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.

After the completion of this course, students are expected to know how:

1. To determine sufficient statistic in a given model and to check whether is minimal sufficient and/or complete statistic,
2. To determine uniformly minimum-variance unbiased estimator in a model with sufficient and complete statistic,
3. To check efficiency of an estimator in a regular model,
4. To estimate parameters by the methods of maximum likelihood, expectation-maximization, and least squares,
5. To check consistency of an estimator,
6. To determine asymptotic law of an estimator and to calculate its asymptotic standard error and confidence interval for given parameter,
7. To calculate posterior distribution of parameters in a Bayesian model and Bayes estimator, and to perform Bayes interval estimation,
8. To perform uniformly most powerful test of parametric statistical hypothesis whenever it is possible,
9. To perform likelihood ratio test of parametric statistical hypothesis,
10. To perform goodness of fit tests.

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).