Abstracts
Wednesday 27 October 2021
1:15 - 2:00
Statistical Analysis of a Changing Dependence Structure of Extremes
Holger Drees
Abstract: Suppose a time series of multivariate random variables is observed. In some applications (like finance and climate) it is questionable whether the dependence structure between the extremes of the coordinates ought to be assumed stationary. Given independent but not necessarily identically distributed regularly varying random vectors, we first discuss estimators of their local angular measure. Since the convergence of these estimators will usually be slow, we consider the angular measure integrated over time and show that, under suitable smoothness assumptions on the underlying extreme value dependence, estimators of the integrated angular measure may converge at a faster rate. Finally, we discuss how these estimators can be used to test whether the angular measure is constant over time.
14:00 - 14:45
Richard Davis
Abstract: This paper provides three results for SVARs under the assumption that the primitive shocks are mutually independent. First, a framework is proposed to study the dynamic effects of disaster-type shocks with infinite variance. We show that the least squares estimates of the VAR are consistent but have non-standard properties. Second, it is shown that the restrictions imposed on a SVAR can be validated by testing independence of the identified shocks. The test can be applied whether the data have fat or thin tails, and to over as well as exactly identified models. Third, the disaster shock is identified as the component with the largest kurtosis, where the mutually independent components are estimated using an estimator that is valid even in the presence of an infinite variance shock. Two applications are considered. In the first, the independence test is used to shed light on the conflicting evidence regarding the role of uncertainty in economic fluctuations. In the second, disaster shocks are shown to have short term economic impact arising mostly from feedback dynamics. (This is joint work with Serena Ng.)
14:45 - 15:30
Gabor Lugosi
Abstract: he dependence structure of high-dimensional distributions is often modeled by graphical models. The problem of learning the graph underlying such distributions has received a lot of attention in statistics and machine learning. In problems of very high dimension, it is often too costly even to store the sample covariance matrix. We propose a new model in which one can query single entries of the covariance matrix. We propose computationally efficient algorithms for structure recovery in Gaussian graphical models with computational complexity that is quasi-linear in the dimension. We present algorithms that work for trees and, more generally, for graphs of small treewidth. The talk is based on joint work with Jakub Truszkowski, Vasiliki Velona, and Piotr Zwiernik.
4:00 - 5:00
Anne Sabourin
Jury: Stéphane Boucheron (reviewer), Richard Davis (reviewer), Holger Drees, Clément Dombry, Matthieu Lerasle (reviewer), Gabor Lugosi, Johan Segers.
Thursday 28 October 2021
9:30 - 10:25
Johan Segers
Abstract: To quantify the dependence between two random vectors of possibly different dimensions, we propose two coefficients. Both of them are based on the Wasserstein distance between the actual distribution and a reference distribution with independent components. The coefficients are normalized to take values between 0 and 1, where 1 represents the maximal amount of dependence possible given the two multivariate margins.
For Gaussian distributions, the two coefficients admit attractive formulas in terms of the joint correlation matrix. Maximal dependence occurs at the Gaussian distribution having minimal differential entropy given the two multivariate margins. The two coefficients can be estimated easily via the empirical correlation matrix. The estimators are asymptotically normal and their asymptotic variances are functions of the correlation matrix, which can thus be estimated consistently too. The results extend to the Gaussian copula case, in which case the estimators are rank-based. The results are illustrated through theoretical examples, Monte Carlo simulations, and a case study involving EEG data.
Preprint available at https://arxiv.org/abs/2104.14023
10:25 - 11:20
Clément Dombry
Abstract: We define infinitesimal gradient boosting as a limit of the popular tree-based gradient boosting algorithm from machine learning. The limit is considered in the vanishing-learning-rate asymptotic, that is when the learning rate tends to zero and the number of gradient trees is rescaled accordingly. For this purpose, we introduce a new class of randomized regression trees bridging totally randomized trees and Extra Trees and using a softmax distribution for
binary splitting. Our main result is the convergence of the associated stochastic algorithm and the characterization of the limiting procedure as the unique solution of a nonlinear ordinary differential equation in a infinite dimensional function space. Infinitesimal gradient boosting defines a smooth path in the space of continuous functions along which the training error decreases, the residuals remain centered and the total variation is well controlled.
11:25 - 12:20
Matthieu Lerasle
Abstract: I will present Bradley-Terry tournaments when players strength are i.i.d. realizations of a probability distribution. In this « typical » situation, I will describe precisely the phase transition between a « Pierre de Coubertin » regime where the best player wins and a « glorious uncertainty one » where one of the aspirants succeeds to beat him. In this last situation, I will also describe the cardinality of the potential winners.