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Advanced Statistical Theory

(0365.4133)


Lecturer Prof. Felix Abramovich ( felix@math.tau.ac.il)
Lecture Hours Tuesday, Shenkar 104

Prerequisites: Statistical Theory


Topics:

  1. Introduction and Background
    • likelihood function, Fisher information, Cramer-Rao inequality
    • sufficiency, minimal sufficiency, Rao-Blackwell theorem
    • completeness, Lehmann-Scheffe theorem
    • exponential family of distributions
  2. Parameter Estimation
    • maximum likelihood estimators (MLE)
    • M-estimators
    • convergence of estimators, consistency and asymptotic normality
    • consistency and asymptotic normality of MLE and M-estimators
    • confidence intervals and confidence regions, Bonferroni approach
    • Wald, Wilks and 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
    • 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
  6. Normal means model
    • introduction
    • MLE, Stein's paradox, James-Stein estimator
    • minimax estimation over ellipsoids: introduction, linear estimators, Pinsker's theorem
    • sparsity, thresholding estimators

Literature

  • Abramovich, F. and Ritov, Y. Statistical Theory: A Concise Introduction
  • Bickel, P.K. and Doksum, K.A. Mathematical Statistics: Basic Ideas and Selected Topics
  • Cox, D.R. and Hinkley, D.V. Theoretical Statistics
  • Tsybakov, A.B. Introduction to Nonparametric Estimation
  • Wasserman, L. All of Statistics
  • Wasserman, L. All of Nonparametric Statistics