# Statistics 1

**Teacher**: dr. Miljenko Huzak, professor

**Semester**: first

**ECTS**: 6

Required course

To prepare students for modelling of statistical data. The primary focus is on parametric statistical models. Students will learn basic concepts and methods for:

- Mathematical analysis of the models
- Understanding, properly applying and analysing optimal statistical procedures
- Statistical inference about the model parameters.

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

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

**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).**Methods of estimation**. Maximum likelihood (examples, regular models), Expectation-maximizations (examples), Least squares (linear regression models, Gauss-Markov theorem).**Asymptotic statistics**. Consistency (Law of large numbers), Asymptotic distributions, Asymptotic normality (Central limit theorem, Cramer theorem), Asymptotic confidence intervals (examples).**Existence, consistency and asymptotic efficiency of maximum likelihood estimators in regular models.**Maximum likelihood estimators of parameters in exponential families of distributions.**Bayesian estimations**. Prior and posterior distributions of model parameter (Bayes theorem), Bayes estimator, Bayes interval estimation.**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).