Speakers:

• Fabienne Comte, Université Paris Descartes 

• Jan Johannes, Ruprecht-Karls-Universität Heidelberg

• Anatoli Juditsky, Université Grenoble Alpes

• Nicolas Klutchnikoff, Université Rennes 2

• Claire Lacour, Université Gustave Eiffel 

• Oleg Lepski, Université Aix-Marseille 

• Timothée Mathieu, Inria

• Elias Ould Said, Université du littoral côte d'opale

• Vincent Rivoirard, Université Paris Dauphine

• Angelina Roche, Université Paris Dauphine

• Flore Sentenac, Ensae Paris

Program: 

Fabienne Comte: Should we estimate a product of density functions by a product of estimators?

Abstract: In this talk, we consider the inverse problem of estimating the product  of two densities, given a d-dimensional n-sample of i.i.d. observations drawn from each distribution. We propose a general method of estimation encompassing both projection estimators with model selection device and kernel estimators with bandwidth selection strategies. The procedures do not consist in making the product of two density estimators, but in plugging an overfitted estimator of one of the two densities, in an estimator based on the second sample. Our findings are a first step toward a better understanding of the good performances of overfitting in regression Nadaraya-Watson estimator.

Jan Johannes: TBA

Anatoli Juditsky: On robust counterpart of linear inverse problems

Abstract: We consider an uncertain linear inverse problem as follows. Given observation omega=Ax+xi where A\in R^{m\times n} and \xi\in R^m is observation noise, we want to recover unknown signal x, known to belong to a convex set X \in R^n. As opposed to the "usual" setting of such problem, we suppose that sensing matrix A, feasible set X, or noise \xi may be uncertain. For instance, observation matrix may satisfy A=A_0+\delta A where the nominal matrix A_0 is known and \delta is unknown perturbation which may be random or belong to a given set A, or observation noise may contain a deterministic or singular component, etc. In a series of problem settings, under various assumptions on the nature of problem uncertainty, we discuss the properties of two types of parameter estimates - linear estimates and polyhedral estimates (A particular class of nonlinear estimates as introduced in Juditsky, A.,  Nemirovski, A. (2020). On polyhedral estimation of signals via indirect observations. Electronic Journal of Statistics, 14(1), 458-502.) We show that in the situation where the signal set is an uncertain ellitope (essentially, a symmetric convex set delimited by quadratic surfaces), nearly minimax optimal (up to a moderate suboptimality factor) estimates can be constructed by means of efficient convex optimization routine.

Joint work with Y. Bekry,  and A. Nemirovski.

Nicolas Klutchnikoff: Adaptive estimation of the regression function with Brownian path covariates

Abstract: In this paper, we are interested in estimating a regression function in the presence of functional covariates. More precisely, we wish to estimate the conditional expectation of a real response variable Y with respect to a standard Wiener coprocess W. Using the Wiener-Itô chaotic decomposition of E(Y|W), we construct natural estimators and obtain minimax rates of convergence over specific regularity classes. We also define a selection procedure for some hyper-parameters, based on the Goldenshluger-Lepski method and obtain an oracle-like inequality and adaptive results.

Claire Lacour: Semiparametric inference for mixtures of circular data

Abstract: We consider a sample of data on the circle S^1, whose distribution is a two-components mixture. The density of the sample is assumed to be g(x)=p f(x-a)+(1-p) f(x-b) where p is the mixing parameter, f a density on the circle, and a and b two angles. The objective is to estimate both the parametric part (p,a,b) and the non-parametric part f. We shall study the specific identifiability problems on the circle, which do not appear for real data. Next we shall present our adaptive estimation procedure, its theoretical performances and some numerical simulations.

Joint work with with T-M Pham Ngoc

Oleg Lepski: Minimax estimation of nonlinear functionals

Abstract: We  deal with the problem of nonparametric estimating the  Lp -norm,  p in (1, infty) , of a probability density on  R^d, d  >= 1,  from independent observations. The unknown density is assumed to belong to a ball in the anisotropic Nikolskii's space. We adopt the minimax approach and demonstrate in particular that accuracy of estimation procedures essentially depends on whether  p is integer or not. Moreover, we develop a general technique for derivation of lower bounds on the minimax risk in the problems of estimating nonlinear functionals. The proposed technique is applicable for a broad class of nonlinear functionals, and it is used for derivation of the lower bounds in the  L_p -norm estimation

Timothée Mathieu:  Robust Multivariate Mean estimation with M-estimators

Abstract: Mean estimation is a fundamental problem in statistics, as it is a tool on which a lot of the statistical procedures are based. In the well-controlled case of Gaussian random variables (or sub-gaussian random variables), it is known that the empirical mean perform fairly well. On the other hand, as soon as the distribution becomes either heavy-tailed or corrupted, things get complicated. This can be a major difficulty because in practice a lot of datasets contains outliers (typically in life sciences there are outliers in most datasets). Estimating the mean optimally for corrupted datasets is still unsolved, and most estimators are either theoretically optimal or computationally efficient but (for now) never both at the same time. In this presentation, I will present how to partially solve the problem with M-estimators that are computable and optimal for Heavy-tail distributions and I will explain the challenges to which we are confronted to design optimal and computable estimators.

Elias Ould Said: Strong uniform consistency of the local linear error regression  estimator under left truncation 

Abstract:  This paper is concerned with a nonparametric estimator of the regression function based on the local linear method when the loss function is the mean squared relative error and the data left truncated. The proposed method avoids the problem of boundary effects and is robust against the presence of outliers. Under suitable assumptions, we establish the uniform almost sure strong consistency with a rate over a compact set. A simulation study is conducted to comfort our theoretical result. This is made according to different cases, sample sizes, rates of truncation, in presence of outliers and a comparison study is made with respect to classical, local linear and relative error estimators. Finally, an experimental prediction is discussed.

Vincent Rivoirard: Bayesian nonparametric inference for nonlinear Hawkes processes

Abstract: Hawkes processes are a specific class of point processes modeling the probability of occurrences of an event depending on past occurrences. Hawkes processes are therefore naturally used when one is interested in graphs for which the temporal dimension is essential. In the linear framework, the statistical inference of Hawkes processes is now well known. We will therefore focus more specifically on the class of nonlinear multivariate Hawkes processes that allow to model both excitation and inhibition phenomena between nodes of a graph. We will present the Bayesian nonparametric estimation of the parameters of the Hawkes model and the posterior contraction rates obtained on Hölder classes. From the practical point of view, since simulating posterior distributions is often out of reach in reasonable time, especially in the mutlivariate framework, we will more specifically use the variational Bayesian approach which provides a direct and fast computation of an approximation of the posterior distributions allowing the analysis in reasonable time of graphs containing several tens of neurons.

Joint work with Déborah Sulem and Judith Rousseau.

Angelina Roche: Minimax rates in regression models for functional data

Abstract: In recent decades, significant research efforts have focused on regression models that involve functional data, which are data that can be modeled as samples of random functions. The minimax rates for the functional linear model and the fully nonparametric model are now well understood, although some aspects of these models still requires further exploration. However, for other models, like the single index model or the models with sparsity, the minimax rates are still unknown. The objective of this presentation is to provide a brief overview of the current state of knowledge regarding these models, as well as ongoning research on them.

Flore Sentenac: Robust Estimation of Discrete Distributions under Local Differential Privacy

Abstract:  Although robust learning and local differential privacy are both widely studied fields of research, combining the two settings is just starting to be explored. We consider the problem of estimating a discrete distribution in total variation from $n$ contaminated data batches under a local differential privacy constraint.  A fraction 1-alpha of the batches contain k i.i.d. samples drawn from a discrete distribution p over d elements. To protect the users' privacy, each of the samples is privatized using an epsilon-locally differentially private mechanism. The remaining alpha n batches are an adversarial contamination. The minimax rate of estimation under contamination alone, with no privacy, is known to be alpha/sqrt{k}+sqrt{d/kn}. Under the privacy constraint alone, the minimax rate of estimation is sqrt{d^2/epsilon^2 kn}. We show, up to a sqrt{log(1/alpha)} factor, that combining the two constraints leads to a minimax estimation rate of alpha sqrt{d/epsilon^2 k} sqrt{d^2/\epsilon^2 kn}$, larger than the sum of the two separate rates. We provide a polynomial-time algorithm achieving this bound, as well as a matching information theoretic lower bound.