Theory of Statistics Topics

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)

(Derived from http://www.math.tau.ac.il/~isaco/TheoStat.html)