Statistics 1 (EN)

Objective

This course introduces the mathematical foundations of statistical modeling, primarily within a parametric framework. Emphasis is placed on inferential approaches, with a particular focus on parameter estimation methods. The properties of estimators, especially in terms of optimality, will be examined.

Learning outcomes: Upon completion of the course, students will be able to:
• Identify the nature of a statistical model: regular, discrete, density-based, and identifiable;
• Apply general parametric estimation methods, such as the method of moments, maximum likelihood estimation, and Bayesian estimation;
• Analyze the asymptotic behavior of estimators;
• Evaluate the optimality of parametric estimation methods;
• Formulate and apply the general principles for constructing confidence regions, and employ them in standard contexts.

Assessment:
• 2/3 of the final grade based on the written end-of-semester examination;
• 1/3 of the final grade based on continuous assessment, consisting of a midterm exam (50%), attendance (25%), and participation (25%).

Planning

Bracketed numbers refer to the corresponding sections of the course notes.

• Lecture 1: Limit theorems [1.4] ; Continuity theorems [1.5]

• Lecture 2: Sample [4.1] ; Empirical distribution function [4.2] ; Glivenko–Cantelli theorem

• Lecture 3: Substitution method [4.3] ; Statistics X?nX?n? and S2S2 [4.5, excluding Prop. 4.4]

• Lecture 4: Identifiable statistical model, with density, discrete [5.1] ; Consistency [5.2]

• Lecture 5: Risk [5.2.1] ; Comparison [5.2.2] ; Cramér–Rao inequality

• Lecture 6: Van Trees inequality and examples

• Lecture 7: Generalized method of moments [5.3] ; Likelihood and log-likelihood

• Lecture 8: MLE: behavior of the log-likelihood [5.5] ; Consistency [5.6]

• Lecture 9: MLE: regularity conditions and asymptotic normality

• Lecture 10: MLE: geometric interpretation, asymptotic efficiency, and examples

• Lectures 11 & 12: Bayesian estimation

• Lecture 13: Confidence sets

References

Lehmann E.L. et G. Casella (2003) Theory of point estimation, 2nd edition, Springer-Verlag [21 LEH 00 D]
Tsybakov A. (2006) Polycopié du cours de Statistique Appliquée, Université Pierre et Marie Curie. Disponible à l'adresse : www.crest.fr/ckfinder/userfiles/files/Pageperso/tsybakov/StatAppli\_tsybakov.pdf
Wasserman L. (2004) All of Statistics, Springer-Verlag [21 WAS 00 A]
van der Vaart, A. W. (1998-10-13). Asymptotic Statistics (1 ed.). Cambridge University Press.