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Course, academic year 2018/2019
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Modern methods of mathematical statistics - NMST603
Title: Moderní metody matematické statistiky
Guaranteed by: Department of Probability and Mathematical Statistics (32-KPMS)
Faculty: Faculty of Mathematics and Physics
Actual: from 2017 to 2018
Semester: winter
E-Credits: 3
Hours per week, examination: winter s.:2/0, Ex [HT]
Capacity: unlimited
Min. number of students: unlimited
4EU+: no
Virtual mobility / capacity: no
State of the course: taught
Language: Czech, English
Teaching methods: full-time
Teaching methods: full-time
Note: you can enroll for the course repeatedly
Guarantor: prof. RNDr. Jana Jurečková, DrSc.
Class: Pravděp. a statistika, ekonometrie a fin. mat.
Classification: Mathematics > Probability and Statistics
Annotation -
Last update: T_KPMS (26.04.2016)
The course for PhD students extends the clasical methods of mathematical statistics for modern procedures. It will be partitioned in two parts, which will alternate year to year. The first part will be devoted to robust statistical methods, to estimation of parameters with heavy and generally unknown ditributions of data, including the regression and multivariate models. The second part of the course will be devoted to nonparametric distribution-free methods based on the ranks and quantiles of observations, and further to estimation of probability densities and regression functions.
Aim of the course -
Last update: prof. RNDr. Jana Jurečková, DrSc. (06.10.2017)

To get acquainted with robust and nonparametric methods of statistics on rigorous mathematical background.

Literature - Czech
Last update: prof. RNDr. Jana Jurečková, DrSc. (06.10.2017)

Základní odborná literatura:

J. Jurečková and J. Picek: Robust Statistical Methods with R. Chapman & Hall/CRC 2006

J. Hájek, Z. Šidák and P.K.Sen: Theory of Rank Tests. 2nd edition. Academic Press 1999.

Doporučená odborná literatura:

J. Jurečková, P. K. Sen and J. Picek: Methodology in Robust and Nonparametric Statistics. Chapman & Hall/CRC, 2013.

H. Oja: Multivariate Nonparametric Methods with R. An Approach Based on Spatial Signs and Ranks. Lecture Notes in Statistics 199, Springer 2010.

P.J. Huber and E. M. Ronchetti: Robust Statistics. 2nd edition. J. Wiley 2009.

Teaching methods -
Last update: prof. RNDr. Jana Jurečková, DrSc. (06.10.2017)


Syllabus -
Last update: prof. RNDr. Jana Jurečková, DrSc. (06.10.2017)

Part 1: Robust Statistical Methods

1. Differentiable statistical functionals, functional deriva tives.

2. Qualitative robustness. Quantitative characteristics of robustness.

3. Robust estimators of real parameter: M -estimators, L-estimators, R-estimators.

4. Robust estimators in linear model: Least squares method; M -estimators, influence function. Leverage points, GM-estimators. L-estimators, regression quantiles, regression rank scores.

5. Multivariate model: M-estimators of location and scatter, admissibility and shrinkage.

6. Some goodness-of-fit tests: Shapiro-Wilk test of normality with nuisance regression and scale.

Part 2: Nonparametric Statistical Methods

1. Invariant tests, order statistics and ranks, their behavior under the hypothesis of randomness.

2. Rank tests of randomness against two samples shift alternative: Wilcoxon test, van der Waerden test, median test.

3. Rank tests of randomness against two samples scale alternative: Siegel-Tukey test, quartile test.

4. Rank tests of randomness based on empirical distribution functions: Kolmogorov-Smirnov test, Cram´er -von Mises test.

5. Hypothesis of symmetry in a bivariate population. One-sample Wilcoxon test, sign test.

6. Hypothesis of independence in bivariate population and its alternatives. Spearman test, Kendall test, quadrant test. Spearman test against alternative of monotone trend.

7. Rank tests of randomness hypothesis against alternative of several samples. Kruskal-Wallis rank test and its application for categorical data.

8. Rank tests of homogeneity of several treatments under the block decomposition: Friedman test.

9. Rank tests under tied observations: Method of randomization, method of midranks.

10. Rank tests in the linear regression model; tests based on regression rank scores.

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