Mathematical Statistics
Code  Completion  Credits  Range  Language 

01MAS  ZK  3  2+0  Czech 
 Lecturer:
 Václav Kůs (guarantor)
 Tutor:
 Supervisor:
 Department of Mathematics
 Synopsis:

The subject is devoted to usage of statistical methods studied in the course of Mathematical statistics. We deal with Fisher information matrix of statistical models, finding unbiased estimators with minimal variance, parameter estimation by method of moments and method of maximum likelihood, derivation of critical regions for hypothesis testing using the NeymanPearson lemma and likelihood ratio, confidence intervals and nonparametric density estimation.
 Requirements:

01MIP nebo 01PRST
 Syllabus of lectures:

1. Unbiased minimum variance estimates, Fisher information matrix, RaoCramér inequality, Bhattacharrya inequality.
2. Moment estimators, Maximum likelihood principle, consistency, asymptotic normality and efficiency of MLE.
3. Testing of simple and composite hypotheses. The NeymanPearson lemma.
4. Uniformly most powerful tests. Randomized testing, generalized NeymanPearson lemma.
5. The likelihood ratio test, ttest, Ftest.
6. Nonparametric models, empirical distribution and density function, their properties.
7. Histogram and kernel density estimates (adaptive), properties.
8. Pearson goodness of fit test, KolmogorovSmirnov test.
9. Confidence sets and intervals, pivotal quantities, acceptance regions, Pratt theorem.
 Syllabus of tutorials:
 Study Objective:

Knowledge:
In frame of the course, to provide students with the knowledge necessary for the following future subjects using stochastic models. To give a deeper insight into the field in the area of point statistical parameter estimation and testing statistical hypothesis in parametric and nonparametric probabilistic models.
Skills:
Basic statistical models and testing hypotheses processing. Orientation in majority of standard notions of the statistics and capabilities of practical applications in actual stochastic computations. Statistical data processing in statistical parametric and nonparametric model estimation and testing.
 Study materials:

Key references:
[1] Shao J., Mathematical Statistics, Springer, 1999.
Recommended references:
[2] Schervish M.J., Theory of Statistics, Springer, 1995.
[3] Lehmann E.L., Point Estimation, Wiley, N.Y., 1984.
[4] Lehmann E.L., Testing Statistical Hypotheses, Springer, N.Y., 1986.
 Note:
 Timetable for winter semester 2019/2020:
 Timetable is not available yet
 Timetable for summer semester 2019/2020:
 Timetable is not available yet
 The course is a part of the following study plans:

 BS Matematické inženýrství  Matematické modelování (compulsory course of the specialization)
 BS Matematické inženýrství  Aplikované matematickostochastické metody (compulsory course of the specialization)