To prepare students for modelling of statistical data. The primary focus is on parametric statistical models. Students will learn basic concepts and methods for:
1. Mathematical analysis of the models
2. Understanding, properly applying and analysing optimal statistical procedures
3. Statistical inference about the model parameters.
1. Introduction to parametric statistical models. Sufficient and complete statistics (conditional distributions, factorization criterion, minimal sufficient statistics, example: exponential family of distributions), Uniformly minimum-variance unbiased estimator (conditional expectation, Rao-Blackwell and Lehmann-Scheffe theorems), Efficient estimation in regular models (Cramer-Rao theorem).
2. Methods of estimation. Maximum likelihood (examples, regular models), Expectation-maximizations (examples), Least squares (linear regression models, Gauss-Markov theorem).
3. Asymptotic statistics. Consistency (Law of large numbers), Asymptotic distributions, Asymptotic normality (Central limit theorem, Cramer theorem), Asymptotic confidence intervals (examples).
4. Existence, consistency and asymptotic efficiency of maximum likelihood estimators in regular models. Maximum likelihood estimators of parameters in exponential families of distributions.
5. Bayesian estimations. Prior and posterior distributions of model parameter (Bayes theorem), Bayes estimator, Bayes interval estimation.
6. Testing statistical hypothesis. Uniformly most powerful tests (Neyman-Pearson Lemma), Likelihood ratio tests (Monotone likelihood ratio tests, examples: t-tests, F-tests, z-tests), Asymptotic distribution of likelihood ratio test, Goodness of fit tests (Pearson-Fisher chi-square test).