## Planning des séminaires 2020-2021

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Le séminaire a lieu le lundi, à 11h, en salle 316 du bâtiment de Métrologie B. Vous trouverez ci-dessous le planning du séminaire de Probabilités-Statistique pour l’année universitaire en cours.

Contacts : yacouba.boubacar_mainassara@univ-fcomte.fr ou romain.biard@univ-fcomte.fr.

### Exposés à venir :

1 mars : Landy Rabehasaina
(LmB, Univ. Bougogne Franche-Comté)

Multitype branching process with nonhomogeneous Poisson and contagious Poisson immigration

Abstract :
In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a contagious Poisson process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient expectation results when there are only two types of particles are provided.

22 février : Quentin Klopfenstein
(IMB, Univ. Bougogne Franche-Comté)

Implicit differentiation for fast hyperparameter selection in non-smooth convex learning

Abstract :
Most modern machine learning models require one hyperparameter to be chosen by the user upstream of the learning phase. Popular approaches use a grid of values on which to evaluate the performance of the model for a given criterion, one can think of grid-search or random-search which means fitting the given model for each value of the grid.
These methods have a major drawback : they scale exponentially with the number of hyperparameters. In this presentation, we will show that the hyperparameter selection problem can be cast as a bilevel optimization problem and will consider non-smooth models (such as the Lasso, the Elastic Net, the SVM).
We propose a first-order method that uses information about the gradient with respect to the hyperparameter to automatically select the best hyperparameter for a given criterion.
We will see that this method is very efficient even when the number of hyperparameters gets large.

### Exposés passés :

11 janvier : Davit Varron
(LmB, Univ. Bougogne Franche-Comté)

Mesure empirique et valeurs extrêmes : une représentation par des lois conditionnelles

Abstract :
Dans de nombreuses méthodes statistiques utilisées en valeurs extrêmes, on construit un estimateur à partir d’un sous échantillon de l’échantillon initial. Ce sous échantillon sélectionnes les $k$ observations $(X_i_X_i)$ pour lesquelles $Y_i$ dépasse sa $n-k$ ième statistique d’ordre. Nous montrons que ces méthodes peuvent être vues comme des images de mesures aléatoires qui se comportent comme des mesure empiriques si on les conditionne correctement. Travail en collaboration avec Dr Benjamin Bobbia et Pr Clément dombry.

14 décembre : Aboubacar Y. Touré
(LmB, Univ. Bougogne Franche-Comté)

On general exponential weight functions and variation phenomenon

Abstract :
General weighted exponential distributions including modified exponential ones are widely used with great ability in statistical applications, particularly in reliability. In this work, we investigate full exponential weight functions and their extensions from any nonnegative continuous reference weighted distribution. Several properties and their
connections with the recent variation phenomenon are then established. In particular, characterizations, weightening operations and dual distributions are set forward.
Illustrative examples and concluding remarks are extensively discussed.

7 décembre : Benjamin Bobbia
(LmB, Univ. Bougogne Franche-Comté)

Titre : Extreme quantile regression : a coupling approach and Wasserstein distance

Abstract :
Résumé : In this work, we develop two coupling approaches for extreme quantile regression. We consider i.i.d copies of $Y \in \mathbb{R}$ and $X \in \mathbb{R}^d$ and we want an estimation of the conditional quantile of $Y$ given $X=x$ of order $1-\alpha$ for a very small $\alpha >0$.

We introduce the proportional tail model, strongly inspired by the heteroscedastic extremes developped by Einmahl, de Haan and Zhou, where $Y$ has an heavy tail $\bar{F}$ with extreme value index $\gamma >0$ and the conditional tails $\bar{F}_x$ are asymptotically equivalent to $\sigma(x)\bar{F}$. We propose and study estimators of both model parameters and conditional quantile with are studied by coupling methods.

The first method is based on coupling of empirical processes while the second is related with optimal transport.
Even if we establish the asymptotic normality of parameters estimators with both methods, the first is focused on the proper quantile estimation whereas the second is more focused on the estimation of $\gamma$ in presence of bias and the elaboration of a validation procedure for our model.

Moreover, we also develop the optimal coupling approach in the general case of univariate extreme value theory.

30 novembre : Mehdi Dagdoug
(LmB, Univ. Bougogne Franche-Comté)

Titre : Model-assisted estimation through random forests in finite population sampling

Abstract :
Résumé : Estimation of finite population totals is of primary interest in survey sampling. Often, additional auxiliary information is available at the population level. The model-assisted approach uses this supplementary source of information to construct improved estimators of finite population totals by assuming a model between the survey variable and the potential predictors. In this work, new classes of model-assisted estimators based on random forests are suggested.
Under mild conditions, the proposed estimators are shown to be asymptotically design unbiased and consistent. Their asymptotic variance is derived, and a consistent variance estimator is suggested. The asymptotic distribution of the estimators is obtained, allowing for the use of normal-based confidence intervals. The high-dimensional behavior of the random forest estimator is also investigated and compared to commonly used estimators. Simulations illustrate that the proposed model is particularly efficient and can outperform state-of-the-art estimators, especially in complex settings such as small sample sizes, high-dimensional regressor space or complex superpopulation models. This is a joint work with Camelia Goga and David Haziza.

9 novembre : Clément Dombry
(LmB, Univ. Bougogne Franche-Comté)

Titre : Behavior of linear L2-boosting algorithms in the vanishing learning rate asymptotic

Abstract :
Résumé : We investigate the asymptotic behaviour of gradient boosting algorithms when the learning rate converges to zero and the number of iterations is rescaled accordingly. We mostly consider L2-boosting for regression with linear base learner as studied in Bühlmann and Yu (2003) and analyze also a stochastic version of the model where subsampling is used at each step (Friedman, 2002). We prove a deterministic
limit in the vanishing learning rate asymptotic and characterize the limit as the unique solution of a linear differential equation in an infinite dimensional function space. Besides, the training and test error of the limiting procedure are thoroughly analyzed.
Joint work with Youssef Esstafa.

21 septembre : Jean-Jil Duchamps
(LmB, Univ. Bougogne Franche-Comté)

La forêt de Moran

Abstract :
On considère la forêt aléatoire obtenue sous la distribution stationnaire de la chaîne de Markov suivante sur les graphes sur 1, ..., n : à chaque étape, un sommet choisi uniformément est déconnecté de ses voisins et reconnecté à un autre sommet choisi uniformément. Cette forêt aléatoire correspond aux liens de parenté direct entre individus dans une population évoluant selon un modèle classique (modèle de
Moran). Elle admet une construction très simple que j’expliciterai, qui permet de révéler les liens intéressants qu’elle présente avec l’arbre uniforme sur 1, ... , n, ainsi qu’avec les "uniform attachment trees". Je donnerai aussi certaines de ses caractéristiques : loi des degrés, d’un arbre uniforme, taille du plus grand degré/arbre (travail en collaboration avec F. Bienvenu et F. Foutel-Rodier).

## Agenda

• ### Lundi 1er mars 11:00-12:00 - Landy Rabehasaina - Univ. Bourgogne Franche-Comté

Séminaire PS : Multitype branching process with nonhomogeneous Poisson and contagious Poisson immigration

Résumé : In a multitype branching process, it is assumed that immigrants arrive according to a nonhomogeneous Poisson or a contagious Poisson process (both processes are formulated as a nonhomogeneous birth process with an appropriate choice of transition intensities). We show that the normalized numbers of objects of the various types alive at time $t$ for supercritical, critical, and subcritical cases jointly converge in distribution under those two different arrival processes. Furthermore, some transient expectation results when there are only two types of particles are provided.