Analyse statistique II [ LSTAT2050 ]

iCampus

A basic course on methodological statistics, including an introductory course in probability and statistics, followed by the course LSTAT2040 "Analyse statistique - I".

The course is a follow-up to the course LSTAT2040 "Analyse statistique - I". Concepts in statistical methodology will be treated in greater depth and will be complemented with more advanced ones.

By the end of the course, the student will have become familiar with the necessary concepts in mathematical statistics in order to follow advanced courses of the finalité approfondie and to perform research in mathematical statistics on the PhD level. The student will be able to put the different themes in a general, abstract context, both regarding their application to problems in statistical analysis and regarding their interpretation. The student will master the technical tools to apply the concepts correctly and will be able to reproduce and to elaborate upon the mathematical arguments underlying the results.

There will be an oral exam, preceded by a written preparation.

The cours consists of both lectures and tutorials.

Part I - Theory of Optimality for Statistical Inference

The concept of sufficiency, in particular when applied to the important and rich class of exponential families, delivers a non-asymptotic theory of optimality of statistical procedures. The applications are numerous: for risk-optimal point estimation one can define the concept of UMV(U) estimators, i.e. "uniformly minimal variance (unbiased)" estimators. For the theory of statistical hypothesis testing, to be more abstractly formalised following the Neyman principle, it is possible to characterise the optimality of existing tests via the concept of UMP(U) tests, i.e., "uniformly most powerful (unbiased)" tests. A particular challenge here is the treatment of multi parameter families. Finally, the results from test theory can be directly transferred to define optimality of confidence regions.

Part II - Asymptotic Theory of Statistical Experiments

A coherent framework to analyse the asymptotic performance of estimators is Lucien Le Cam's theory of statistical experiments. For regular parametric models, the associated experiments are locally asymptotically normal. This means that the limit experiment is equal to a Gaussian shift experiment, the problem being of estimating the mean of a Gaussian distribution with known covariance matrix. The proof of this result relies on the concept of contiguity of sequences of statistical experiments and in particular on Le Cam's lemmas. The Hajek--Le Cam convolution theorem theorem then stipulates a lower bound for the covariance matrix of regular estimators of the model parameters. This result provides a precise statement about the optimality of maximum likelihood estimators in smooth parametric models.

Apart from the course notes, excerpts from the following books constitute recommended reading:

'          Casella, G., Berger, R.L. (2001). Statistical Inference (2nd ed). Cengage Learning.

'          Lehmann, E.L. (1999). Elements of Large-Sample Theory. Springer.

'          Lehmann, E.L., Romano, J. (2005). Testing Statistical Hypotheses (3rd ed). Springer.

'          Monfort, A. (1997). Cours de statistique mathématique (3rd ed). Economica.

'          van der Vaart, A.W. (1998). Asymptotic Statistics. Cambridge University Press, Cambridge. Chapters 6-9.

The course notes will be distributed during the lectures itself.