In the fist part of the talk, we will look to some statistical inverse problems for which the natural framework is no more an Euclidian one.
In the second part we will try to give the initial construction of (not orthogonal) wavelets -of the 80 - by Frazier, Jawerth,Weiss, before the Yves Meyer ORTHOGONAL wavelets theory.
In the third part we will propose a construction of a geometric wavelet theory. In the Euclidian case, Fourier transform plays a fundamental role. In the geometric situation this role is given to some "Laplacian operator" with some properties.
In the last part we will show that the previous theory could help to revisit the topic of regularity of Gaussian processes, and to give a criterium only based on the regularity of the covariance operator.[-]

In this talk we introduce a class of statistics for spatial data that is observed on an irregular set of locations. Our aim is to obtain a unified framework for inference and the statistics we consider include both parametric and nonparametric estimators of the spatial covariance function, Whittle likelihood estimation, goodness of fit tests and a test for second order spatial stationarity. To ensure that the statistics are computationally feasible they are defined within the Fourier domain, and in most cases can be expressed as a quadratic form of a discrete Fourier-type transform of the spatial data. Evaluation of such statistic is computationally tractable, requiring $O(nb)$ operations, where $b$ are the number Fourier frequencies used in the definition of the statistic (which varies according to the application) and $n$ is the sample size. The asymptotic sampling properties of the statistics are derived using mixed spatial asymptotics, where the number of locations grows at a faster rate than the size of the spatial domain and under the assumption that the spatial random field is stationary and the irregular design of the locations are independent, identically distributed random variables. We show that there are quite intriguing differences in the behaviour of the statistic when the spatial process is Gaussian and non-Gaussian. In particular, the choice of the number of frequencies $b$ in the construction of the statistic depends on whether the spatial process is Gaussian or not. If time permits we describe how the results can also be used in variance estimation. And if we still have time some simulations and real data will be presented.[-]

This work presents the impact of a class of transformations of copulas in their upper and lower multivariate tail dependence coefficients. In particular we focus on multivariate Archimedean copulas. In the first part, we calculate multivariate transformed tail dependence coefficients when the generator of the considered transformed copula exhibits some regular variation properties, and we investigate the behavior of these coefficients in cases that are close to tail independence. We obtain new results under specific conditions involving regularly varying hazard rates of components of the transformation. These results are also valid for non-transformed Archimedean copulas. In the second part we deal with a class of particular hyperbolic transformations. We show the utility of using transformed Archimedean copulas, as they permit to build Archimedean generators exhibiting any chosen couple of lower and upper tail dependence coefficients.[-]

Les processus de Hawkes forment une classe des processus ponctuels pour lesquels l'intensité s'écrit comme :

$\lambda(t)= \int_{0}^{t^-} h(t-s)dN_s +\nu$

où $N$ représente le processus de Hawkes, et $\nu > 0$. Les processus de Hawkes multivariés ont une intensité similaire sauf que des interractions entre les différentes composantes du processus de Hawkes sont autorisées. Les paramètres de ce modèle sont donc les fonctions d'interractions $h_{k,\ell}, k, \ell \le M$ et les constantes $\nu_\ell, \ell \le M$. Dans ce travail nous étudions une approche bayésienne nonparamétrique pour estimer les fonctions $h_{k,\ell}$ et les constantes $\nu_\ell$. Nous présentons un théorème général caractérisant la vitesse de concentration de la loi a posteriori dans de tels modèles. L'intérêt de cette approche est qu'elle permet la caractérisation de la convergence en norme $L_1$ et demande assez peu d'hypothèses sur la forme de la loi a priori. Une caractérisation de la convergence en norme $L_2$ est aussi considérée. Nous étudierons un exemple de lois a priori adaptées à l'étude des interractions neuronales. Travail en collaboration avec S. Donnet et V. Rivoirard.[-]

In many health studies, interest often lies in assessing health effects on a large set of outcomes or specific outcome subtypes, which may be sparsely observed, even in big data settings. For example, while the overall prevalence of birth defects is not low, the vast heterogeneity in types of congenital malformations leads to challenges in estimation for sparse groups. However, lumping small groups together to facilitate estimation is often controversial and may have limited scientific support.
There is a very rich literature proposing Bayesian approaches for clustering starting with a prior probability distribution on partitions. Most approaches assume exchangeability, leading to simple representations in terms of Exchangeable Partition Probability Functions (EPPF). Gibbs-type priors encompass a broad class of such cases, including Dirichlet and Pitman-Yor processes. Even though there have been some proposals to relax the exchangeability assumption, allowing covariate-dependence and partial exchangeability, limited consideration has been given on how to include concrete prior knowledge on the partition. We wish to cluster birth defects into groups to facilitate estimation, and we have prior knowledge of an initial clustering provided by experts. As a general approach for including such prior knowledge, we propose a Centered Partition (CP) process that modifies the EPPF to favor partitions close to an initial one. Some properties of the CP prior are described, a general algorithm for posterior computation is developed, and we illustrate the methodology through simulation examples and an application to the motivating epidemiology study of birth defects.[-]

We discuss the problem of positive-semidefinite extension: extending a partially specified covariance kernel from a subdomain Ω of a rectangular domain I x I to a covariance kernel on the entire domain I x I. For a broad class of domains Ω called serrated domains, we present a complete theory. Namely, we demonstrate that a canonical completion always exists and can be explicitly constructed. We characterise all possible completions as suitable perturbations of the canonical completion, and determine necessary and sufficient conditions for a unique completion to exist. We interpret the canonical completion via the graphical model structure it induces on the associated Gaussian process. Furthermore, we show how the determination of the canonical completion reduces to the solution of a system of linear inverse problems in the space of Hilbert-Schmidt operators, and derive rates of convergence when the kernel is to be empirically estimated. We conclude by providing extensions of our theory to more general forms of domains, and by demonstrating how our results can be used in statistical inverse problems associated with stochastic processes.[-]

In this talk, we consider two-component mixture models having one single known component. This type of model is of particular interest when a known random phenomenon is contaminated by an unknown random effect.
We propose in this setup to test the equality in distribution of the unknown random sources involved in two separate samples generated from such a model. For this purpose, we introduce the so-called IBM (Inversion-Best Matching) approach resulting in a tuning-free relaxed semiparametric Cramér-von Mises type two-sample test requiring minimal assumptions about the unknown distributions. The accomplishment of our work lies in the fact that we establish, under some natural and interpretable mutual-identifiability conditions specific to the two-sample case, a functional central limit theorem about the proportion parameters along with the unknown cumulative distribution functions of the model. An intensive numerical study is carried out from a large range of simulation setups to illustrate the asymptotic properties of our test. Finally, our testing procedure, implemented in the admix R package, is applied to a real-life situation through pairwise post COVID-19 mortality excess profil testing across a panel of European countries.[-]