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Advanced Statistical Theory
(0365.4133)
Lecturer
Prof. Felix Abramovich
(
felix@tauex.tau.ac.il
)
Lecture Hours
Wednesday 15-18, Schreiber 008
Prerequisites
:
Statistical Theory
Course Requirements
:
homework's average grade is 10% of the final grade
syllabus
literature
exercises
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
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
Literature
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
Homework Exercises:
Exercise 1
(9 November)
Exercise 2
(14 December)
Exercise 3
(28 December)
Exercise 4
(11 January)
Exercise 5
(25 January)