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

• Bickel, P.K. and Doksum, K.A. Mathematical Statistics
• Hogg, R. and Craig, A. Mathematical Statistics
• Larsen, R.J. and Marx, M.L. An Introduction to Mathematical Statistics and its Applications
• Lindgren, B.W. Statistical Theory
• Samuel-Cohen, E. Statistical Theory (in Hebrew)