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
o Fisher information
o Cramer-Rao inequality
o 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
o One- and two-sample t-tests
o Chi-squared test for variance
o 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)