Estimation asymptotiquement sans biais de l'estimateur des valeurs extrêmes en présence de censure
Armelle Guillou (09h30-10h15)
Nous nous plaçons dans le cadre de la théorie des valeurs extrêmes, et plus particulièrement de distributions de type Pareto, en présence de données censurées à droite. L'objectif est de proposer un estimateur de l'indice des valeurs extrêmes asymptotiquement sans biais.
Partant d'un estimateur basé sur une intégrale de l'estimateur de Kaplan-Meier, nous proposons un estimateur asymptotiquement sans biais qui a la même variance que l'estimateur initial. La convergence faible de notre estimateur est établie.
Nous illustrons ses performances par simulations et appliquons notre méthodologie sur un jeu de données en assurance.
Facteurs d'influence de la réalisation de la trace graphique en écriture manuscrite : anticipation et prise en compte du temps
Cyril Perret (10h30-11h00)
Abstract: A venir.
Estimation robuste de la distribution ex-gaussienne en présence de données manquantes dans l’analyse des temps de réaction
Alandra Zakkour (11h00-11h30)
Le temps de réaction (RT) constitue un indicateur clé du traitement cognitif humain, et son analyse statistique joue un rôle important en psychologie expérimentale. La distribution ex-gaussienne est largement utilisée pour modéliser les données de temps de réaction, car elle permet de capturer efficacement leur asymétrie caractéristique.
Cependant, l’estimation des paramètres devient difficile en présence de petits échantillons et de données manquantes, situations fréquemment rencontrées dans les contextes expérimentaux.
Dans cet exposé, nous présentons un cadre statistique pour une estimation robuste de la distribution ex-gaussienne sous ces contraintes. Nous discutons des moyens d’améliorer la précision des estimations et de traiter correctement les données incomplètes tout en préservant les propriétés essentielles de la distribution.
A Factorial Semi-Markov Mixture Model for Clustering Patients’ Chronic Pain Treatment Trajectories with Concurrent Therapies and Irregular Follow-Up
Naomi Debataraja (11h30-12h00)
Abstract:
This study proposes a factorial semi-Markov mixture model for clustering of multichannel longitudinal treatment sequences characterized by concurrent therapies, varying sequence lengths, and irregular follow-up intervals. The proposed framework integrates a covariate-dependent multinomial logistic mixing structure with cluster-specific hidden semi-Markov dynamics, allowing treatment states to persist for variable durations through explicit duration distributions.
Parameter estimation is carried out using a variational expectation-maximization (VEM) method, which relies on a log-space forward-backward recursion for hidden semi-Markov models and integrates right-censoring adjustments for terminal segments. The efficacy of the proposed methodology is evaluated through simulation experiments based on synthetic patient data designed to mimic longitudinal clinical treatment records. Parameter recovery is analyzed for sample sizes of N = 100, 200, 500, and 1000.
The VEM approach exhibits consistent monotonic convergence across all settings, as assessed by the evidence lower bound (ELBO), while estimation accuracy improves with increasing sample size, as indicated by the root mean squared error (RMSE). The theoretical properties of the proposed estimator are also established, including identifiability up to label permutation, monotonic increase of the ELBO, and asymptotic consistency under standard regularity conditions.
Keywords: variational inference; semi-Markov mixture model; hidden semi-Markov model; longitudinal treatment sequences; patient trajectory clustering.
Après-midi - Titres et résumés
Titre: Parameter Estimation in Ergodic Jump-Diffusions: From JCIR to BAJD under Continuous-Time Observation
Hamdi Fathallah (14h00-14h30)
Abstract: This presentation investigates parameter estimation for ergodic jump-type Cox–Ingersoll–Ross (JCIR) processes driven by a Brownian motion and a subordinator. We consider continuous-time observations over long time horizons and address the joint estimation of drift and jump components.
First, we extend the results of Barczy et al. (2018) on maximum likelihood estimation of the drift parameters (a, b) by analyzing the inverse-path functional
∫0T (1 / Ys) ds in the ergodic regime b > 0. Under the Feller condition 2a > σ2, we establish strong consistency and asymptotic normality of the maximum likelihood estimator. To overcome this restriction, we introduce a moment-based estimator that remains well-defined and consistent for all positive parameters, providing a robust alternative outside the Feller region.
We then specialize to the basic affine jump-diffusion (BAJD) model, where the jump component is a compound Poisson process with exponentially distributed jumps. In this setting, we develop estimation procedures for the full parameter vector (a, b, λ), derive the maximum likelihood estimator, and prove its consistency and asymptotic normality under the same ergodicity and Feller conditions.
Since the likelihood depends on latent jump quantities that are not directly observable in practice, we propose an alternative estimator based on stationary moments. This method avoids explicit use of jump counts and sizes, remains valid beyond the Feller regime, and relies on a skewness-based identification of the jump parameter. We establish its strong consistency and asymptotic normality via a delta-method argument.
Nonparametric Recursive Estimation for the coefficients of Stable-Driven Stochastic Differential Equations
Gilles Christ Dansou (14h30-15h00)
Stochastic differential equations driven by α-stable Lévy processes provide an effective framework for modeling dynamical systems subject to jumps and heavy-tailed fluctuations, overcoming the limitations of Gaussian models when facing extreme events. However, statistical inference for these models remains challenging due to the infinite variance of the driving noise and the dependence structure of discretely observed trajectories.
We propose a nonparametric recursive methodology for estimating the drift and diffusion coefficients of such equations, with stability index α ∈ (1,2). Our approach extends the Nadaraya-Watson estimator to a stochastic approximation framework, enabling sequential updating of estimates and offering significant computational advantages for large samples.
The proposed framework allows for weaker regularity conditions than existing approaches and accommodates unbounded diffusion functions, thereby aligning with standard assumptions for existence and uniqueness of solutions.
Asymptotic analysis of the recursive kernel estimator is conducted under general conditions on bandwidth and step-size sequences, accounting for the invariant distribution and mixing properties of the process.
We further extend the methodology to estimate the stable noise parameters via a recursive characteristic function estimator within a regression framework.
This work thus unifies drift and noise parameter estimation in a computationally efficient recursive procedure, while providing a rigorous theoretical foundation for inference in models with jumps and heavy tails.
Asymptotic properties for dilute Wigner random matrices
Slim Ayadi (15h45-16h15)
We define the dilute Wigner random matrix ensemble Hn,p and formulate some results.
We study the spectral properties and improve the leading term of the correlation function of the resolvent Gn,p(z) = (Hn,p − zI)−1 with sufficiently large |Im z| in the limit p, n → ∞, with p = O(nα).
We show that increasing the order in the cumulant expansion formula decreases the value of α.
Titre: A venir
Salah Khardani (16h15-16h45)
Abstract: A venir
Deep Learning for Stable-Driven Stochastic Differential Equations with Physics-Informed Neural Networks
Stefana Tabera Tsilefa (16h45-17h15)
Abstract: In this talk, we propose the framework of Physics-Informed Neural Networks (PINNs) to track stochastic model parameters. By embedding the governing dynamics into the neural network’s loss function, our method simultaneously tackles forward and inverse problems: it approximates the evolution of probability densities while recovering unknown model parameters. This unified approach offers an effective tool for learning and inference both in Gaussian and non-Gaussian, heavy-tailed stochastic systems.
