Statistics is concerned with the development and use of mathematical and computational methods for the collection, analysis, and interpretation of data in support of scientific inquiry, informed decision-making, and risk management. It calls on a broad range of tools from probability theory to computer-intensive techniques. The main areas of research by statisticians in the ISM network include
Statistical research is largely motivated by collaboration with other disciplines. It finds applications in many fields, including biology, environmental science, finance and insurance, health sciences, hydrology, market research, and social sciences. With the abundance of very large and complex data sets coming, for example, from the social media and digital processes, financial transactions, astronomy, genomics, meteorology or Big Science like the Giant Hadron Collide, the statistical treatment and analysis of Big Data has become a major challenge of modern statistics.
The statistics program gives an opportunity to graduate students to study in these two major areas of modern statistics. The curriculum allows the students to get well acquainted with the basic elements of mathematical statistics, decision theory and applied statistics. Furthermore, advanced graduate courses can be offered in some more specialized areas.
This program welcomes graduate students with a good background in calculus, mathematical statistics, numerical analysis, and probability (all at the undergraduate level). To get strong training in decision theory and mathematical statistics students should take the basic course in measure and integration (for PhD students) and at least three courses at the intermediate and advanced levels.
This course is an introduction to statistical inference for parametric models. The following topics will be covered:
1. Distribution of functions of several random variables (distribution function and change of variable techniques), sampling distribution of mean and variance of a sample from Normal distribution.
2. Distribution of order statistics and sample quantiles.
3. Estimation: unbiasedness, CramÈr-Rao lower bound and efficiency, method of moments and maximum likelihood estimation, consistency, limiting distributions, delta-method.
4. Sufficiency, minimal sufficiency, completeness, UMVUE, Rao-Blackwell and Lehman-Scheffe theorems.
5. Hypothesis-testing: likelihood-ratio tests.
6. Elements of Bayesian estimation and hypothesis-testing.
Statistical software (R) will be used for the analysis of real‑life data sets. Topics may include techniques from generalized linear models, model selection, log‑linear models for categorical data, logistic regression, survival models.
This course introduces multivariate statistical analysis, both theory and methods, with focus on the multivariate Normal distribution. It can be seen as a preparatory course, although not a formal prerequisite, for Statistical Learning. Topics covered include:
This course is an introduction to reinforcement learning techniques. It requires extensive programming with the R language. Topics covered include: Multi-armed bandit problem, Markov Decision Problems, Dynamic Programming, Monte-Carlo solution methods, Temporal difference methods, Multi-period Approximation methods, Policy gradient.
Multivariate normal and chi-squared distributions; quadratic forms. Multiple linear regression estimators and their properties. General linear hypothesis tests. Prediction and confidence intervals. Asymptotic properties of least squares estimators. Weighted least squares. Variable selection and regularization. Selected advanced topics in regression. Applications to experimental and observational data.
Distribution theory, stochastic models and multivariate transformations. Families of distributions including location-scale families, exponential families, convolution families, exponential dispersion models and hierarchical models. Concentration inequalities. Characteristic functions. Convergence in probability, almost surely, in Lp and in distribution. Laws of large numbers and Central Limit Theorem. Stochastic simulation.
Subjective probability, Bayesian statistical inference and decision making, de Finetti’s representation. Bayesian parametric methods, optimal decisions, conjugate models, methods of prior specification and elicitation, approximation methods. Hierarchical models. Computational approaches to inference, Markov chain Monte Carlo methods, Metropolis—Hastings. Nonparametric Bayesian inference.
The course will develop the probabilistic foundations of large-sample theory and applies them to classical and modern problems in statistics. Emphasis will be placed on stochastic convergence, asymptotic expansions, uniform convergence, rates of convergence, and optimality. The final part of the course will introduce minimax rates in nonparametric regression and some semiparametric ideas such as influence functions and orthogonality. The course will be split into three main parts: (A) Mathematical and probabilistic foundations; (B) Classical large-sample theory; (C) Introduction to minimax and semiparametric theory.
In part (A), the focus will be on mathematical reminders, stochastic convergence, uniform integrability, laws of large numbers, central limit theorems, triangular arrays, and asymptotic linearity.
Part (B) will focus on studying the asymptotic properties of various classical estimators, such as the empirical distribution function and sample quantiles, maximum likelihood, M-estimation, Z-estimation, with various applications to estimation and testing.
Part (C) will be an introduction to selected modern topics such as minimax upper/lower bounds for nonparametric regression and semiparametric theory, as time permits.
Description: This course introduces the fundamental concepts, methodologies, and computational tools of sparse statistical learning and related areas, with an emphasis on supervised learning problems in both low- and (ultra-)high-dimensional settings. Topics include:
Numerical algorithms and computational aspects will be discussed throughout the course as they naturally arise in the development of the material.
Method of evaluation: Assignments (40%), final term project (40%), final presentation (20%)
General introduction to computational methods in statistics; optimization methods; EM algorithm; random number generation and simulations; bootstrap, jackknife, cross-validation, resampling and permutation; Monte Carlo methods: Markov chain Monte Carlo and sequential Monte Carlo; computation in the R language.
Conditional probability and Bayes’ Theorem, discrete and continuous univariate and multivariate distributions, conditional distributions, moments, independence of random variables. Modes of convergence, weak law of large numbers, central limit theorem. Point and interval estimation. Likelihood inference. Bayesian estimation and inference. Hypothesis testing.
Tableaux de contingence à plusieurs dimensions. Mesures d'association. Risque relatif, rapport de cote. Tests exacts et asymptotiques. Régression logistique, de Poisson, multinomiale, logistique cumulative. Modèles log-linéaires. Modèles graphiques.
Principes d'inférence : estimation ponctuelle, distribution des estimateurs, test d’hypothèse, région de confiance. Approche bayésienne. Méthodes de rééchantillonnage. Estimation non paramétrique. Applications modernes de la statistique.
Fonctions de variables aléatoires, fonction génératrice des moments, quelques inégalités et identités en probabilité, familles de distributions dont la famille exponentielle, vecteurs aléatoires, loi multinormale, espérances conditionnelles, mélanges et modèles hiérarchiques. Théorèmes de convergence, méthodes de simulation, statistiques d'ordre, exhaustivité, vraisemblance. Estimation ponctuelle et par intervalles : construction d'estimateurs et critères d'évaluation, méthodes bayésiennes. Normalité asymptotique et efficacité relative asymptotique.
Espérance conditionnelle. Prédiction. Modèles statistiques, familles exponentielles, exhaustivité. Méthodes d'estimation: maximum de vraisemblance, moindres carrés etc. Optimalité: estimateurs sans biais à variance minimum, inégalité de l'information. Propriétés asymptotiques des estimateurs. Intervalles de confiance et précision. Éléments de base de la théorie des tests. Probabilité critique, puissance en relation avec la taille d'échantillon. Relation entre tests et intervalles de confiance. Tests pour des données discrètes.
Étude des distributions échantillonnales classiques: T2 de Hotelling; loi de Wishart; distribution des valeurs et des vecteurs propres; distribution des coefficients de corrélation. Analyse de variance multivariée. Test d'indépendance de plusieurs sous-vecteurs. Test de l'égalité de matrices de covariance. Sujets spéciaux.
Nombre aléatoire. Simulation de lois classiques. Méthodes d'inversion et de rejet. Algorithmes spécifiques. Simulation des chaines de Markov à temps discret et continu. Solution numérique des équations différentielles ordinaires et stochastiques. Méthode numérique d'Euler et de Runge-Kutta. Formule de Feynman-Kac. Discrétisation. Approximation faible et forte, explicite et implicite. Réduction de la variance. Analyse des données simulées. Sujets spéciaux.
Rappel sur les principales notions de statistique mathématique et sur la statistique asymptotique. Introduction à la théorie des copules. Description des modèles de dépendance bidimensionnels et multidimensionnels les plus populaires et exploration exhaustive des propriétés de ces copules. Inférence statistique dans les modèles de copules : estimation de paramètres, copule empirique, tests d'adéquation et tests d'hypothèses composites. La méthode delta fonctionnelle et ses nombreuses applications, notamment en inférence de copules. Survol des avancées récentes, incluant les tests de rupture, l'étude de la dépendance conditionnelle, la modélisation de la dépendance spatiale et l'utilisation de la fonction caractéristique. Les objectifs spécifiques de ce cours sont : de maîtriser la théorie des copules, de connaître les principales méthodes d'inférence concernant les copules, d'être au fait des principaux développements récents, de bien connaître la littérature sur les copules, d'être capable de mettre en oeuvre les méthodes statistiques avec le logiciel Matlab (estimation de la puissance de tests, analyse de jeux de données).
This course introduces the theory and practice of time series analysis. Both time and frequency domain methods will be discussed. The objective of this course is to learn and apply statistical methods for the analysis of data that have been observed over time. The Analyses will be performed using the freely available package ITSM, which accompanies the textbook. Topics covered include:
This course is an introduction to the theory of prediction with neural networks. Several applications of neural networks to common problems faced in practice are finally explored. Students will also be exposed to the implementation of methods seen in class; programming assignments use the Python or R programming languages.
Topics covered: Review of predictive analytics and numerical computation concepts: Supervised learning, cross-validation, hyperparameters; Overflow and underflow; Feed-forward neural networks, Motivation, Non-linear predictions, Universality property.
Classification versus regression problems: Architecture specification, Parameter estimation, Objective function; Steepest gradient descent; Backpropagation, saturation, Hessian computation; Parameter initialization strategies.
Advanced estimation topics: Adaptive learning rates: Regularization, Dataset augmentation and noise injection, Alternative neural network types, Recurrent neural networks (RNN), Long-short term (LSTM) neural networks, Convolutional neural networks, Implementations and Applications.
This course is an introduction to statistical learning techniques. Topics covered include: cross- validation, regression methods, classification methods, tree-based methods, introduction to neural networks, unsupervised learning.
Probability distributions for categorical data, Analysis of 2X2 contingency tables, Multiway contingency tables, The Logistic regression, Logistic regression for categorical predictors, Logit models for nominal and ordinal responses, Log-linear models and modelling ordinal associations in contingency tables, Unsupervised learning techniques for categorical data, Non Linear Principal component analysis, Applications of unsupervised learning techniques using R, Item Response Theory, Rasch model. Some topics may be included or excluded as the time permits.
Exponential families, link functions. Inference and parameter estimation for generalized linear models; model selection using analysis of deviance. Residuals. Contingency table analysis, logistic regression, multinomial regression, Poisson regression, log-linear models. Multinomial models. Overdispersion and Quasilikelihood. Applications to experimental and observational data.
In-depth study of survival analysis, covering foundational concepts and advanced techniques in time-to-event data analysis. Exploration of censoring and truncation, survival and hazard functions, and nonparametric methods like the Kaplan-Meier estimator. Core topics include hypothesis testing for survival distributions, parametric and semiparametric modeling, and covariate inclusion through the Cox proportional hazards model. Emphasis is placed on model diagnostics, validation, and variable selection techniques, including best-subset selection, LASSO, and nonconcave penalized likelihood approaches. Practical applications and hands-on analysis using R for survival data in research and applied settings.
Stationary processes; estimation and forecasting of ARMA models; non-stationary and seasonal models; state-space models; financial time series models; multivariate time series models; introduction to spectral analysis; long memory models.
Sufficiency, minimal and complete sufficiency, ancillarity. Fisher and Kullback-Leibler information. Elements of decision theory. Theory of estimation and hypothesis testing from the Bayesian and frequentist perspective. Elements of asymptotic statistics including large-sample behaviour of maximum likelihood estimators, likelihood-ratio tests, and chi-squared goodness-of-fit tests.
Introduction to concepts in statistically designed experiments. Randomization and replication. Completely randomized designs. Simple linear model and analysis of variance. Introduction to blocking. Orthogonal block designs. Models and analysis for block designs. Factorial designs and their analysis. Row-column designs. Latin squares. Model and analysis for fixed row and column effects. Split-plot designs, model and analysis. Relations and operations on factors. Orthogonal factors. Orthogonal decomposition. Orthogonal plot structures. Hasse diagrams. Applications to real data and ethical issues.
Principes de l’analyse bayésienne; loi à priori et à postériori, inférence statistique et théorie de la décision. Méthodes computationnelles; méthodes de Monte Carlo par chaînes de Markov. Applications.
Rappels sur les modèles linéaires généralisés (inférence, tests, validation, choix de modèle). Géométrie de la régression. Étude asymptotique des estimateurs et réduction de variance. Régression robuste. Régression non paramétrique.
Distributions elliptiques. Estimateurs de localisation et dispersion. Estimateur robuste. Corrélations multiple, partielle, canonique. Tests paramétriques, de permutation, du bootstrap. Classification. Analyse en composantes principales. Prévision.
Analyse en composantes principales. Analyse des corrélations canoniques et régression multidimensionnelle. Analyse des correspondances. Discrimination. Classification. Analyse factorielle d'opérateurs.
Régression linéaire, modèles mixtes, modèles linéaires généralisés, régression de copule, méthodes non paramétriques, sélection de modèle, erreurs de mesure, données manquantes. Utilisation du logiciel R.
Théorie des modèles linéaires généraux. Théorie des modèles linéaires généralisés. Régression logistique. Modèles log-linéaires.
Classes d'hypothèse. Fonctions de perte et de risque. Décomposition biais-complexité. Complexité algorithmique. Régularisation, stabilité et surapprentissage. Optimisation convexe. Révision des modèles d'apprentissage statistique classiques, tels les réseaux de neurones, à travers cette nouvelle perspective. Programmation dans un langage tel que R ou Python.