1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2019/2020
4. Computer science and econometrics - pierwszego stopnia z tyt. licencjata
5. Statystyka matematyczna

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Statystyka matematyczna - opis przedmiotu

Cel przedmiotu

Knowledge about theoretical foundations and methods of mathematical statistics

 

Wymagania wstępne

Zapisz zmiany

Mathematical analysis and  probability theory

Zakres tematyczny

Lecture:

1. 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)
2. 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)
3. Estimaction Theory
   Unbiased estimators with minimal variance, Lehmann-Sheffe's theorem, Rao-Blacwell's theorem (4 h)
   Moments method. Maximal likehood method (3 h)
   Confidence intervals (2 h)
4. 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

1. 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)
2. Independence. The notion of the statistical sample and its distribution. Applications of Fisher's theorem (3 h)
3. Conditional distributions. Calculations of sufficient statistics. Applications of factorization theorem for evaluations of sufficient statistics (3 h)
4. Examples of exponential families of distributions and applications of Lehmann's theorem for evaluations of sufficient and complete statistics (3 h)
5. Estimators-biased and unbiased. Calculations of mean and varianve of selected estimators (1 h)
6. Control work (2 h)
7. Applications of the Lehmann-Sheffe Thm and  Rao-Blackwell Thm for constructions of unbiased and with minimal variance estimators (2 h)
8. The method of moments and the maximal likehood method in constructions of selected parameters estimators (3 h)
9. Confidence intervals and the real data analysis (4 h)
10. Testing statistical hypotheses. Probabilities of Type I and Type II errors. Power test function (2 h)
11. Uniformly most powerful tests-prectical excercises (3 h)
12. Control work (2 h)

Metody kształcenia

Lectures: traditional or online form

Classes: exercises (theoretical and computational) traditional or online form

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

Students' activities during classes, tests with exercises, exam

Literatura podstawowa

1. R.V. Hogg, A.T. Craig, Introduction to mathematical statistics, Macmillan Publ. 1978

2. F. Bijma, M. Jonker, A, van der Vaart, An introduction to mathematical statistics. Epsilon Uitgaven, 2016 

3. M. Krzyśko, Statystyka matematyczna, UAM 1996

4. J. Bartoszewicz, Wykłady ze statystyki matematycznej, PWN 1989

Literatura uzupełniająca

Uwagi

Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 21-11-2020 06:10)