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)

Statistical Theory

The theory of statistics includes a number of topics:

Statistical models of the sources of data and typical problem formulation:

1. Sampling from a finite population
2. Measuring observational error and refining procedures
3. Studying statistical relations

Planning statistical research to measure and control observational error:

1. Design of experiments to determine treatment effects
2. Survey sampling to describe natural populations

Summarizing statistical data in conventional forms (also known as descriptive statistics)

1. Choosing summary statistics to describe a sample
2. Fitting probability distributions to sample data

Interpreting statistical data is the final objective of all research:

1. Common assumptions that we make
2. Likelihood principle
3. Estimating parameters
4. Testing statistical hypotheses
5. Revising opinions in statistics

(http://en.wikipedia.org)