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Over the past few decades a broad variety of models has been developed for graphs. However, modern applications in various fields highlighted the need to account for higher-order interactions, to include the information deriving from groups of three or more nodes. Simple examples include group interactions in social networks, scientific co-authorship, interactions between more than two species in ecological models or high-order correlations between neurons in brain networks. Hypergraphs provide the most general formalization of higher-order interactions: similarly to a graph, a hypergraph is defined as a set of nodes and a set of hyperedges, the latter specifying nodes taking part in each interaction. We propose a stochastic block model for hypergraphs to perform model-based clustering, capturing the information deriving from higher-order interactions. The formulation is sufficiently flexible to account for possible simplified latent structures. A variational expectation-maximization algorithm is developed to perform parameter estimation and model selection is explored using the ICL criterion. The model is applied to both simulated and real data, and the performance of the proposal is assessed in terms of parameter estimation and ability to recover the clusters. The estimation algorithm was implemented in C++ language and it was made available for the R software.

International audience; Comparing the architecture of interaction networks in space or time is essential for understanding the assembly, trajectory, functioning and persistence of species communities. Graph embedding methods, which position networks into a vector space where nearby networks have similar architectures, could be ideal tools for this purposes.Here, we evaluated the ability of seven graph embedding methods to disentangle architectural similarities of interactions networks for supervised and unsupervised posterior analytic tasks. The evaluation was carried out over a large number of simulated trophic networks representing variations around six ecological properties and size.We did not find an overall best method and instead showed that the performance of the methods depended on the targeted ecological properties and thus on the research questions. We also highlighted the importance of normalizing the embedding for network sizes for meaningful posterior unsupervised analyses.We concluded by orientating potential users to the most suited methods given the question, the targeted network ecological property, and outlined links between those ecological properties and three ecological processes: robustness to extinction, community persistence and ecosystem functioning. We hope this study will stimulate the appropriation of graph embedding methods by ecologists.

The use of comparative genomics for functional, evolutionary, and epidemiological studies requires methods to classify gene families in terms of occurrence in a given species. These methods usually lack multivariate statistical models to infer the partitions and the optimal number of classes and don’t account for genome organization. We introduce a graph structure to model pangenomes in which nodes represent gene families and edges represent genomic neighborhood. Our method, named PPanGGOLiN, partitions nodes using an Expectation-Maximization algorithm based on multivariate Bernoulli Mixture Model coupled with a Markov Random Field. This approach takes into account the topology of the graph and the presence/absence of genes in pangenomes to classify gene families into persistent, cloud, and one or several shell partitions. By analyzing the partitioned pangenome graphs of isolate genomes from 439 species and metagenome-assembled genomes from 78 species, we demonstrate that our method is effective in estimating the persistent genome. Interestingly, it shows that the shell genome is a key element to understand genome dynamics, presumably because it reflects how genes present at intermediate frequencies drive adaptation of species, and its proportion in genomes is independent of genome size. The graph-based approach proposed by PPanGGOLiN is useful to depict the overall genomic diversity of thousands of strains in a compact structure and provides an effective basis for very large scale comparative genomics. The software is freely available at https://github.com/labgem/PPanGGOLiN., Author summary Microorganisms have the greatest biodiversity and evolutionary history on earth. At the genomic level, it is reflected by a highly variable gene content even among organisms from the same species which explains the ability of microbes to be pathogenic or to grow in specific environments. We developed a new method called PPanGGOLiN which accurately represents the genomic diversity of a species (i.e. its pangenome) using a compact graph structure. Based on this pangenome graph, we classify genes by a statistical method according to their occurrence in the genomes. This method allowed us to build pangenomes even for uncultivated species at an unprecedented scale. We applied our method on all available genomes in databanks in order to depict the overall diversity of hundreds of species. Overall, our work enables microbiologists to explore and visualize pangenomes alike a subway map.

International audience; We consider a dynamic version of the stochastic block model, in which the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution is modeled through a hidden Markov chain on the nodes memberships. We prove the consistency (as the number of nodes and time steps increase) of the maximum likelihood and variational estimators of the model parameters, and obtain upper bounds on the rates of convergence of these estimators. We also explore the particular case where the number of time steps is fixed and connectivity parameters are allowed to vary.

International audience; The aim of this paper is to explore the robustness of the parsimonious host-symbiont tree reconciliation method under editing or small perturbations of the input. The editing involves making different choices of unique symbiont mapping to a host in the case where multiple associations exist. This is made necessary by the fact that the tree reconciliation model is currently unable to handle such associations. The analysis performed could however also address the problem of errors. The perturbations are re-rootings of the symbiont tree to deal with a possibly wrong placement of the root specially in the case of fast-evolving species. In order to do this robustness analysis, we introduce a simulation scheme specifically designed for the host-symbiont cophylogeny context, as well as a measure to compare sets of tree reconciliations, both of which are of interest by themselves.

Summary Statistical node clustering in discrete time dynamic networks is an emerging field that raises many challenges. Here, we explore statistical properties and frequentist inference in a model that combines a stochastic block model for its static part with independent Markov chains for the evolution of the nodes groups through time. We model binary data as well as weighted dynamic random graphs (with discrete or continuous edges values). Our approach, motivated by the importance of controlling for label switching issues across the different time steps, focuses on detecting groups characterized by a stable within-group connectivity behaviour. We study identifiability of the model parameters and propose an inference procedure based on a variational expectation–maximization algorithm as well as a model selection criterion to select the number of groups. We carefully discuss our initialization strategy which plays an important role in the method and we compare our procedure with existing procedures on synthetic data sets. We also illustrate our approach on dynamic contact networks: one of encounters between high school students and two others on animal interactions. An implementation of the method is available as an R package called dynsbm.

We propose an approach for multiple sequence alignment (MSA) derived from the dynamic time warping viewpoint and recent techniques of curve synchronization developed in the context of functional data analysis. Starting from pairwise alignments of all the sequences (viewed as paths in a certain space), we construct a median path that represents the MSA we are looking for. We establish a proof of concept that our method could be an interesting ingredient to include into refined MSA techniques. We present a simple synthetic experiment as well as the study of a benchmark dataset, together with comparisons with 2 widely used MSA softwares.

International audience; The aim of this paper is to explore the robustness of the parsimonious host-symbiont tree reconciliation method under editing or small perturbations of the input. The editing involves making different choices of unique symbiont mapping to a host in the case where multiple associations exist. This is made necessary by the fact that no tree reconciliation method is currently able to handle such associations. The analysis performed could however also address the problem of errors. The perturbations are re-rootings of the symbiont tree to deal with a possibly wrong placement of the root specially in the case of fast-evolving species. In order to do this robustness analysis, we introduce a simulation scheme specifically designed for the host-symbiont cophylogeny context, as well as a measure to compare sets of tree reconciliations, both of which are of interest by themselves.

International audience; We focus on the parametric estimation of the distribution of a Markov environment from the observation of a single trajectory of a one-dimensional nearest-neighbor path evolving in this random environment. In the ballistic case, as the length of the path increases, we prove consistency, asymptotic normality and efficiency of the maximum likelihood estimator. Our contribution is two-fold: we cast the problem into the one of parameter estimation in a hidden Markov model (HMM) and establish that the bivariate Markov chain underlying this HMM is positive Harris recurrent. We provide different examples of setups in which our results apply, in particular that of DNA unzipping model, and we give a simple synthetic experiment to illustrate those results.

International audience; We consider a one dimensional ballistic random walk evolving in an i.i.d. parametric random environment. We provide a maximum likelihood estimation procedure of the parameters based on a single observation of the path till the time it reaches a distant site, and prove that the estimator is consistent as the distant site tends to infinity. Our main tool consists in using the link between random walks and branching processes in random environments and explicitly characterising the limiting distribution of the process that arises. We also explore the numerical performance of our estimation procedure.

International audience; In a multiple testing context, we consider a semiparametric mixture model with two components where one component is known and corresponds to the distribution of p-values under the null hypothesis and the other component f is nonparametric and stands for the distribution under the alternative hypothesis. Motivated by the issue of local false discovery rate estimation, we focus here on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of the unknown proportion. of true null hypotheses. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Holder densities. To our knowledge, this is the first result establishing convergence as well as corresponding rate for the estimation of the unknown component in this nonparametric mixture. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate. Their respective performances are then compared on synthetic data.

International audience; This article proposes a novel approach to statistical alignment of nucleotide sequences by introducing a context dependent structure on the substitution process in the underlying evolutionary model. We propose to estimate alignments and context dependent mutation rates relying on the observation of two homologous sequences. The procedure is based on a generalized pair-hidden Markov structure, where conditional on the alignment path, the nucleotide sequences follow a Markov distribution. We use a stochastic approximation expectation maximization (saem) algorithm to give accurate estimators of parameters and alignments. We provide results both on simulated data and vertebrate genomes, which are known to have a high mutation rate from CG dinucleotide. In particular, we establish that the method improves the accuracy of the alignment of a human pseudogene and its functional gene.

Summary Random-graph mixture models are very popular for modelling real data networks. Parameter estimation procedures usually rely on variational approximations, either combined with the expectation–maximization (EM) algorithm or with Bayesian approaches. Despite good results on synthetic data, the validity of the variational approximation is, however, not established. Moreover, these variational approaches aim at approximating the maximum likelihood or the maximum a posteriori estimators, whose behaviour in an asymptotic framework (as the sample size increases to ∞) remains unknown for these models. In this work, we show that, in many different affiliation contexts (for binary or weighted graphs), parameter estimators based either on moment equations or on the maximization of some composite likelihood are strongly consistent and √n convergent, when the number n of nodes increases to ∞. As a consequence, our result establishes that the overall structure of an affiliation model can be (asymptotically) caught by the description of the network in terms of its number of triads (order 3 structures) and edges (order 2 structures). Moreover, these parameter estimates are either explicit (as for the moment estimators) or may be approximated by using a simple EM algorithm, whose convergence properties are known. We illustrate the efficiency of our method on simulated data and compare its performances with other existing procedures. A data set of cross-citations among economics journals is also analysed.

We prove identifiability of parameters for a broad class of random graph mixture models. These models are characterized by a partition of the set of graph nodes into latent (unobservable) groups. The connectivities between nodes are independent random variables when conditioned on the groups of the nodes being connected. In the binary random graph case, in which edges are either present or absent, these models are known as stochastic blockmodels and have been widely used in the social sciences and, more recently, in biology. Their generalizations to weighted random graphs, either in parametric or non-parametric form, are also of interest in many areas. Despite a broad range of applications, the parameter identifiability issue for such models is involved, and previously has only been touched upon in the literature. We give here a thorough investigation of this problem. Our work also has consequences for parameter estimation. In particular, the estimation procedure proposed by Frank and Harary for binary affiliation models is revisited in this article.

Summary: The R package SIMoNe (Statistical Inference for MOdular NEtworks) enables inference of gene-regulatory networks based on partial correlation coefficients from microarray experiments. Modelling gene expression data with a Gaussian graphical model (hereafter GGM), the algorithm estimates non-zero entries of the concentration matrix, in a sparse and possibly high-dimensional setting. Its originality lies in the fact that it searches for a latent modular structure to drive the inference procedure through adaptive penalization of the concentration matrix. Availability: Under the GNU General Public Licence at http://cran.r-project.org/web/packages/simone/ Contact: julien.chiquet@genopole.cnrs.fr

Our concern is selecting the concentration matrix's nonzero coefficients for a sparse Gaussian graphical model in a high-dimensional setting. This corresponds to estimating the graph of conditional dependencies between the variables. We describe a novel framework taking into account a latent structure on the concentration matrix. This latent structure is used to drive a penalty matrix and thus to recover a graphical model with a constrained topology. Our method uses an $\ell_1$ penalized likelihood criterion. Inference of the graph of conditional dependencies between the variates and of the hidden variables is performed simultaneously in an iterative \textsc{em}-like algorithm. The performances of our method is illustrated on synthetic as well as real data, the latter concerning breast cancer., Comment: 35 pages, 15 figures

While hidden class models of various types arise in many statistical applications, it is often difficult to establish the identifiability of their parameters. Focusing on models in which there is some structure of independence of some of the observed variables conditioned on hidden ones, we demonstrate a general approach for establishing identifiability utilizing algebraic arguments. A theorem of J. Kruskal for a simple latent-class model with finite state space lies at the core of our results, though we apply it to a diverse set of models. These include mixtures of both finite and nonparametric product distributions, hidden Markov models and random graph mixture models, and lead to a number of new results and improvements to old ones. In the parametric setting, this approach indicates that for such models, the classical definition of identifiability is typically too strong. Instead generic identifiability holds, which implies that the set of nonidentifiable parameters has measure zero, so that parameter inference is still meaningful. In particular, this sheds light on the properties of finite mixtures of Bernoulli products, which have been used for decades despite being known to have nonidentifiable parameters. In the nonparametric setting, we again obtain identifiability only when certain restrictions are placed on the distributions that are mixed, but we explicitly describe the conditions., Comment: Published in at http://dx.doi.org/10.1214/09-AOS689 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org)

In a convolution model, we observe random variables whose distribution is the convolution of some unknown density f and some known noise density g. We assume that g is polynomially smooth. We provide goodness-of-fit testing procedures for the test H0: f=f0, where the alternative H1 is expressed with respect to $\mathbb{L}_{2}$-norm (i.e. has the form $\psi_{n}^{-2}\|f-f_{0}\|_{2}^{2}\ge \mathcal{C}$). Our procedure is adaptive with respect to the unknown smoothness parameter τ of f. Different testing rates (ψn) are obtained according to whether f0 is polynomially or exponentially smooth. A price for adaptation is noted and for computing this, we provide a non-uniform Berry–Esseen type theorem for degenerate U-statistics. In the case of polynomially smooth f0, we prove that the price for adaptation is optimal. We emphasise the fact that the alternative may contain functions smoother than the null density to be tested, which is new in the context of goodness-of-fit tests.

This paper deals with order identification for Markov chains with Markov regime (MCMR) in the context of finite alphabets. We define the joint order of a MCMR process in terms of the number k of states of the hidden Markov chain and the memory m of the conditional Markov chain. We study the properties of penalized maximum likelihood estimators for the unknown order (k, m) of an observed MCMR process, relying on information theoretic arguments. The novelty of our work relies in the joint estimation of two structural parameters. Furthermore, the different models in competition are not nested. In an asymptotic framework, we prove that a penalized maximum likelihood estimator is strongly consistent without prior bounds on k and m . We complement our theoretical work with a simulation study of its behaviour. We also study numerically the behaviour of the BIC criterion. A theoretical proof of its consistency seems to us presently out of reach for MCMR, as such a result does not yet exist in the simpler case where m = 0 (that is for hidden Markov models).

We consider a semiparametric convolution model. We observe random variables having a distribution given by the convolution of some unknown density $f$ and some partially known noise density $g$. In this work, $g$ is assumed exponentially smooth with stable law having unknown self-similarity index $s$. In order to ensure identifiability of the model, we restrict our attention to polynomially smooth, Sobolev-type densities $f$, with smoothness parameter $\beta$. In this context, we first provide a consistent estimation procedure for $s$. This estimator is then plugged-into three different procedures: estimation of the unknown density $f$, of the functional $\int f^2$ and goodness-of-fit test of the hypothesis $H_0:f=f_0$, where the alternative $H_1$ is expressed with respect to $\mathbb {L}_2$-norm (i.e. has the form $\psi_n^{-2}\|f-f_0\|_2^2\ge \mathcal{C}$). These procedures are adaptive with respect to both $s$ and $\beta$ and attain the rates which are known optimal for known values of $s$ and $\beta$. As a by-product, when the noise density is known and exponentially smooth our testing procedure is optimal adaptive for testing Sobolev-type densities. The estimating procedure of $s$ is illustrated on synthetic data., Comment: Published in at http://dx.doi.org/10.1214/08-EJS225 the Electronic Journal of Statistics (http://www.i-journals.org/ejs/) by the Institute of Mathematical Statistics (http://www.imstat.org)

We consider a semiparametric convolution model where the noise has known Fourier transform which decays asymptotically as an exponential with unknown scale parameter; the deconvolution density is less smooth than the noise in the sense that the tails of the Fourier transform decay more slowly, ensuring the identifiability of the model. We construct a consistent estimation procedure for the noise level and prove that its rate is optimal in the minimax sense. Two convergence rates are distinguished according to different smoothness properties for the unknown density. If the tail of its Fourier transform does not decay faster than exponentially, the asymptotic optimal rate and exact constant are evaluated, while if it does not decay faster than polynomially, this rate is evaluated up to a constant. Moreover, we construct a consistent estimator of the unknown density, by using a plug-in method in the classical kernel estimation procedure. We establish that the rates of estimation of the deconvolution density are slower than in the case of an entirely known noise distribution. In fact, nonparametric rates of convergence are equal to the rate of estimation of the noise level, and we prove that these rates are minimax. In a few specific cases the plug-in method converges at even slower rates.

In this paper, we consider the consistency and asymptotic normality of the maximum likelihood estimator for a possibly non-stationary hidden Markov model where the hidden state space is a separable and compact space not necessarily finite, and both the transition kernel of the hidden chain and the conditional distribution of the observations depend on a parameter θ. For identifiable models, consistency and asymptotic normality of the maximum likelihood estimator are shown to follow from exponential memorylessness properties of the state prediction filter and geometric ergodicity of suitably extended Markov chains.

This thesis deals with maximum likelihood estimation in dynamic and spatial extensions of the stochastic block model (SBM), based respectively on hidden Markov chains and fields. First, we consider a dynamic version of the stochastic block model, suited for the observation of networks at multiple time steps. In this dynamic SBM, the nodes are partitioned into latent classes and the connection between two nodes is drawn from a Bernoulli distribution depending on the classes of these two nodes. The temporal evolution of the nodes memberships is modeled by a hidden Markov chain. We prove the consistency (as the numbers of nodes and time steps increase) of the maximum likelihood and variational estimators of the parameters and obtain upper bounds on their rates of convergence. We also explore the case where the number of time steps is fixed and the connectivity parameters are allowed to vary. Besides, we obtain some results regarding parameter identifiability. Second, we introduce a spatial version of the SBM, suited for the observation of networks at different spatial locations. As before, the nodes are partitioned into latent classes and the connection is drawn from a Bernoulli distribution depending on the classes of these two nodes. The spatial evolution of the nodes memberships is modeled through hidden Markov random fields. We first prove that the parameter is generically identifiable under certain conditions. For the estimation of the parameters, we propose an algorithm based on the simulated field Expectation-Maximisation (EM) algorithm, a variation of the EM algorithm relying on a mean field like approximation based on the simulation of latent configurations.; Cette thèse porte sur le maximum de vraisemblance dans des versions dynamiques et spatiales du modèle à blocs stochastiques (SBM) fondées respectivement sur des chaînes et champs de Markov cachés. D’abord, on considère une version dynamique du SBM adaptée à l’observation de réseaux à différents pas de temps. Dans ce modèle, les nœuds sont répartis dans des groupes latents et la connexion entre deux nœuds suit une loi de Bernoulli dont le paramètre dépend du groupe de ces nœuds. L’évolution temporelle des appartenances aux groupes est modélisée par une chaîne de Markov cachée. On prouve la consistance (lorsque les nombres de nœuds et pas de temps augmentent) des estimateurs du maximum de vraisemblance et variationnels des paramètres, et on obtient des bornes supérieures pour leur taux de convergence. On explore aussi le cas où le nombre de pas de temps est fixé et les probabilités de connexion varient dans le temps. On obtient également des résultats concernant l’identifiabilité des paramètres. Ensuite, on introduit une version spatiale du SBM adaptée à l’observation de réseaux à différentes localisations. Les nœuds sont répartis dans des groupes latents et la connexion entre deux nœuds suit une loi de Bernoulli dont le paramètre dépend du groupe de ces nœuds. L’évolution spatiale des appartenances aux groupes des nœuds est modélisée par des champs de Markov cachés. On montre que le paramètre est génériquement identifiable sous certaines conditions. Pour l’estimation des paramètres, on propose d’adapter à notre modèle une variante de l’algorithme Espérance-Maximisation (EM) reposant sur une approximation de type champ moyen grâce à la simulation de configurations latentes.

In this Ph.D. work, we proposed models and algorithms to study the common evolutionary history of hosts and symbionts. The first goal was to analyse the robustness of the methods of phylogenetic tree reconciliations, which are a common way of performing such study. This involves mapping one tree, most often the symbiont’s, to the other using a so-called event-based model. The events considered in general are cospeciation, duplication, host switch, and loss. The host and the symbiont phylogenies are usually considered as given and without any errors. The objective here was to understand the strengths and weaknesses of the parsimonious model used in such mappings of one tree to another, and how the final results may be influenced when small errors are present, or are introduced in the input datasets. This may correspond either to a wrong choice of present-day symbiont-host associations in the case where multiple ones exist, or to small errors related to a wrong rooting of the symbiont tree. Our results show that the choice of leaf associations and of root placement may have a strong impact on the variability of the reconciliation output. We also noticed that the host switch event has an important role in particular for the rooting problem. The second goal of this Ph.D. was to introduce some events that are little or not formally considered in the literature. One of them is the spread, which corresponds to the invasion of different hosts by a same symbiont. In this case, as when spreads are not considered, the optimal reconciliations obtained will depend on the choice made for the costs of the events. The need to develop statistical methods to assign the most appropriate ones therefore remains of actuality. Two types of spread are introduced: vertical and horizontal. The first case corresponds to what could be called also a freeze in the sense that the evolution of the symbiont “freezes” while the symbiont continues to be associated with a host and with the new species that descend from this host. The second includes both an invasion, of the symbiont which remains with the initial host but at the same time gets associated with (“invades”) another one incomparable with the first, and a freeze, actually a double freeze as the evolution of the symbiont “freezes” in relation to the evolution of the host to which it was initially associated and in relation to the evolution of the second one it “invaded”. Our results show that the introduction of these events makes the model more realistic, but also that it is now possible to directly use datasets with a symbiont that is associated with more than one host at the same time, which was not feasible before; Lors de cette thèse, je me suis intéressée aux modèles et aux algorithmes pour étudier l'histoire évolutive commune des hôtes et des symbiotes. Le premier objectif était d'analyser la robustesse des méthodes de réconciliation des arbres phylogénétiques, qui sont très utilisées dans ce type d'étude. Celles-ci associent (ou lient) un arbre, d'habitude celui des symbiotes, à l'autre, en utilisant un modèle dit basé sur des évènements. Les évènements les plus utilisés sont la cospéciation, la duplication, le saut et la perte. Les phylogénies des hôtes et des symbiotes sont généralement considérés comme donnés, et sans aucune erreur. L'objectif était de comprendre les forces et les faiblesses du modèle parcimonieux utilisé et comprendre comment les résultats finaux peuvent être influencés en présence de petites perturbations ou d'erreurs dans les données en entrée. Ici deux cas sont considérés, le premier est le choix erroné d'une association entre les feuilles des hôtes et des symbiotes dans le cas où plusieurs existent, le deuxième est lié au mauvais choix de l'enracinement de l'arbre des symbiotes. Nos résultats montrent que le choix des associations entre feuilles et le choix de l'enracinement peuvent avoir un fort impact sur la variabilité de la réconciliation obtenue. Nous avons également remarqué que l'evènement appelé “saut” joue un rôle important dans l'étude de la robustesse, surtout pour le problème de l'enracinement. Le deuxième objectif de cette thèse était d'introduire certains evènements peu ou pas formellement considérés dans la littérature. L'un d'entre eux est la “propagation”, qui correspond à l'invasion de différents hôtes par un même symbiote. Dans ce cas, lorsque les propagations ne sont pas considérés, les réconciliations optimales sont obtenues en tenant compte seulement des coûts des évènements classiques (cospeciation, duplication, saut, perte). La nécessité de développer des méthodes statistiques pour assigner les coûts les plus appropriés est toujours d'actualité. Deux types de propagations sont introduites : verticaux et horizontaux. Le premier type correspond à ce qu'on pourrait appeler aussi un gel, à savoir que l'évolution du symbiote s'arrête et “gèle” alors que le symbiote continue d'être associé à un hôte et aux nouvelles espèces qui descendent de cet hôte. Le second comprend à la fois une invasion, du symbiote qui reste associé à l'hôte initial, mais qui en même temps s'associe (“envahit”) un autre hôte incomparable avec le premier, et un gel par rapport à l'évolution des deux l'hôtes, celui auquel il était associé au début et celui qu'il a envahi. Nos résultats montrent que l'introduction de ces evènements rend le modèle plus réaliste, mais aussi que désormais il est possible d'utiliser directement des jeux de données avec un symbiote qui est associé plusieurs hôtes au même temps, ce qui n'était pas faisable auparavant

In a multiple testing context, we consider a semiparametric mixture model with two components. One component is assumed to be known and corresponds to the distribution of p-values under the null hypothesis with prior probability p. The other component f is nonparametric and stands for the distribution under the alternative hypothesis. The problem of estimating the parameters p and f of the model appears from the false discovery rate control procedures. In the first part of this dissertation, we study the estimation of the proportion p. We discuss asymptotic efficiency results and establish that two different cases occur whether f vanishes on a non-empty interval or not. In the first case, we exhibit estimators converging at parametric rate, compute the optimal asymptotic variance and conjecture that no estimator is asymptotically efficient (i.e. attains the optimal asymptotic variance). In the second case, we prove that the quadratic risk of any estimator does not converge at parametric rate. In the second part of the dissertation, we focus on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of p. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Holder densities. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate.; Dans un contexte de test multiple, nous considérons un modèle de mélange semi-paramétrique avec deux composantes. Une composante est supposée connue et correspond à la distribution des p-valeurs sous hypothèse nulle avec probabilité a priori p. L'autre composante f est nonparamétrique et représente la distribution des p-valeurs sous l'hypothèse alternative. Le problème d'estimer les paramètres p et f du modèle apparaît dans les procédures de contrôle du taux de faux positifs (``false discovery rate'' ou FDR). Dans la première partie de cette dissertation, nous étudions l'estimation de la proportion p. Nous discutons de résultats d'efficacité asymptotique et établissons que deux cas différents arrivent suivant que f s'annule ou non surtout un intervalle non-vide. Dans le premier cas (annulation surtout un intervalle), nous présentons des estimateurs qui convergent \`{a} la vitesse paramétrique, calculons la variance asymptotique optimale et conjecturons qu'aucun estimateur n'est asymptotiquement efficace (i.e atteint la variance asymptotique optimale). Dans le deuxième cas, nous prouvons que le risque quadratique de n'importe quel estimateur ne converge pas à la vitesse paramétrique. Dans la deuxième partie de la dissertation, nous nous concentrons sur l'estimation de la composante inconnue nonparamétrique f dans le mélange, en comptant sur un estimateur préliminaire de p. Nous proposons et étudions les propriétés asymptotiques de deux estimateurs différents pour cette composante inconnue. Le premier estimateur est un estimateur à noyau avec poids aléatoires. Nous établissons une borne supérieure pour son risque quadratique ponctuel, en montrant une vitesse de convergence nonparamétrique classique sur une classe de Holder. Le deuxième estimateur est un estimateur du maximum de vraisemblance régularisée. Il est calculé par un algorithme itératif, pour lequel nous établissons une propriété de décroissance d'un critère. De plus, ces estimateurs sont utilisés dans une procédure de test multiple pour estimer le taux local de faux positifs (``local false discovery rate'' ou lfdr).

In a multiple testing context, we consider a semiparametric mixture model with two components. One component is assumed to be known and corresponds to the distribution of p-values under the null hypothesis with prior probability p. The other component f is nonparametric and stands for the distribution under the alternative hypothesis. The problem of estimating the parameters p and f of the model appears from the false discovery rate control procedures. In the first part of this dissertation, we study the estimation of the proportion p. We discuss asymptotic efficiency results and establish that two different cases occur whether f vanishes on a non-empty interval or not. In the first case, we exhibit estimators converging at parametric rate, compute the optimal asymptotic variance and conjecture that no estimator is asymptotically efficient (i.e. attains the optimal asymptotic variance). In the second case, we prove that the quadratic risk of any estimator does not converge at parametric rate. In the second part of the dissertation, we focus on the estimation of the nonparametric unknown component f in the mixture, relying on a preliminary estimator of p. We propose and study the asymptotic properties of two different estimators for this unknown component. The first estimator is a randomly weighted kernel estimator. We establish an upper bound for its pointwise quadratic risk, exhibiting the classical nonparametric rate of convergence over a class of Holder densities. The second estimator is a maximum smoothed likelihood estimator. It is computed through an iterative algorithm, for which we establish a descent property. In addition, these estimators are used in a multiple testing procedure in order to estimate the local false discovery rate., Dans un contexte de test multiple, nous considérons un modèle de mélange semi-paramétrique avec deux composantes. Une composante est supposée connue et correspond à la distribution des p-valeurs sous hypothèse nulle avec probabilité a priori p. L'autre composante f est nonparamétrique et représente la distribution des p-valeurs sous l'hypothèse alternative. Le problème d'estimer les paramètres p et f du modèle apparaît dans les procédures de contrôle du taux de faux positifs (``false discovery rate'' ou FDR). Dans la première partie de cette dissertation, nous étudions l'estimation de la proportion p. Nous discutons de résultats d'efficacité asymptotique et établissons que deux cas différents arrivent suivant que f s'annule ou non surtout un intervalle non-vide. Dans le premier cas (annulation surtout un intervalle), nous présentons des estimateurs qui convergent \`{a} la vitesse paramétrique, calculons la variance asymptotique optimale et conjecturons qu'aucun estimateur n'est asymptotiquement efficace (i.e atteint la variance asymptotique optimale). Dans le deuxième cas, nous prouvons que le risque quadratique de n'importe quel estimateur ne converge pas à la vitesse paramétrique. Dans la deuxième partie de la dissertation, nous nous concentrons sur l'estimation de la composante inconnue nonparamétrique f dans le mélange, en comptant sur un estimateur préliminaire de p. Nous proposons et étudions les propriétés asymptotiques de deux estimateurs différents pour cette composante inconnue. Le premier estimateur est un estimateur à noyau avec poids aléatoires. Nous établissons une borne supérieure pour son risque quadratique ponctuel, en montrant une vitesse de convergence nonparamétrique classique sur une classe de Holder. Le deuxième estimateur est un estimateur du maximum de vraisemblance régularisée. Il est calculé par un algorithme itératif, pour lequel nous établissons une propriété de décroissance d'un critère. De plus, ces estimateurs sont utilisés dans une procédure de test multiple pour estimer le taux local de faux positifs (``local false discovery rate'' ou lfdr).