Accordingly, by Proposition 7, 9 and 11, the three tests can be summarized by

n λ 2 2 compared to v 2 χ 2 1 , n λ 2 2 v 2 compared to χ 2 2 , n min λ 2 1 v 1 , λ 2 2 v 2 compared to χ 2 2 ,
where v k = var(λ k,1 ). Assume there is less variance on the estimate of the smallest eigenvalue, i.e. v 1 > v 2 such that

λ 2 1 v 1 < λ 2 2
v 2 , this situation may arise when λ 1 and λ 2 have similar values but different variances. Then to conduct the test, the statistic

λ 2 1 v 1 is a better choice than λ 2 2 v 2 .
As a consequence, unlike Λ 1 and Λ 2 , the statistic Λ 3 appears as a coherent choice because its associated minimization takes into account the variance of the estimation.

Application to sufficient dimension reduction

We focus on a particularly famous method in SDR called sliced inverse regression (SIR) which has been introduced in [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF] to deal with the regression model

Y = f (P X, ε) (21)
where ε ⊥ ⊥ X ∈ R p , Y ∈ R, and P is a projector on the vector space E with dimension d 0 < p, called the central subspace. The objective is to estimate E. If X is elliptically distributed, then we have that Σ

-1 (E[(X -E[X])ψ(Y )
] ∈ E with Σ = var(X), for any measurable function ψ. Accordingly, in order to recover the whole central subspace one needs to consider many functions ψ. For a given family of functions (ψ h ) 1≤h≤H we define Ψ = (ψ 1 (Y ), ..., ψ H (Y )) T . Under some additional conditions [START_REF] Portier | Optimal transformation: A new approach for covering the central subspace[END_REF], the image of the matrix Σ -1/2 cov(X, Ψ(Y )) is equal to Σ 1/2 E. Then one can make the svd of an estimator of this matrix to obtain d 0 vectors that form an estimated basis of Σ 1/2 E. Motivated by the curse of dimensionality, the estimation of d 0 is one of the most crucial points in SDR. To make that possible, a popular way consists in estimating the rank of Σ -1/2 cov(X, Ψ) using the hypothesis testing framework given by (4) (see for example [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF], [START_REF] Bura | Extending sliced inverse regression: the weighted chi-squared test[END_REF] and [START_REF] Cook | Sufficient dimension reduction via inverse regression: a minimum discrepancy approach[END_REF]). Since we are interested in estimating the rank, we prefer to deal directly with cov(X, Ψ) to avoid the introduction of an additional noise due to the estimation of the matrix Σ. Assume that (( 21), denote by P its associated empirical c.d.f. and define the quantity

X 1 , Y 1 ), • • • , (X n , Y n )) is a i.i.d. sequence from model (
C = E[K], with K = (X -E[X])(Ψ(Y ) -E[Ψ(Y )]) T ,
associated with its empirical estimator

C = K, with K i = (X i -X)(Ψ i -Ψ) T , and Ψ i = Ψ(Y i ).
We apply the CS bootstrap to calculate the quantiles of each statistic. Facing ( 19) and ( 20), we use an independent weighted bootstrap to reproduce the asymptotic law of √ n( C -C), that is we define the bootstrap matrix

C * = C c + K * , with K * i = w i ( K i -K) (22)
where C c stands for the solution of an optimization problem depending on the selected statistic Λ 1 , Λ 2 or Λ 3 (see Section 3 for the details) and (w i ) is a sequence of i.i.d. random variables. We also define

V = var(vec(K)) and V * = 1 n n i=1 vec(K * i -K * ) vec(K * i -K * ) T .
To apply propositions 7, 9, and 11, we need the following result which is of particular interest since it provides a new bootstrap procedure for SIR that is different than the one proposed in [START_REF] Barrios | A bootstrap method for assessing the dimension of a general regression problem[END_REF].

Proposition 12. Assume that E[ X 2 ] < +∞, E[ Ψ(Y ) 2
] and E[ K 4 F ] are finites, if moreover (w i ) is a i.i.d. sequence of real random variables with mean 0 and variance 1, then we have

L ∞ (n 1/2 K * | P ) = L ∞ (n 1/2 ( C -C)) a.s. and V * P → V conditionally a.s.. Remark 5.
Taking a partition {I(h), h = 1, . . . , H} of the range of Y we recover the original SIR method with the family formed by the p

-1/2 h 1 {Y ∈I(h)} 's with p h = P(Y ∈ I(h)). Then C SIR = Σ -1/2 cov(X, 1)D -1/2 with 1 = (1 {Y i ∈I(1)} , . . . , 1 {Y i ∈I(H)} ) T and D = diag(p h ), is estimated by C SIR = Σ -1/2 (X -X)1 T D -1/2 with D = diag( p h ), p h = 1 {Y ∈I(h)} , Σ = (X -X)(X -X) T . We have the expansion n -1/2 ( C SIR -C SIR ) = n -1/2 Σ -1/2 ((X -E[X])1 T -cov(X, 1))D -1/2 -Σ -1/2 n -1/2 ( Σ 1/2 -Σ 1/2 )C SIR -C SIR n -1/2 ( D 1/2 p -D 1/2 p )D -1/2 p + o P (1).
As a consequence, the matrix Σ -1/2 and the weights p h 's are playing an important role on the asymptotic of the matrix SIR. They introduce some other terms in the asymptotic distribution and clearly the simple bootstrap presented before does not work for SIR as it was originally defined. Even if we believe that a more evolved weighted bootstrap works to bootstrap √ n( C SIR -C SIR ), we emphasize that it may be less accurate than the one we propose since it complicates the asymptotic without being necessary for testing the rank.

Recall that m is a non-negative integer, for k ∈ {1, 2, 3} and B ∈ N * we calculate independent copies Λ * k,1 , ..., Λ * k,B with the CS bootstrap algorithm corresponding to each statistic. Then we estimate the quantile with

q * k (α) = inf t∈R {F * k (t) > α} = Λ * k,(⌈Bα⌉) , where F * k (t) = 1 B B b=1
H 0 is rejected if Λ k > q * k (α). ( 23
)
On the other hand, the traditional test is conducted by comparing the statistic Λ 2 and Λ 3 to the quantile of their asymptotic law respectively given by propositions 8 and 10. For Λ 1 , in general the limit in law is quite complicated 9 (see Proposition 6), so that we use approximations: the Wood's approximation (see [START_REF] Andrew | An f approximation to the distribution of a linear combination of chi-squared variables[END_REF]) as it is computed in the R software, an adjusted version

Λ 1,adj. = Λ 1 /a d → χ 2 b , with a = s k=1 ω 2 / s k=1 ω k , b = ( s k=1 ω k ) 2 / s k=1 ω 2 k , and a re-scaled version Λ 1,sc = Λ 1 /c d → χ 2
s , c = ω (see [START_REF] Bentler | Corrections to test statistics in principal hessian directions[END_REF] for these two corrections). In all the simulations we compute the matrix C by taking Ψ(t) = (1 {y∈I(1),...,y∈I(H)} ) where the I(h)'s form an equi-partition of the range of the data Y 1 , . . . , Y n . In the whole study we put (p, H) = (6, 5), B = 1000 and we consider n = 50, 100, 200, 500. Although the parameter H does not really affect the SIR method, we choose it globally good with respect to all the situations.

The first model we study is the following standard model: In order to highlight guidelines (A) and (B), we produce in figure 1 two graphics each representing situation under H 1 and H 0 for the statistic Λ 3 . Similar graphics dealing with Λ 2 have been drawn but are not presented here. On the first one we see that even if the sample is under H 1 the bootstrap distribution reflects H 0 . As a consequence, guideline (A) is satisfied and the power of the bootstrap test is going to 1. The second graph shows that the statistic distribution is closer to the bootstrap distribution than its asymptotic distribution. This has no reason to occur when the statistic is not pivotal (see the introduction and [START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF] for the details). As a consequence, we believe that this good fitting is due to Guideline B.

In figure 2 we analyse the asymptotic distribution of q(α) in model I for each statistic. To measure the error we consider the behaviour of

F n ( q(α)),
which is optimally equal to 1α. To make that possible, F n is estimated with a large sample size so that the estimation error is negligible. Then we run over 100 samples the CS bootstrap to provide, for each sample, a bootstrap estimation of the quantile q(α). The associated boxplot for n = 100, 200, 500 are provided in Figure 2. As a consequence, we may notice that the behaviour of Λ 2 and Λ 3 are quite similar facing the one of Λ 1 . Even if every boxplot argues for convergence to 1α, testing with Λ 1 seems a better choice when n is small because of a quasi immediate convergence of the bias. When n increase, this is no longer evident because the variance of either Λ * 2 or Λ * 3 is smaller. Furthermore, we go into details in Table 2 by running Model I over 5000 samples. For each of them and every statistic, we conduct the bootstrap test [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF] and its traditional version. The table presents for each m ≤ d 0 , the proportion of rejected tests. This corresponds to either estimate of the power or estimate of the level.

Λ 2 Λ 1 Λ 3 Λ 1 Λ 2 Λ 3
Although it has not the best power, the clear winner is the tests based on Λ 1 . Inside this group, for any sample number, the bootstrap and the rescaled version are the closest to the nominal level. Concerning Λ 2 and Λ 3 the result are quite impressive when n is small: for n = 100, whereas traditional testing makes a type I error 30% of the time, the bootstrap testing goes wrong around 7%. This confirms observation on the second graph of Figure 1. In Table 3 and Table 4 we consider the same model than Model I excepted that we change the distribution of the predictors: in Model Ia, X has independent coordinates with a student distribution with 5 degrees of freedom, in Model Ib,

n m Λ 1 Λ 2 Λ 3 Wood Resc. Adj. CB Λ 1 Λ 2 CB Λ 2 Λ 3 CB Λ 3 50
X d = .1X 1 ǫ + X 2 (1 -ǫ) with ǫ d = B(1/2), X 1 d = N ((6, 0, • • • , 0), I), X 2 d
= N (0, I). For this two models, we have similar conclusions than model I with two new things. First, the rescaled version is not robust to the distribution of the predictors (Table 4). Second, the algorithm employed to optimized Λ 3 could failed at very small sample size.

We introduce a non linear relationship by considering the model In Table 5, we present similar results as in tables 3-5 with the difference that the nominal level is α = 1% in order to highlight differences in the power of each test. Again, the CS bootstrap induces a large improvement of the accuracy of the test with Λ 2 and Λ 3 . At n = 50, the test based on Λ 1 is less powerful than the others but it is more accurate under H 0 . The winner remains the CS bootstrap with Λ 1 . A new important things is that at n = 500, it seems better to use the CS bootstrap with Λ 2 and Λ 3 . Actually this is due to the variance of the formers which is smaller than the variance of Λ * 1 as it was already highlighted in Figure 2. We conclude by increasing difficulty considering the following model, introduced in [START_REF] Li | Sliced inverse regression for dimension reduction[END_REF],

Model III: Y = X 1 .5 + (X 2 + 2) 2 + e e ⊥ ⊥ X, X d = N (0, I)

We still present in Table 6 the estimated level and power with the nominal level α = 2% for each test. For such a model the conclusions are quite mitigated because it induces a trade-off between high power and accurate level. Indeed when n is small, the better powers are provided by the traditional tests with Λ 2 and Λ 3 . Nevertheless the more accurate levels can be found looking at the CS bootstrap with Λ 2 (n = 100) or Λ 1 (n = 200). Moreover the tests associated to Λ 1 without bootstrap are the worst concerning this model. Accordingly, the simulation study highlighted the good behaviour of the CS bootstrap: in every model it improves the accuracy of the traditional test for each statistic. One may remember that the bias of the CS bootstrap with Λ 1 has the faster rate of convergence with respect to the CS bootstrap of Λ 2 or Λ 3 . Otherwise, the variance of Λ * 1 may be greater than the variance of Λ * 2 or Λ * 3 . Finally, for the simple models it seems better to use the CS bootstrap with the statistic Λ 1 .

Concluding remarks

Along this study, we found that the main advantages of the CS bootstrap are: with g continuously differentiable on θ c and J g (θ c ) full rank. By assumption on g, θ * c , at least for n large enough, satisfies the first order conditions, that are

A * (θ * 0 -θ * c ) -J T g (θ * c )λ * n = 0 g(θ * c ) = 0
where λ * n is the Lagrange multiplier. Using a Taylor expansion of g around θ c , we get g(θ * c ) = g( θ c ) + J T g ( θ c )(θ * cθ c ) + o P ( θ * cθ c ), and with the previous equations we have

A * J T g (θ * c ) J g ( θ c ) 0 θ * c -θ c λ * n = A * (θ * 0 -θ c ) o P ( θ * c -θ c )
.

Now by Slutsky's lemma, we get A J T g (θ c ) J g (θ c ) 0

n 1/2 (θ * c -θ c ) n 1/2 λ * n = n 1/2 A(θ * 0 -θ c ) 0 + o P (1),
and the conclusion follows by multiplying on the left by the matrix A -1 -P A -1 , A -1 J T g (θ c )(J g (θ c )A -1 J T g (θ c )) -1

with P = A -1 J T g (θ c )(J g (θ c )A -1 J T g (θ c )) -1 J g (θ c ).

Proof of Theorem 4

The proof is divided in two parts each corresponding to the level and the power of the test. Assume H 0 and define F n and F ∞ respectively as the c.d.f. of Λ and the weak limit of F n . Note that we can apply Proposition 2 to get In other words, with probability 1, F converges pointwise to F ∞ . As in [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF] chapter 23, Lemma 3, consider ∆ the set of discontinuity of F -1 ∞ . For every α ∈]0, 1[\∆, we have q(α) -→ q(α) a.s. (see for instance [START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF], chapter 21). Using Slutsky's theorem, we get L ∞ ( Λq(α)) = L ∞ ( Λq(α)), accordingly P( Λ ≤ q(α)) -→ F ∞ (q(α)) for all α ∈]0, 1[\∆.

Because F ∞ is continuous F ∞ (q(α)) = α. Since F ∞ is non-decreasing, ∆ is denumerable, since α → P( Λ ≤ q(α)) is non-decreasing with continuous limit, the convergence is uniform and so

  X i , θ) where Θ is an open set, then the bootstrap of √ n( θ Mθ M ) is carried out by the quantity √ n(θ * Mθ M ) with

Theorem 3 .

 3 Let M be a submanifold. Assume that θ a.s. → θ 0 with θ 0 M-nonsingular and A P → A hold. If moreover (17) holds and conditionally a.s. A * P

Theorem 4 .

 4 Let M be a manifold. Assume that θ a.s.

Proposition 8 . 2 d-→ χ 2

 822 If (1) and (2) hold, we have Λ s , with s = min(rank(Γ), (pd)(Hd)).

Figure 1 :

 1 Figure 1: Plot of the asymptotic distribution, and the estimated distribution of the statistic and the bootstrap statistic for Λ 3 in the case of Model I.

Figure 2 :

 2 Figure 2: Bowplot over 100 samples of q(α) for Λ 1 , Λ 2 , Λ 3 and α = 0.95 in the case of Model I for different values of n .

  Model II: Y = tanh(X 1 ) + .1e with e ⊥ ⊥ X, X d = N (0, I), e d = N (0, 1).

n 1 / 2 θ

 12 θ 0 θ cθ 0 = n 1/2 I I -P ( θθ 0 ) + o P (1),and Theorem 3 to get conditionally a.s.n 1/2 θ * 0θ c θ * cθ c = n 1/2 I I -P (θ * 0θ c ) + o P (1).with P detailed in the statement of Proposition 2. Using (11), (18) and Slutsky's theorem we haveL ∞ (Λ * | P ) = L ∞ ( Λ) a. s. .

Table 1 :

 1 Values of A and B in (8) for Λ 1 , Λ 2 and Λ 3 .

at level α if

  * p the smallest singular values of M * . The following proposition is a straightforward application of Theorem 4 with the submanifold {rank(M ) = m}.

	2 k ,
	with λ * m+1 , ..., λ Proposition 7. If (1), (19) and M

a.s.

Table 2 :

 2 Estimated levels and power in Model i for α = 5%.

	n	m	Wood	Resc.	Λ 1	Adj.	CB Λ 1	Λ 2	Λ 2	CB Λ 2	Λ 3	Λ 3 CB Λ 3
	50	0 0.9988 0.9998 0.9988 0.9988 1.0000 1.0000 1.0000 1.0000 1 0.0326 0.0590 0.0336 0.0494 0.3466 0.0744 0.3098 0.07
	100	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0386 0.052 0.0388 0.0456 0.1494 0.0676 0.1466 0.0722
	200	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0474 0.055 0.0476 0.0514 0.096 0.0646 0.0954 0.0664
	500	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0492 0.0514 0.0494 0.0516 0.0656 0.0584 0.0654 0.0584
	n	m	Wood	Resc.	Λ 1	Adj.	CB Λ 1	Λ 2	Λ 2	CB Λ 2	Λ 3	Λ 3 CB Λ 3
	50	0 0.9646 0.9928 0.9656 0.9682 1.0000 1.0000 1.0000 1.0000 1 0.0318 0.0628 0.0324 0.0496 0.3412 0.0588 0.3042 0.0628
	100	0 0.9996 1.0000 0.9996 0.9996 1.0000 1.0000 1.0000 1.0000 1 0.0336 0.0486 0.0344 0.0412 0.1516 0.0696 0.1432 0.0718
	200	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0378 0.0486 0.038 0.0424 0.0844 0.0602 0.0832 0.0604
	500	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0454 0.0502 0.0458 0.0474 0.0638 0.0606 0.0634 0.0608

Table 3 :

 3 Estimated levels and power in Model ia for α = 5%.

Table 4 :

 4 Estimated levels and power in Model ib for α = 5%.

		0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000
		1	0.034 0.1072 0.034 0.0378 0.2122 0.0396 0.1394 0.015
	100	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.037 0.0904 0.0374 0.0404 0.0986 0.0572 0.0614 0.0284
	200	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0484 0.096 0.0488 0.0518 0.0708 0.066 0.056 0.0506
	500	0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.0486 0.0912 0.0486 0.0490 0.0598 0.0664 0.0612 0.0674

Table 5 :

 5 Estimated levels and power in Model ii for α = 1%.

A bootstrap with a Delta-method approach (see[START_REF] Van Der | Asymptotic statistics, volume 3 of Cambridge Series in Statistical and Probabilistic Mathematics[END_REF], chapter

23, Theorem 5) fails because x → min θ =1xθ is not continuously differentiable on the unit circle.

We refer to[START_REF] Hall | Two guidelines for bootstrap hypothesis testing[END_REF] for a study of this bootstrap in order to test θ0 = µ.

The bootstrap procedure to get W * is not specified because it depends on θ. For instance, if θ is a mean over some i.i.d. random variables, one can use the Efron's traditional bootstrap and if θ is a M-estimator, one should use a bootstrap as detailed by equation[START_REF] Hall | The bootstrap and Edgeworth expansion[END_REF].

Assumptions about A * and B * are provided further in the statements of the propositions.

A similar proposition can be stated applying Proposition 12. Following this way, the asymptotic depends on g which is difficult to estimate for rank constraints (see Remark 1).

We no longer need the results of[START_REF] Eaton | The asymptotic distribution of singular values with applications to canonical correlations and correspondence analysis[END_REF] about the asymptotic behaviour of singular values.

We write the expression of Λ2 another way for the reasons explained in Remark 2 but one can recover the original expression by noting that for any symmetric matrix A, A + H = (AH) + if H is an orthonormal basis of a vector subspace of Im(A).

See[START_REF] Cragg | Inferring the rank of a matrix[END_REF] for the original proof.

When the predictors are normally distributed, it has been shown that Λ1 is asymptotically chi-squared distributed (see[START_REF] Bura | Extending sliced inverse regression: the weighted chi-squared test[END_REF]). The authors also pointed out that it was less robust than the weighted chi-squared asymptotic as soon as the predictors distribution deviates from normality. As a result, we keep in the nonparametric framework by avoiding such asymptotic in this simulation study.

CB Λ 3 50 0 0.9950 0.9992 0.9962 0.9960 1.0000 0.9966 1.0000 0.9966 1 0.3750 0.5342 0.3990 0.4676 0.9074 0.5066 0.8344 0.3270 2 0.0078 0.0156 0.0086 0.0240 0.0620 0.0164 0.0344 0.0136 100 0 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 1 0.9330 0.9556 0.9368 0.9446 0.9952 0.9842 0.9934 0.9806 2 0.0134 0.0176 0.0138 0.0210 0.0306 0.0228 0.0266 0.0278 200 0 1.000 1.0000 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 1 1.000 1.0000 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0.0154 0.0182 0.0158 0.0198 0.025 0.024 0.0244 0.026 500 0 1.0000 1.000 1.0000 1.0000 1.0000 1.000 1.0000 1.0000 1 1.0000 1.000 1.0000 1.0000 1.0000 1.0000 1.0000 1.0000 2 0.0184 0.0194 0.0184 0.02 0.0228 0.0228 0.0228 0.023 Table 6: Estimated levels and power in Model ii for α = 2%.

1. Alternative to the asymptotic comparison. This argument is even stronger since the asymptotic law can be unknown (or difficult to estimate) or the asymptotic law remains too much different from the statistic law (e.g. large matrix inversion).

2. By Theorem 4, which provides its consistency, the CS bootstrap works under mild assumptions. Essentially, we ask the manifold to be locally smooth, and we require a bootstrap 18 of the unconstrained estimator.

3. The CS bootstrap is computationally as simple than the considered statistic.

4. In the case of rank testing, the CS bootstrap clearly improves the accuracy of traditional testing (cf. the simulation study).

Besides, there exists some natural extensions of the previous work. First although it is suitable for testing, the form of the objective function bQ is quiet restrictive. For example, we believe that the CS bootstrap could be extended to M and Z estimation. Secondly, conditions that guarantee

have not been provided yet. This would valid theoretically the use of the CS bootstrap with respect to traditional testing.

Appendix Proof of Lemma 1

The whole proof is made conditionally on the sample. By definition of θ c , with high probability, A * is full rank for n large enough, we have

Then since θ * 0θ c P → 0, θ c → θ c and because A * P → A is full rank, one gets that θ * c P → θ c . Therefore, since θ c is M-nonsingular and reffering to Definition 1, we get

holds for every α ∈]0, 1[. This concludes the proof for the level. It remains to show that the power of the test goes to 1. Assume H 1 and let α ∈]0, 1[, the statistic Λ goes to infinity in probability and it suffices to show that with probability 1 the bootstrap quantile q(α) remains bounded. This means exactly that conditionally a.s. the sequence Λ * is tight. Note that conditionally a.s. we have

where Λ * converges in distribution by [START_REF] Hall | Two guidelines for bootstrap hypothesis testing[END_REF], and is therefore tight.

Proof of Proposition 6

We have

By the Delta method and because H 0 is realized, we can apply convergence results about eigenprojectors to both matrices M T M and M M T to obtain the √ n-convergence for Q 1 and Q 2 . Then we write

which suffices to obtained the first statement of the theorem. For the second statement, the symmetric matrix

and so are its eigenvalues.

Proof of Proposition 8

We can notice that

Proof of Proposition 12

Recall that

). First note that, by Slutsky's theorem, √ n K * has the same asymptotic law than

). Then we can develop

Checking a Lindeberg condition as bellow to ensure the weak convergence of n -1/2 n i=1 w i (X i -E[X]) and n -1/2 n i=1 w i (Ψ i -Ψ) T , and using the Slutsky's theorem we get conditionally a.s.

We can apply the multidimensional version of the Lindeberg's central limit theorem (see for instance [START_REF] Bhattacharya | Normal approximation and asymptotic expansions[END_REF], Corollary 18.2), provided that

-→ 0,

where

The above convergence is a consequence of the Lebesgue domination theorem which ensure that each term of the sum goes to 0, afterwards we can conclude by the Cesaro's Lemma. Thus we have proved that conditionally a.s.

and it remains to note that V a.s.

→ V the variance of the limit in law of √ n( C -C) provided that K has a finite order 2 moment. For the second convergence, we note that conditionally a.s.

(w 2 i -1)ξ i ξ T i + o P (1), then by noting v i a coordinate of ξ i ξ T i we calculate

which goes to 0 a.s. provided that K has a finite order 4 moment. We conclude by using the Markov inequality to get that V * P → V conditionally a.s..