Statistics 1

Objective

This course presents the theoretical bases of statistical modelling, essentially in a parametric framework. Preference is given to the inferential approach, and we deal primarily with parameter estimation methods and their properties, particularly in terms of optimality (asymptotic or finite distance). The theory of hypothesis testing will also be examined.

Planning

1. General principles- The aims of statistics, the various approaches (inferential, Bayesian). Types of statistical models (parametric, semi- and non-parametric). Sampling, information given by a sample (Fisher, Kullback), statistics (exhaustive, free), exponential models.
2. Estimation- Estimation problem. Decision-making approach: admissibility. Eliminating estimation bias: optimality, Cramér-Rao bound, efficiency. Asymptotic estimation: maximum likelihood, moments method, asymptotic efficiency. Bayesian estimation: Bayes' formula, Bayes estimator, subjective and objective approaches.
3. Hypothesis testing- Neyman-Pearson approach (confidence region, power, level, risks). Simple tests, Neyman-Pearson lemma. Student's t-test. Asymptotic tests (Wald, Score, likelihood ratio). Goodness-of-fit tests (chi-squared, K-S).
4. Resampling techniques- Bootstrap, permutation tests.

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]