This course develops the material in the statistics 1 course. Firstly, new models (the generalised linear model, for example) and general estimation methods (M-estimation, moments method) are introduced. The asymptotic properties of the corresponding estimators are studied in detail. The course also sets out the foundation of statistical test theory and approaches the properties of goodness-of-fit tests. A few non-parametric tests are also presented. Finally, the course returns to the construction of confidence intervals. Without limiting itself to a constraining and profitless formalism, but without compromising on conceptual rigour, the course seeks to clarify several key ideas in both statistics and mathematics. These ideas are illustrated with application examples and exercises.
- Estimation objectives and methods –Reminder about efficiency and optimality, Stein counterexample. General estimation methods: minimum contrast (maximum likelihood and least squares estimators), moments method. Generalised linear models. Algorithmic questions.
- Asymptotic estimation –M-estimators: convergence, asymptotic normality.
- Tests– Foundations. Neyman-Pearson lemma. Monotone likelihood ratio. UMP tests, without bias and similar.
- Goodness-of-fit tests –Problem of goodness-of-fit tests. Chi-squared test and empirical measurement. Modifications resulting from the use of the maximum likelihood estimator. Principle of goodness-of-fit tests relying on an empirical distribution function (Kolmogorov-Smirnov and Cramér-von Mises).
- Non-parametric tests –Sample comparison: permutation tests, rank and sign tests.
- Confidence intervals and regions. Size, asymptotic or duality criteria with a test. Example of a semi-parametric confidence interval.
Bickel P., et K. Doksum (2001) Mathematical statistics. Basic Ideas and selected topics, vol I. Prentice Hall.
Hettmansperger T. P. (1984) Statistical Inference based on Ranks, Wiley.
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
Van der Vaart A. W. (1998), Asymptotic statistics, Cambridge University Press.