Knowledge about theoretical foundations and methods of mathematical statistics
Prerequisites
Mathematical analysis and probability theory
Scope
Lecture:
Normal (Gaussian) probability distribution and probability distribustions related to the normal one.
Random variables and their basic probabilistic characteristics, Random variable with Gaussian distribution (2 h)
Chi-square distribution, t-Student's distribution and F-Snedecor's distribution (1 h)
Statistical Models.
Aims of statistical analysis, statistical space, statistical sample, badań statystycznych, limit theorems for the empirical distribution function (3 h)
Probability distributions of selected sample statistics, Fisher's theorem (2 h)
Sufficient statistics, factorization theorem, completness of statistics (4 h)
The family of exponential distributions, space of parameters, Lehmann's theorem (2 h)
Statistical Hypotheses.
Basic notions(2 h)
Uniformly most powerful tests, Neyman-Pearson's Lemma (3 h)
Uniformly most powerful tests in models with a monotonic likehood ratio, Karlin-Rubin's theorem (2h)
Classes
Repetition of elements of probability theory. Normal distribution and its properties. Statistical tables. Distributions of random vectors, multivariate normal distribution and its characteristics. Functions of random variables and their distributions (2 h)
Independence. The notion of the statistical sample and its distribution. Applications of Fisher's theorem (3 h)
Conditional distributions. Calculations of sufficient statistics. Applications of factorization theorem for evaluations of sufficient statistics (3 h)
Examples of exponential families of distributions and applications of Lehmann's theorem for evaluations of sufficient and complete statistics (3 h)
Estimators-biased and unbiased. Calculations of mean and varianve of selected estimators (1 h)
Control work (2 h)
Applications of the Lehmann-Sheffe Thm and Rao-Blackwell Thm for constructions of unbiased and with minimal variance estimators (2 h)
The method of moments and the maximal likehood method in constructions of selected parameters estimators (3 h)
Confidence intervals and the real data analysis (4 h)
Testing statistical hypotheses. Probabilities of Type I and Type II errors. Power test function (2 h)
Uniformly most powerful tests-prectical excercises (3 h)
Control work (2 h)
Teaching methods
Lectures: traditional or online form
Classes: exercises (theoretical and computational) traditional or online form
Learning outcomes and methods of theirs verification
Outcome description
Outcome symbols
Methods of verification
The class form
Assignment conditions
Students' activities during classes, tests with exercises, exam
