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Random graph models are used to describe the complex structure of real-world networks in diverse fields of knowledge. Studying their behaviour and fitting properties are still critical challenges that, in general, require model-specific techniques. An important line of research is to develop generic methods able to fit and select the best model among a collection. Approaches based on spectral density (i.e. distribution of the graph adjacency matrix eigenvalues) appeal to that purpose: they apply to different random graph models. Also, they can benefit from the theoretical background of random matrix theory. This work investigates the convergence properties of model fitting procedures based on the graph spectral density and the corresponding cumulative distribution function. We also review the convergence of the spectral density for the most widely used random graph models. Moreover, we explore through simulations the limits of these graph spectral density convergence results, particularly in the case of the block model, where only partial results have been established. random graphs, spectral density, model fitting, model selection, convergence.

To model recurrent interaction events in continuous time, an extension of the stochastic block model is proposed where every individual belongs to a latent group and interactions between two individuals follow a conditional inhomogeneous Poisson process with intensity driven by the individuals' latent groups. The model is shown to be identifiable and its estimation is based on a semiparametric variational expectation-maximization algorithm. Two versions of the method are developed, using either a nonparametric histogram approach (with an adaptive choice of the partition size) or kernel intensity estimators. The number of latent groups can be selected by an integrated classification likelihood criterion. Finally, we demonstrate the performance of our procedure on synthetic experiments, analyse two datasets to illustrate the utility of our approach and comment on competing methods.

We present a selective review on probabilistic modeling of heterogeneity in random graphs. We focus on latent space models and more particularly on stochastic block models and their extensions that have undergone major developments in the last five years., Nous présentons une revue non exhaustive de la modélisation probabiliste de l'hétérogénéité des graphes aléatoires. Nous décrivons les modèles à espaces latents en nous intéressant plus particulièrement au modèle à blocs stochastiques et ses extensions, qui a connu des développements majeurs au cours des cinq dernières années.

We consider the problem of estimating the proportion $\theta$ of true null hypotheses in a multiple testing context. The setup is classically modeled through a semiparametric mixture with two components: a uniform distribution on interval $[0,1]$ with prior probability $\theta$ and a nonparametric density $f$. We discuss asymptotic efficiency results and establish that two different cases occur whether $f$ vanishes on a set with non null Lebesgue measure 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 (\emph{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. We illustrate those results on simulated data.

This paper deals with parameter estimation in pair hidden Markov models (pair-HMMs). We first provide a rigorous formalism for these models and discuss possible definitions of likelihoods. The model being biologically motivated, some restrictions with respect to the full parameter space naturally occur. Existence of two different Information divergence rates is established and divergence property (namely positivity at values different from the true one) is shown under additional assumptions. This yields consistency for the parameter in parametrization schemes for which the divergence property holds. Simulations illustrate different cases which are not covered by our results., Comment: corrected typos

This paper deals with semiparametric convolution models, where the noise sequence has a Gaussian centered distribution, with unknown variance. Non-parametric convolution models are concerned with the case of an entirely known distribution for the noise sequence, and they have been widely studied in the past decade. The main property of those models is the following one: the more regular the distribution of the noise is, the worst the rate of convergence for the estimation of the signal's density g is [5]. Nevertheless, regularity assumptions on the signal density g improve those rates of convergence [15]. In this paper, we show that when the noise (assumed to be Gaussian centered) has a variance σ2 that is unknown (actually, it is always the case in practical applications), the rates of convergence for the estimation of g are seriously deteriorated, whatever its regularity is supposed to be. More precisely, the minimax risk for the pointwise estimation of g over a class of regular densities is lower bounded by a constant over log n. We construct two estimators of σ2, and more particularly, an estimator which is consistent as soon as the signal has a finite first order moment. We also mention as a consequence the deterioration of the rate of convergence in the estimation of the parameters in the nonlinear errors-in-variables model.

Resume Nous etablissons la consistance et la normalite asymptotique de l'estimateur du maximum de vraisemblance dans un modele de Markov cache non stationnaire, lorsque l'espace d'etats de la chaine est separable et compact. Nous prouvons ces convergences sous de faibles hypotheses de regularite et d'identifiabilite du modele, en utilisant l'oubli exponentiel du filtre de prediction et l'ergodicite geometrique d'une chaine etendue bien choisie.

We consider a one dimensional ballistic random walk evolving in a parametric independent and identically distributed random environment. We study the asymptotic properties of the maximum likelihood estimator of the parameter based on a single observation of the path till the time it reaches a distant site. We prove an asymptotic normality result for this consistent estimator as the distant site tends to infinity and establish that it achieves the Cram\'er-Rao bound. We also explore in a simulation setting the numerical behaviour of asymptotic confidence regions for the parameter value.