Advanced Statistical Theory

Lecturer	Prof. Felix Abramovich (felix@math.tau.ac.il)
Lecture Hours	Tuesday 16-19, Schreiber 006

Topics:

Introduction
- likelihood function
- sufficiency, minimal sufficiency
- completeness
- exponential family of distributions
Parameter Estimation
- maximum likelihood estimation
- other methods of estimation: method of moments, least squares
- criteria for estimators, mean squared error
- unbiased estimators
  - Fisher information
  - Cramer-Rao inequality
  - Rao-Blackwell theorem
  - Lehmann-Scheffe theorem
Confidence Intervals & Confidence Regions
Hypotheses Testing
- introduction, basic concepts
- simple hypotheses, Neyman-Pearson lemma
- composite hypotheses, uniformly most powerful tests
- generalized likelihood ratio test
Large-Sample Theory
- different types of convergence of estimators
- consistency of estimators
- asymptotic normality
- asymptotic distribution of maximum likelihood estimators
- asymptotic confidence regions
- asymptotic distribution of generalized likelihood ratio, Wilks theorem
Bayesian Inference
- introduction, basic concepts
- choices for prior distributions
- Bayes estimation, credible sets and hypotheses testing
Elements of Statistical Decision Theory
- introduction, basic concepts
- minimax risk and minimax rules
- Bayes risk and Bayes rules
- posterior expected loss and Bayes actions

Abramovich, F. and Ritov, Y. Statistical Theory: A Concise Introduction (to appear)
Beaumont, G.P. Intermediate Mathematical Statistics
Bickel, P.K. and Doksum, K.A. Mathematical Statistics
Cox, D.R. and Hinkley, D.V. Theoretical Statistics
Schervish, M.J. Theory of Statistics
Shao, J. Mathematical Statistics
more...