Course detail

Probability and Statistics III

FSI-SP3-AAcad. year: 2023/2024

This course is concerned with the following topics: theory of estimation, maximum likelihood, method of moments, Bayesian methods of estimation, testing statistical hypotheses, nonparametric methods, exponential family of distribution, asymptotic tests.

Language of instruction

English

Number of ECTS credits

4

Mode of study

Not applicable.

Guarantor

doc. Mgr. Zuzana Hübnerová, Ph.D.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Rudiments of probability theory and mathematical statistics, linear models.

Rules for evaluation and completion of the course

Course-unit credit requirements: active participation in seminars, mastering the subject matter, passing both written exams, and semester assignment acceptance. Design and defense of the project. Writing of the classification papers (4-5 examples from the discussed topics).
Evaluation by points obtained from the project (max: 20 points) and from the classification letter (maximum 80 points): excellent (90 - 100 points), very good (80 - 89 points), good (70 - 79 points), satisfactory (60 - 69 points), sufficient (50 - 59 points), failed (0 - 49 points).

Participation in the exercise is mandatory and the teacher decides on the compensation for absences.

Aims

The course objective is to make students majoring in Mathematical Engineering acquainted with methods of estimation theory, an asymptotic approach to statistical hypotheses testing, and prepare students for independent applications of these methods for statistical analysis of real data.

Students acquire needed knowledge from important parts of mathematical statistics, which will enable them to evaluate and develop stochastic models of technical phenomena and processes based on these methods and realize them on PC.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Dobson, A. J. An introduction to generalized linear models. Chapman & Hall/CRC Boca Raton. 2002 (EN)
Hogg, V.R., McKean J.W., and Craig A.T. Introduction to Mathematical Statistics. Seventh Edition, 2013. New York: Pearson. ISBN: 978-0-321-79543-4. (EN)
Lehmann, E.L., Casella G.: Theory of Point Estimation. New York: Springer. 1998. (EN)

Recommended reading

Montgomery, D.C, Runger, G.: Applied Statistics and Probability for Engineers. New York: John Wiley & Sons. 2002. (EN)

Classification of course in study plans

  • Programme N-MAI-A Master's 1 year of study, winter semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

doc. Mgr. Zuzana Hübnerová, Ph.D.

Syllabus

Unbiased and consistent estimates
Regular family of distributions, Rao - Cramér theorem, efficient estimates
Fisher information and Fisher information matrix
Sufficient statistics, Neuman factorization criterion
Rao - Blackwell theorem and its applications
Method of moments, maximum likelihood method
Bayesian approach
Testing statistical hypotheses
Principles of nonparametric methods
Exponential family of distribution
Asymptotic tests based on likelihood function
Tests with nuisance parameters, examples
Tests of hypotheses on parameters
Generalized linear models

Computer-assisted exercise

13 hod., compulsory

Teacher / Lecturer

doc. Mgr. Zuzana Hübnerová, Ph.D.

Syllabus

Survey of probability distributions, graphs of densities
Unbiased and consistent estimates - examples of estimates and verification of their properties
Computation of the lower bound for variance of unbiased estimates
Determination of Fisher information and Fisher information matrix for given distributions
Applications of Neuman factorization criterion
Findings estimates by Rao - Blackwell theorem
Estimator’s determination by method of moments and by maximum likelihood method
Estimator’s determination by Bayes method
Project setting - finding parameters estimates for given distribution - application at least two approaches, verification properties of the estimates and their numerical computation
Verification of exponential family for a given distribution
Application of asymptotic tests based on likelihood function
Tests with nuisance parameters, estimates of parameters for Weibull and gamma distribution
Tests of hypotheses on parameters of generalized linear model