1. SylabUZ
2. Wydział Matematyki, Informatyki i Ekonometrii
3. semestr zimowy 2018/2019
4. Mathematics - pierwszego stopnia z tyt. licencjata
5. Statystyka matematyczna

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Statystyka matematyczna - opis przedmiotu

Cel przedmiotu

Theoretical background of statistical inference.

Wymagania wstępne

Zapisz zmiany

Passed lecture on probability theory.

Zakres tematyczny

Lecture
1. Normal distribution and other connected distributions.
Random variable and its basic characteristics, normal random variable (revision). (2 hours)
Chi-square distribution, t-Student’s distribution, F-Snedecor’s distribution. (1 hour)
2. Statistical model.
Aim of statistical research, statistical space, random sample, theorem on convergence of empirical distribution functions. (3 hours)
Distributions of same sample statistics, Fisher theorem. (2 hours)
Sufficient statistics, factorial theorem. Complete statistics. (4 hours)
Exponential family of distributions, natural space of parameters, a general form of sufficient statistic, Lehmann theorem. (2 hours)
3. Theory of estimation.
The best unbiased estimator, Lehmann-Scheffe theorem, Rao-Blackwell theorem. (4 hours)Moments method, maximum likelihood method. (3 hours)
Confident intervals. (2 hours)
4. Theory of testing statistical hypothesis.
Introduction. (2 hour)
Most powerful tests, Neyman-Pearson lemma. (3 hours)
Most powerful tests for models with monotonic likelihood ratio, Karlin-Rubin Theorem. (2 hour)

Class
1. Revision and amplification information from probability theory. Normal distribution and its properties. Using statistical tables. Distribution of random vectors and its basic
characteristics. Distribution of a function of random vector. (2 hours)
2. Independence of random variables. Distribution of a random sample. Examples of random variables, which are not statistic. Application of Fisher theorem. (3 hours)
3. Conditional distribution. Proving sufficiency by definition and by factorial criterion. (3 hours)
4. Examples of models from the exponential family of distributions, a form of sufficient statistics, problem of completeness.
5. Calculation of the expected value and the variance of selected estimators, problem of unbiasedness. (1 hour)
6. Test. (2 hours)
7. Construction of the best unbiased estimator using Lehmann-Scheffe theorem and Rao-Blackwell theorem. (2 hours)
8. Application of moments method and maximum likelihood method to estimation.(3 hours)
9. Construction of confidence intervals for parameters of selected statistical models.
Calculations of interval estimates with using proper statistical tables. (4 hours)
10. Calculation of power functions. (2 hour)
11. Construction of the most powerful tests for testing selected statistical hypothesis. (3 hours)
12. Test. (2 hours)

Laboratory

1. An introduction to chosen statistical package (e.g. R-project). (3 teaching hrs.)
2. Properties of some probability distributions. Calculation of probabilities. Calculation of critical values and quantiles random variables. (5)
3. Use of the central limit theorem and illustration of its effects. (3)
4. Illustrate the impact of parameters of the normal distribution on sample.(simulations). (1)
5. Test. (2)
6. Illustrate the theorem on the convergence of empirical distribution function. (1)
7. Use and illustrate the Fisher theorem. (2)
8. Confidence intervals for parameters of a normal distribution. Analysis of the impact of the confidence level and the size of the sample on length of confidence intervals. (3)
9. Calculating the probability of type I error and the probability of type II error. The power of a test. (3)
10. Tests for the mean and the variance of a normal distribution. The definition of p-value. Use of confidence intervals for testing. (5)
11. Test. (2)

Metody kształcenia

Lecture traditional. Class - solving problems from prepared lists. Laboratory - using the statistical package (e.g. R-project) to analysis data.

Efekty uczenia się i metody weryfikacji osiągania efektów uczenia się

Opis efektu	Symbole efektów	Metody weryfikacji	Forma zajęć

Warunki zaliczenia

1. Class - tests with problems on different level of difficulties, which allow to assess, that student posses learning outcomes on minimal level.

2. Laboratory - checking students prepare for class and their active participation in class; tests with the tasks of different difficulty (The condition of a positive grade from laboratory is to obtain of at least 50% of the maximum sum of points from the written tests).

3. Lecture – exam (I – written, II – oral) with questions from theory (definitions, theorems and its applications).

To take an exam student has to obtain positive grade from class. To complete the course one has to obtain positive grade form exam. The course grade consists of a grade from class (30%), a grade from laboratory (20%) and a grade from exam (50%).

Literatura podstawowa

1. Jarosław Bartoszewicz, Wykłady ze statystyki matematycznej, PWN, Warszawa 1989.
2. Mirosław Krzyśko, Statystyka matematyczna, Wydawnictwo Naukowe UAM, Poznań 1996.

Literatura uzupełniająca

1. J. B. Barra, Matematyczne podstawy statystyki, PWN, Warszawa 1982.
2. W. Krysicki, J. Bartos, W. Dyczka, W. Królikowska, W. Wasilewski, Rachunek prawdopodobieństwa i statystyka matematyczna w zadaniach, część I i II, wydanie V, PWN, Warszawa 1995.
3. E. L. Lehmann, Testing statistical hypothesis, Second edition. Wiley, New York 1986 (polski przekład pierwszego wydania: Testowanie hipotez statystycznych, PWN, Warszawa1968).

Uwagi

Zmodyfikowane przez dr Alina Szelecka (ostatnia modyfikacja: 08-07-2018 07:30)