*Conducted in terms:*2019L, 2020L

*Erasmus code:*14.3

*ISCED code:*0311

*ECTS credits:*5

*Language:*Polish

*Organized by:*Faculty of Economic Sciences

*Related to study programmes:*

# Mathematical Statistics 2400-PP2ST

The course consists of lectures and discussions. The discussions will be devoted to solving practical problems related to theoretical issues introduced during lectures. The topics covered include

Lectures 1,2:

1. Introduction to statistics and statistical concepts (population, characteristic etc.). Basic population characteristics (empirical CDF, mean, median, variance and other measures of location and dispersion).

2. Graphical presentation of data.

3. Statistical indices.

Lecture 3:

4. The statistical model and statistical inference. Concepts of estimation, hypothesis testing and prediction. The notion of a statistic. Basic probability distributions commonly used in statistics.

Lecture 4:

5. Point estimation (method of moments, quantiles, maximum likelihood).

Lectures 5,6,7:

6. Estimator properties: bias, measures of the quality of an estimator, estimator risk, information inequality, estimator efficiency.

7. Asymptotic estimator properties (consistency, asymptotic normality, asymptotic efficiency).

Lecture 8:

8. Interval estimation: confidence intervals for the parameters in a normal model, two-point distribution model, asymptotic confidence intervals.

Lecture 9:

9. Verification of statistical hypotheses. Notions of hypothesis, test, critical region, error of 1st and 2nd type, level of significance, p-value.

Lecture 10:

10. Test power, most powerful tests, Neyman-Pearson Lemma.

Lecture 11:

11. Tests based on likelihood ratio. Testing hypotheses about the parameters of a normal model. Comparing populations.

Lecture 12:

12. Comparing more than two populations (ANOVA). Asymptotic properties of the likelihood ratio tests.

Lecture 13:

13. Tests of consistency and independence. Kolmogorov tests, the chi-square test.

Lectures 14, 15:

14. The Bayesian model and Bayesian statistics. Prior and posterior distributions. The Bayesian estimator.

15. Summary.

## Type of course

## Course coordinators

## Learning outcomes

KNOWLEDGE

The student knows and understands selected concepts of probability calculus and mathematical statistics, the most important of which is a random variable, distribution of a random variable, basic characteristics of the distribution of a random variable and types of random variables. Knows the theory of statistical inference, point estimation, interval estimation, the theory of verification of statistical hypotheses. The student knows parametric and nonparametric models for verification of hypotheses regarding theoretical distribution.

SKILLS

The student is able to use the tools of mathematical statistics. He can use selected statistical procedures. Student is able to describe models in formal statistical language. The student is able to use analytical methods to correctly formulate and solve tasks in the field of mathematical statistics. The student is able to construct an unbiased and effective parameter estimator using the chosen method. The student is able to estimate the parameter using the confidence interval. He can verify the hypothesis regarding theoretical distribution.

COMMON SKILLS

The student knows the applications of theories and methods of mathematical statistics in economics and related sciences

KU03, KK01, KU02

## Assessment criteria

The student counts exercises (100%) based on 2 tests (60%), unannounced small tests (20%) and homework (20%), and the subject ends with a written exam. The final grade is 1/3 of the exercise grade + 2/3 of the exam grade.

## Bibliography

- W.Niemiro, Rachunek Prawdopodobieństwa i Statystyka Matematyczna, wyd. SNS, 1999 (część II: Statystyka Matematyczna). [Sygn. Bibl. WNE UW: 33103]

- J.Koronacki i J. Mielniczuk, Statystyka, WNT 2004

- J.Jóźwiak i J. Podgórski, Statystyka od podstaw, PWE 1994

ZBIORY ZADAŃ

- W. Krysicki i in., Rachunek prawdopodobieństwa i statystyka matematyczna w zadaniach, PWN, 1998 (część II: Statystyka Matematyczna). [Sygn. Bibl. WNE UW: 27978/2 (1994 r.), S-9275 a-z (1998 r.), S-8969 a-n, 30479/2]

- H. Kassyk-Rokicka, Statystyka, zbiór zadań, 2005 lub inne wydania

- J. Greń, Statystyka Matematyczna, modele i zadania, PWN, 1978. [Sygn. Bibl. WNE UW: S-1060 b (1976 r.), 15489 (1978 r.)]

A. Boratyńska, Zadania ze statystyki matematycznej, skrypt w ksero na WNE UW

## Additional information

Information on *level* of this course, *year of study* and semester when the course
unit is delivered, types and amount of *class hours* - can be found in course structure
diagrams of apropriate study programmes. This course is related to
the following study programmes:

Additional information (*registration* calendar, class conductors,
*localization and schedules* of classes), might be available in the USOSweb system: