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Statistical Theory

(0365.2103)

Lecturer Prof. Felix Abramovich ( felix@math.tau.ac.il)
Teaching Assistant Iman Jaljuli ( imanjljule@gmail.com)
Lecture Hours Sunday 15-18
Exercises Wednesday 14-16

Prerequisites: Probability
Course Requirements: each student has to submit at least 2/3 of homework execrises.


Topics:

  1. Introduction
    • statistical models
    • likelihood function
    • sufficient statistic
    • exponential family of distributions
  2. Parameter Estimation
    • maximum likelihood estimation
    • the method of moments
    • criteria for estimators, mean squared error
    • unbiased estimators
      • Fisher information
      • Cramer-Rao inequality
      • Rao-Blackwell theorem
  3. Confidence Intervals
  4. Hypotheses Testing
    • introduction, basic concepts
    • simple hypotheses, Neyman-Pearson lemma
    • composite hypotheses, uniformly most powerful tests
    • generalized likelihood ratio tests
    • statistical inference for normal samples
      • one- and two-sample t-tests
      • χ 2 test for variance
      • comparison of variances ( F-test)
    • hypotheses testing and confidence intervals
    • sequential Wald test
  5. Large-Sample Theory
    • convergence in mean and in probablity
    • consistency of estimators
    • asymptotic normality
    • asymptotic distribution of maximum likelihood estimators, Wald confidence intervals
    • asymptotic distribution of generalized likelihood ratio, Wilks' theorem
    • tests for goodness of fit and independence
  6. Bayesian Inference
    • introduction, basic concepts
    • Bayes' theorem, prior and posterior distributions
    • Bayes estimation, credible sets and hypotheses testing

Literature

  • Abramovich, F. and Ritov Y. Statistical Theory: A Concise Introdutcion
  • Bickel, P. and Doksum, K.A. Mathematical Statistics
  • Casella, G. and Berger, R.L. Statistical Inference
  • Hogg, R. and Craig, A. Introduction to Mathematical Statistics
  • Larsen, R.J. and Marx, M.L. An Introduction to Mathematical Statistics and Its Applications
  • Lindgren, B.W. Statistical Theory
  • Young, G.A. and Smith, R.L. Essentials of Statistical Inference
  • Samuel-Cohen, E. Statistical Theory (in Hebrew)
  • more...