Advanced Statistical Theory

Lecturer	Prof. Felix Abramovich ( felix@tauex.tau.ac.il)
Exercise grader	Tomer Levy (tmrlvi@gmail.com)

Lecture Hours	Wednesday 15-18, Shreiber 008

Prerequisites:	Statistical Theory
Course Requirements:	homework's average grade is 10% of the final grade

Topics:

Introduction and Background
- sufficiency, minimal sufficiency
- completeness
- exponential family of distributions
Parameter Estimation
- maximum likelihood estimators (MLE)
- M-estimators
- mean squared error (MSE), unbiased estimators, uniformly minimal variance unbiased estimators (UMVUE)
  - Fisher information
  - Cramer-Rao inequality
  - Rao-Blackwell theorem
  - Lehmann-Scheffe theorem
- confidence intervals and confidence regions, Bonferroni approach
- large sample theory
  - convergence of estimators, consistency and asymptotic normality
  - consistency and asymptotic normality of MLE and M-estimators
  - Wald, Wilks and Rao/score asymptotic confidence regions and tests
Nonparametric density estimation
- histograms
- kernel density estimation, asymptotic properties, bandwidth choice
Bayesian Inference
- choices for prior distributions: conjugate and non-informative priors, fully Bayes and empirical Bayes approaches, posterior prediction distribution
- Bayes estimation, credible sets and hypotheses testing
- large sample properties of Bayesian procedures, Bernstein-von Mises theorem
Elements of Statistical Decision Theory
- introduction, basic concepts
- minimax risk and minimax rules
- Bayes risk and Bayes rules
- posterior expected loss and Bayes actions
- admissibility and minimaxity of Bayes rules

Abramovich, F. and Ritov, Y. Statistical Theory: A Concise Introduction (second edition)
Bickel, P.K. and Doksum, K.A. Mathematical Statistics: Basic Ideas and Selected Topics

Cox, D.R. and Hinkley, D.V. Theoretical Statistics
Hogg, R.V., McKean, J.W. and Craig, A.T. Introduction to Mathematical Statistics
Wasserman, L. All of Statistics
Wasserman, L. All of Nonparametric Statistics