Theory of Statistics

(0365-2103-03)

Lecturer Prof. Isaac Meilijson (isaco@math.tau.ac.il)
Teaching Assistant Ala Berlin (isaco@math.tau.ac.il)
Lecture Hours Sunday 10:10-12:00, Schreiber 007; Wednesday 11:10-12:00, Schreiber 007.
Office Hours Sunday 15:00-16:00, Schreiber 311.
Exercise Section Monday 11:10-13:00, Dan David 202

Pre-requisites: Probability
Course Requirements: submission of at least 2/3 of homework exercises, final exam.


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. Large-Sample Theory
    • Convergence in mean and in probablity
    • Consistency of estimators
    • Asymptotic normality
    • Asymptotic distribution of maximum likelihood estimators
  5. Hypotheses Testing
    • Introduction, basic concepts
    • Simple hypotheses, Neyman-Pearson lemma
    • Composite hypotheses, uniformly most powerful tests
    • Statistical inference for normal samples
      • One- and two-sample t-tests
      • Chi-squared test for variance
      • Comparison of variances (F-test)
    • Hypotheses testing and confidence intervals
    • Tests for goodness of fit and independence
    • Sequential probability ratio test (Wald)
  6. Bayesian Inference
    • Parameters as random variables
    • Bayes' theorem, prior and posterior distributions
    • Bayes estimation
    • Bayes choice between hypotheses

Literature

Last update: February 2004