Statistical Theory

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 estimates
   • 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
   • generalized likelihood ratio tests
   • statistical inference for normal samples
     • one- and two-sample t-tests
     • χ² test for variance
     • comparison of variances (F-test)
   • hypotheses testing and confidence intervals
   • asymptotic distribution of generalized likelihood ratio, Wilks theorem
   • tests for goodness of fit and independence
   • sequential Wald test
6. Bayesian Inference
   • parameters as random variables
   • Bayes' theorem, prior and posterior distributions
   • Bayes estimation, credible sets and hypotheses testing

Literature

• Bickel, P. and Doksum, K.A. Mathematical Statistics.
• 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...