# Statistical Theory

### (0365.2103)

 Lecturer Prof. Felix Abramovich ( felix@math.tau.ac.il) Teaching Assistant Iman Jaljuli ( jaljuli.iman@gmail.com) Lecture Hours Sunday 15-18, Schreiber 6 Exercises Wednesday 14-16, Schreiber 6

 Prerequisites: Probability Course Requirements: homework's average grade is 10% of the final grade; a student has to submit at least 10 homework exercises to get it

## 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 asymptotic confidence intervals and tests
• score (Rao) asymptotic confidence intervals and tests
• 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...