Séminaire Statistique
organisé par l'équipe Statistique
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Éléonore Blanchard
Approximate Bayesian Calibration of Option Pricing Models
7 février 2025 - 11:00Salle de séminaires IRMA
Nous proposons une méthodologie pour la calibration de modèles financiers complexes basée sur l'algorithme Approximate Bayesian Computation (ABC). Cette approche permet de calibrer les paramètres de modèles nécessitant un pricing par Monte-Carlo. Nous appliquons la méthodologie à un modèle exponentiel de Levy et un modèle exponentiel de Levy modulé par une chaîne de Markov. Nos expériences, réalisées sur données simulées et réelles (options européennes sur le S&P500), montrent que l'algorithme ABC fournit des calibrations précises. En complément, une approche par processus gaussiens améliore la stabilité des calibrations dans le temps. Cette étude démontre l'efficacité et la flexibilité d'ABC pour la calibration de modèles complexes, ouvrant des perspectives pour son application à des produits dérivés plus sophistiqués. -
Hidden Markov Models (HMMs) are powerful tools for modeling time-dependent phenomena governed by underlying dynamic processes that are only partially observable. However, standard estimation techniques for HMMs — such as maximum likelihood, least square and spectral methods — often suffer from sensitivity to model misspecification, outliers, and data contami- nation, resulting in unreliable parameter estimates. To overcome these challenges, we introduce a novel estimation method based on ρ-estimators, a class of robust estimators developed by Baraud et al. [1, 2] for independent set- tings. We prove a non-asymptotic bound on the Hellinger distance between the target distribution and our estimator. It allows us to derive a minimax convergence rate (up to a logarithmic factor) in the well-specified case. We establish the robustness of the estimator by demonstrating that its performance remains stable under contamination, as long as the contamination rate is moderate, regardless of the type of contamination. We can notice a posteriori that the proposed method is not restricted to HMMs and can be generalized to other models satisfying similar properties. For instance, we obtain results for the estimation of the stationary distribution for a class of Langevin diffusions.
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Yiye Jiang
Graph learning from time series in complex data settings
7 mars 2025 - 11:00Salle de séminaires IRMA
Graph learning is an active research domain in statistics, highlighted by well-known models such as Gaussian graphical models for i.i.d. data and autoregressive models for time series. In this talk, we will present two new models developed to address distinct analytical challenges. The first model tackles a non-classical data setting where the data points are probability distributions. Here, the graph is inferred to represent the dependence structure of a set of distributional time series. Leveraging Wasserstein space theory, we develop a novel autoregressive model, which is then applied to a demographic dataset. The second model is designed to meet an application-driven need: inferring a functional connectivity graph for a single subject’s brain fMRI time series while quantifying uncertainty. We adopt a Bayesian modeling approach to infer these graphs, with posterior distributions over edges providing uncertainty estimates. In particular, we introduce a prior for correlation matrices that facilitates the integration of expert knowledge. The model is applied to a rat fMRI dataset, where two follow-up analyses—edge detection and subject comparison—are conducted. The results highlight the robustness gained through uncertainty quantification. -
Yves Ngounou Bakam
Advancing Copula Methods: Nonparametric Estimation, Smooth Testing, and Data-Driven Clustering
14 mars 2025 - 11:00Salle de séminaires IRMA
Copulas, introduced in the 1950’s and rediscovered in recent years, are powerful tools for modeling dependence structures between multidimensional variables. These tools are particularly valuable in fields like finance, insurance, economics, and biology, where understanding the relationships between variables is critical. Despite their generality, copulas can present significant challenges, particularly when estimating dependence structures in complex datasets, especially when dealing with data from different sources, scales, and shapes. This work addresses three core challenges in copula modeling: estimation, testing, and clustering. We first propose a nonparametric copula density estimator based on Legendre orthogonal polynomials. A nonparametric copula estimator is then deduced by integration. Both estimates are based on a set of moments that define the copulas, and we’ll call them the copula coefficients. Flexible modeling is possible even when copula densities may not exist due to the complete characterization of these coefficients. A data-driven method is introduced to select the optimal number of copula coefficients to use, and extensive simulations show the superior performance of our approach compared to existing methods. Next, we propose a smooth test for comparing K ≥ 2, copulas simultaneously, based on differences in their copula coefficients. The procedure involves a two-step data-driven procedure. In the first step, the most significantly different coefficients are selected for all pairs of populations and the subsequent step utilizes these coefficients to identify populations that exhibit significant differences. Finally, we use this test to develop a clustering method that automatically identifies populations with similar dependence structures. They approaches, implemented in the Kcop R package, are demonstrated through numerical studies and real-world applications. This approach can be extended to the independent clustering in high dimension where work is ongoing. -
Gaspard Bernard
Testing for sphericity using spatial signs under elliptical directions
21 mars 2025 - 11:00Salle de séminaires IRMA
In this talk, we consider the problem of testing for the sphericity of a collection of random vectors. It is well known that in the classical elliptical model, testing for rotational symmetry of the underlying distribution is equivalent to testing that a scatter parameter is a multiple of the identity matrix. We consider the more general model of random vectors with elliptical directions introduced by R.H. Randles and present a few scenarios where testing for sphericity is still equivalent to testing that the scatter parameter is a multiple of the identity. These new scenarios include, for instance, non-classical settings where some dependence of a rather general form studied here for the first time may be present between observations. We study, under these new assumptions, the behavior of the classical spatial sign test and show that under certain mild assumptions, the test is asymptotically valid and has the same local asymptotic power as in the classical elliptical scenario. We then show that, contrary to some commonly held belief, the spatial sign test enjoys some local asymptotic optimality properties when it comes to testing for sphericity when the underlying distribution is strongly heavy-tailed. -
Lennart Martens
Pas encore disponible, mais autour de la stat et de la bioinfo
4 avril 2025 - 11:00Salle de séminaires IRMA
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Bertrand Michel
Introduction à l’Analyse Topologique des Données
25 avril 2025 - 11:00Salle de séminaires IRMA
L'analyse topologique des données (Topological Data Analysis - TDA) est un domaine récent et en plein essor, à l'intersection des mathématiques, de l'informatique et de la statistique. Elle est principalement motivée par l'idée que la topologie et la géométrie offrent des approches puissantes pour inférer, analyser et exploiter des informations qualitatives et quantitatives robustes dans la structure des données, généralement représentées sous forme de nuages de points ou d'échantillons dans des espaces euclidiens ou des espaces métriques plus généraux. Cet exposé proposera une introduction à la TDA et à l'homologie persistante. J’aborderai aussi des aspects plus statistiques de la TDA. -
Roberto Molinari
Fiducial Matching: Differentially Private Inference for Categorical Data
13 juin 2025 - 11:00Salle de séminaires IRMA
The task of statistical inference, which includes building confidence intervals and tests for parameters and effects of interest to a researcher, is still an open area of investigation in a differentially private (DP) setting. Indeed, in addition to the randomness due to data sampling, DP delivers another source of randomness consisting in the noise added to protect an individual’s data from being disclosed to a potential attacker. As a result of this convolution of noises, in many cases it is too complicated to determine the stochastic behavior of the statistics and parameters resulting from a DP procedure. In this work we contribute to this line of investigation by employing a simulation-based matching approach, solved through tools from the fiducial framework, which aims to replicate the data generation pipeline (including the DP step) and retrieve an approximate distribution of the estimates resulting from this pipeline. For this purpose we focus on the analysis of categorical (nominal) data that is common in national surveys, for which sensitivity is naturally defined, and on additive privacy mechanisms. We prove the validity of the proposed approach in terms of coverage and highlight its good computational and statistical performance for different inferential tasks in simulated and applied data settings.