Séance du 4 juin 2018

Séance organisée par Cécile Durot et Etienne Roquain. Exceptionnellement 4 exposés. La séance débutera donc à 13h00

Lieu : IHP, Amphi Hermite

13.00 : Vladimir Koltchinskii (Georgia Tech)

Titre : Efficient Estimation of Smooth Functionals in Gaussian Shift Models

Résumé : We will discuss a problem of efficient estimation of a smooth functional $f(\theta)$ of unknown

parameter $\theta$ of a general Gaussian shift model. In this model, $\theta$ is a vector in a separable Banach space $E$ observed in an additive Gaussian noise $\xi$ with mean zero and arbitrary (but known) covariance operator $\Sigma.$ It is assumed that the noise level, characterized by the operator norm $\|\Sigma\|$ of $\Sigma$ is small and, moreover, ${\mathbb E}\|\xi\|^2$ is also small, but the effective rank $$r(\Sigma):=\frac{{\mathbb E}\|\xi\|^2}{\|\Sigma\|}$$

of covariance operator $\Sigma$ could be large. This framework includes, in particular,

(a) usual Gaussian shift model, in which $E={\mathbb R}^d$ is a ``high-dimensional" Euclidean space and $\xi=\varepsilon Z,$ $Z$ being a standard normal random vector;

(b) matrix Gaussian shift model, in which $E$ is the space of symmetric $d\time d$ matrices equipped with the operator norm and $\xi=\varepsilon Z,$ $Z$ being sampled, for instance, from Gaussian orthogonal ensemble;

(c) Gaussian functional data model, in which $E=C[0,1]$ with usual uniform norm and $\xi$ is a mean zero continuos Gaussian stochastic process.The goal is to find optimal conditions on the smoothness of functional $f(\theta)$ (depending on $\|\Sigma\|$ and $r(\Sigma)$) for which there exists its estimator with ``approximately normal” distribution and with an optimal variance (comparable to the noise level).

The talk is based on a joint work with Mayya Zhilova.

14.00 : Gilles Blanchard (Universität Potsdam)

Titre : Is adaptive early stopping possible in statistical inverse problems?

Résumé : Consider a statistical estimation problem where different estimators $\hat{f}_1, \ldots , \hat{f}_K$ are available (ranked in order of increasing "complexity", here identified with variance) and one would like to select one that is the best (or close to the best) for the statistical task at hand: this is the classical problem of estimator selection. For a (data-dependent) choice of the estimator index $\hat{k}$, there exist a number of well-known methods achieving oracle adaptivity in a wide range of contexts, for instance penalization methods, or Lepski's method.

However, they have in common that the estimators for {\em all} possible values of $k$ have to be computed first, and only then compared to each other in some way to determine the final choice. Contrast this to an "early stopping" approach where we are able to compute iteratively the estimators for $k= 1, 2, \ldots $ and have to decide to stop at some point without being allowed to compute the following estimators. Is oracle adaptivity possible then? This question is motivated by settings where computing estimators for larger $k$ requires more computational cost; furthermore, some form of early stopping is most often used in practice.

We propose a precise mathematical formulation of this question -- in the idealized framework of a Gaussian sequence model with $D$ observed noisy coefficients. In this model, we provide upper and lower bounds on what is achievable using linear regularized filter estimators commonly used for statistical inverse problems.

Joint work with M. Hoffmann and M. Reiss

15.00 : Viet Chi Tran (Université de Lille 1)

Titre : Renouvellement pour des processus de Hawkes avec auto-excitation et inhibition

Résumé : We consider Hawkes processes on the positive real line exhibiting both self-excitation and inhibition. Each point of the Hawkes process impacts the intensity of the random point process by the addition of a signed reproduction function. The case of a non-negative reproduction function corresponds to self-excitation; it has been largely investigated in the literature and is well understood. In particular, there then exists a cluster representation of the self-excited Hawkes processes which allows to apply results known for continuous-time age-structured Galton-Watson trees to these random point processes. In the case we study, the cluster representation is no longer valid, and we use renewal techniques. We establish limit results for Hawkes process with signed reproduction functions, notably generalizing exponential concentration inequalities proved byReynaud-Bouret and Roy (2007) for non-negative reproduction functions. An

important step is to establish the existence of exponential moments for the distribution of renewal times of M/G/1 queues that appear naturally in our problem.

Joint work with M. Costa, C. Graham and L. Marsalle.

16.00 : Eni Musta (Delft University of Technology)

Titre : Central limit theorems for global errors of smooth isotonic estimators.

Résumé : A typical problem in nonparametric statistics is estimation of an unknown function on the real line, which may, for instance, be a probability density, a regression function or a failure rate. In many cases, it is natural to assume that this function is monotone and incorporating such a prior knowledge in the estimation procedure leads to more accurate results. We consider smooth isotonic estimators, constructed by combining an isotonization step with a smoothing step. We investigate the global behavior of these estimators and obtain central limit theorems for their L_p losses. We also perform a simulation study for testing monotonicity on the basis of the L_2 distance between the kernel estimator and a smooth isotonic estimator.

Joint work with Hendrik P. Lopuhaä.