Probability and Statistics III (FSI-SP3)

Academic year 2023/2024
Supervisor: doc. Mgr. Zuzana Hübnerová, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:
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.
Learning outcomes and competences:

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 use them on PC.
Prerequisites:
Rudiments of probability theory and mathematical statistics, linear models.
Course contents:

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, generalized linear models.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:
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).
Controlled participation in lessons:
Participation in the exercise is mandatory and the teacher decides on the compensation for absences.
Type of course unit:
    Lecture  13 × 2 hrs. optionally                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

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

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
Powers of test and derivation of uniformly most powerful tests
Application of nonparametric methos in data analysis
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
Literature - fundamental:
1. Anděl, J. Základy matematické statistiky. Matfyzpress. Praha 2005
2. 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
4. Lehmann, E.L., Casella G.: Theory of Point Estimation. New York: Springer. 1998
5. Dobson, A. J. An introduction to generalized linear models. Chapman & Hall/CRC Boca Raton. 2002.
Literature - recommended:
2. Militký, J.: Statistické techniky v řízení jakosti. Pardubice : TriloByte, 1996.
5. Montgomery, D.D, Runger, G.: Applied Statistics and Probability for Engineers, New York : John Wiley & Sons. 2002
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-P full-time study --- no specialisation -- GCr 4 Compulsory 2 1 W