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.[-]

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.[-]

Many modern applications seek to understand the relationship between an outcome variable of interest and a high-dimensional set of covariates. Often the first question asked is which covariates are important in this relationship, but the immediate next question, which in fact subsumes the first, is \emph{how} important each covariate is in this relationship. In parametric regression this question is answered through confidence intervals on the parameters. But without making substantial assumptions about the relationship between the outcome and the covariates, it is unclear even how to \emph{measure} variable importance, and for most sensible choices even less clear how to provide inference for it under reasonable conditions. In this paper we propose \emph{floodgate}, a novel method to provide asymptotic inference for a scalar measure of variable importance which we argue has universal appeal, while assuming nothing but moment bounds about the relationship between the outcome and the covariates. We take a model-X approach and thus assume the covariate distribution is known, but extend floodgate to the setting that only a \emph{model} for the covariate distribution is known and also quantify its robustness to violations of the modeling assumptions. We demonstrate floodgate's performance through extensive simulations and apply it to data from the UK Biobank to quantify the effects of genetic mutations on traits of interest.[-]

This paper addresses the following question: “Can regression trees do what other machine learning methods cannot?” To answer this question, we consider the problem of estimating regression functions with spatial inhomogeneities. Many real life applications involve functions that exhibit a variety of shapes including jump discontinuities or high-frequency oscillations. Unfortunately, the overwhelming majority of existing asymptotic minimaxity theory (for density or regression function estimation) is predicated on homogeneous smoothness assumptions which are inadequate for such data. Focusing on locally Holder functions, we provide locally adaptive posterior concentration rate results under the supremum loss. These results certify that trees can adapt to local smoothness by uniformly achieving the point-wise (near) minimax rate. Such results were previously unavailable for regression trees (forests). Going further, we construct locally adaptive credible bands whose width depends on local smoothness and which achieve uniform coverage under local self-similarity. Unlike many other machine learning methods, Bayesian regression trees thus provide valid uncertainty quantification. To highlight the benefits of trees, we show that Gaussian processes cannot adapt to local smoothness by showing lower bound results under a global estimation loss. Bayesian regression trees are thus uniquely suited for estimation and uncertainty quantification of spatially inhomogeneous functions.[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

Estimation of conditional quantiles is requiered for many purposes, in particular when the conditional mean is not suffisiant to describe the impact of covariates on the dependent variable. For example, one may estimate the quantile of one financial index (e.g. WisdomTree Japan Hedged Equity Fund) knowing financial indeces from other countries. It is also requiered to estimated conditional quantiles in Quantile Oriented Sensitivity Analysis (QOSA). QOSA indices are relevant in order to quantify uncertainty on quantiles, for example in insurance operational risk contexts. We shall present several view points on conditional quantile estimation: quantile regression and improvements, Kernel based estimation, random forest estimation. We shall focus on applications to QOSA.[-]

In extreme value statistics, the tail index is used to measure the occurrence and the intensity of extreme events. In many applied fields, the tail behavior of such events depends on explanatory variables. This article proposes an ensemble learning method for tail index regression which is called Hill random forests and combines Hill's approach on tail index estimation (Hill (1975)) with the aggregation of randomized decision trees based on the gamma deviance. We prove a consistency result when the tail index function is a multiplicative function.[-]

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.[-]

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.[-]