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Advanced Statistical Theory


Lecturer Prof. Felix Abramovich (
Lecture Hours Wednesday 15-18, Schreiber 008

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


  1. Introduction and Background
    • sufficiency, minimal sufficiency
    • completeness
    • exponential family of distributions
  2. 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
  3. Nonparametric density estimation
    • histograms
    • kernel density estimation, asymptotic properties, bandwidth choice
  4. 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
  5. 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

Homework Exercises: