Statistical Theory, Advanced Course

Aim

The course gives an introduction to the general theory of statistical inference. After a completed course the student is expected to be able to:

• describe advanced concepts and theorems of theoretical statistics, e.g., sufficiency, completeness, and the Neyman-Pearson lemma, and to prove some of the theorems.
• construct suitable, in some cases optimal, point estimators, hypothesis tests, and confidence sets, in general situations where the data are observations from a parametric family of probability distributions.
• carry out Bayesian inference in general situations where the data are observations from a parametric family of probability distributions.
• derive asymptotic results for point estimators, hypothesis tests, and confidence sets.
• understand and assess statistical inference occurring in other undergraduate courses, research reports, or the media.

Content

Exponential families. Location and scale families. Sufficient, mimimal sufficient, ancillary, and complete statistics. Methods for point estimation, e.g. maximum likelihood. Evaluation of point estimates using e.g. the Cramer-Rao inequality and the Rao-Blackwell theorem. Likelihood ratio tests. Uniformly most powerful tests and the Neyman-Pearson lemma. The correspondence between tests and confidence sets. Pivotal variables. Optimality for confidence sets. Bayesian inference and decision theory. Asymptotic theory.