Introduction to mathematical statistics, Bayesian and frequentistic.

Lectures

written and oral examinations

The course is intended as a first course in mathematical statistics. Students in this course are assumed to have a good knowledge of the fundamental material taught in the first course in probability theory, in addition to that of advanced calculus.

- Statistics in inductive logic : brief historical survey.
- Bayes-Laplace paradigm. Conditional law of a sequence of observations given an unknown random parameter ; initial distribution .
- Final and predictive distributions : their deducrion and use to solve hypothetical and predictive problems within the theory of statistical decisions.
- Asymptotics for the above distributions, as the number of observations goes to infinity, in connection with the frequentistic interpretation of probability and statistics.
- The Fisherian criticism to the Bayes-Laplace paradigm, and the rise of objective methods based on the likelihood random function.
- Sufficient statistic: definition and characterization (factorization theorem); the likelihood as example of minimal sufficient statistic.
- Fisher information; ancillary statistic and Basu theorem. A concise analysis of the exponential statistical model.
- Point estimation. Maximum likelihood estimators: definition, examples and asymptotic properties. Uniformly minimum variance unbiased estimators: Kolmogorov-Rao-Blackwell and Lehmann-Scheffé theorems.
- Testing statistical hypotheses. Fisherian criteria : spirit and applications to Gaussian samples and to nonparametric settings. The Neyman-Pearson approach ; fundamental lemma for simple hypotheses and its use also for composite hypotheses in a remarkable kind of statistical models. Estimation by confidence sets.
- Linear statistical model. Estimation and testing statistical hypotheses in distinguished forms of the linear statistical model.

-Bickel, P.J. and Doksum, K. A. Mathematical statistics, Holden-Day Inc.