Syllabus
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.
Annotation
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.